dataset stringlengths 6 23 | prompt stringlengths 10 1.99k | initial_reason_steps stringlengths 33 2.35k | chosen stringlengths 15 2.57k | rejected stringlengths 14 3.72k | full_chosen stringlengths 45 3.04k | full_rejected stringlengths 42 4.59k | answer stringlengths 1 80 | chosen_steps_rejected_answer stringlengths 37 4.46k | rejected_steps_chosen_answer stringlengths 18 4.48k | chosen_steps_substitute_answer stringlengths 48 3.04k | rejected_steps_substitute_answer stringlengths 18 4.48k | score_initial_reason_steps_only float64 -13.88 3 | score_chosen float64 -13.5 9.69 | score_rejected float64 -14.69 10.5 | score_full_chosen float64 -9.19 9.69 | score_full_rejected float64 -12.69 10 | score_rejected_steps_chosen_answer float64 -10.31 5.66 | score_chosen_steps_rejected_answer float64 -11.81 7.63 | score_chosen_steps_substitute_answer float64 -10 5 | score_rejected_steps_substitute_answer float64 -11.69 5.03 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
MATH_Rephrased | In the month of July, the bookstore has a sale on days that are multiples of 5, and the shoe store has a sale every 6 days. If the shoe store has a sale on July 3, how many times in July do both stores have sales on the same date? | Let's think step by step.
Step 1: | Find the dates on which both stores have sales.
The bookstore has a sale on days that are multiples of 5. So the dates are 5, 10, 15, 20, 25, 30 (for July).
The shoe store has a sale every 6 days. If it has a sale on July 3, then the next sale will be on July 9, then on July 15, and so on. So the dates of its sales ar... | Find the dates when the shoe store has a sale.
The shoe store has a sale every 6 days. If they have a sale on July 3, then they will have a sale on July 3 + 6 = 9, 15, 21, 27, 33, 39, 45, 51, 57. | Find the dates on which both stores have sales.
The bookstore has a sale on days that are multiples of 5. So the dates are 5, 10, 15, 20, 25, 30 (for July).
The shoe store has a sale every 6 days. If it has a sale on July 3, then the next sale will be on July 9, then on July 15, and so on. So the dates of its sales ar... | Find the dates when the shoe store has a sale.
The shoe store has a sale every 6 days. If they have a sale on July 3, then they will have a sale on July 3 + 6 = 9, 15, 21, 27, 33, 39, 45, 51, 57.
Step 2: Find the dates when the bookstore has a sale.
The bookstore has a sale on days that are multiples of 5. So the dat... | 1 | Find the dates on which both stores have sales.
The bookstore has a sale on days that are multiples of 5. So the dates are 5, 10, 15, 20, 25, 30 (for July).
The shoe store has a sale every 6 days. If it has a sale on July 3, then the next sale will be on July 9, then on July 15, and so on. So the dates of its sales ar... | Find the dates when the shoe store has a sale.
The shoe store has a sale every 6 days. If they have a sale on July 3, then they will have a sale on July 3 + 6 = 9, 15, 21, 27, 33, 39, 45, 51, 57.
Step 2: Find the dates when the bookstore has a sale.
The bookstore has a sale on days that are multiples of 5. So the dat... | Find the dates on which both stores have sales.
The bookstore has a sale on days that are multiples of 5. So the dates are 5, 10, 15, 20, 25, 30 (for July).
The shoe store has a sale every 6 days. If it has a sale on July 3, then the next sale will be on July 9, then on July 15, and so on. So the dates of its sales ar... | Find the dates when the shoe store has a sale.
The shoe store has a sale every 6 days. If they have a sale on July 3, then they will have a sale on July 3 + 6 = 9, 15, 21, 27, 33, 39, 45, 51, 57.
Step 2: Find the dates when the bookstore has a sale.
The bookstore has a sale on days that are multiples of 5. So the dat... | -10.125 | -6.84375 | -7.21875 | -1.679688 | -3.78125 | -5.53125 | -5.1875 | -6.125 | -6.21875 |
gpt-4-1106-preview-MATH | Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$ | Let's think step by step.
Step 1: Choosing two different prime numbers between $4$ and $18$.
The prime numbers between $4$ and $18$ are $5, 7, 11, 13, 17$.
Step 2: We will calculate the product and sum of two different prime numbers.
Let's calculate the product and sum of some pairs of prime numbers:
- For $5$ and... | Comparing the results with the given options.
Comparing the results with the given options, we see that the only number that could be obtained is $119$.
So, the correct answer is C. $119$.
The answer is: 119 | Comparing the results with the given options.
We can see that the result $119$ is obtained from the pair of prime numbers $7$ and $13$.
So, the correct answer is $\boxed{C}$.
The answer is: C | Comparing the results with the given options.
Comparing the results with the given options, we see that the only number that could be obtained is $119$.
So, the correct answer is C. $119$.
The answer is: 119 | Comparing the results with the given options.
We can see that the result $119$ is obtained from the pair of prime numbers $7$ and $13$.
So, the correct answer is $\boxed{C}$.
The answer is: C | 119 | Comparing the results with the given options.
Comparing the results with the given options, we see that the only number that could be obtained is $119$.
So, the correct answer is C. $119$.
The answer is: C | Comparing the results with the given options.
We can see that the result $119$ is obtained from the pair of prime numbers $7$ and $13$.
So, the correct answer is $\boxed{C}$.
The answer is: 119 | Comparing the results with the given options.
Comparing the results with the given options, we see that the only number that could be obtained is $119$.
So, the correct answer is C. $119$.
The answer is: -772 | Comparing the results with the given options.
We can see that the result $119$ is obtained from the pair of prime numbers $7$ and $13$.
So, the correct answer is $\boxed{C}$.
The answer is: -772 | -1.671875 | 0.396484 | 1.914063 | 0.396484 | 1.710938 | 2.171875 | 0.59375 | -1.414063 | 0.298828 |
gpt-3.5-turbo-GSM | If $h(x) = x^3 + f(x)$ where $f(x) = 2x^2 - 3x + 5$, what is $h(g(1)) + h(g^{-1}(2))$? | Let's think step by step.
Step 1:
First, we need to find $g(1)$ and $g^{-1}(2)$.
Step 2: |
We are given that $g(x) = x^2 - 2x + 3$. So, to find $g(1)$, we substitute $x = 1$ into the expression for $g(x)$:
\[g(1) = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2.\] |
Given $g(x) = 2x - 3$, we can find $g(1)$ by substituting $x = 1$ into the equation: $g(1) = 2(1) - 3 = -1$. |
We are given that $g(x) = x^2 - 2x + 3$. So, to find $g(1)$, we substitute $x = 1$ into the expression for $g(x)$:
\[g(1) = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2.\]
Step 3:
Next, we need to find $g^{-1}(2)$. To do this, we set $g(x) = 2$ and solve for $x$:
\[2 = x^2 - 2x + 3.\]
Rearranging, we get the quadratic equation ... |
Given $g(x) = 2x - 3$, we can find $g(1)$ by substituting $x = 1$ into the equation: $g(1) = 2(1) - 3 = -1$.
Step 3:
To find $g^{-1}(2)$, we set $g(x) = 2$ and solve for $x$: $2 = 2x - 3$, so $2x = 5$ and $x = \frac{5}{2}$. Thus, $g^{-1}(2) = \frac{5}{2}$.
Step 4:
Now we can find $h(g(1))$ and $h(g^{-1}(2))$.
St... | 20 |
We are given that $g(x) = x^2 - 2x + 3$. So, to find $g(1)$, we substitute $x = 1$ into the expression for $g(x)$:
\[g(1) = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2.\]
Step 3:
Next, we need to find $g^{-1}(2)$. To do this, we set $g(x) = 2$ and solve for $x$:
\[2 = x^2 - 2x + 3.\]
Rearranging, we get the quadratic equation ... |
Given $g(x) = 2x - 3$, we can find $g(1)$ by substituting $x = 1$ into the equation: $g(1) = 2(1) - 3 = -1$.
Step 3:
To find $g^{-1}(2)$, we set $g(x) = 2$ and solve for $x$: $2 = 2x - 3$, so $2x = 5$ and $x = \frac{5}{2}$. Thus, $g^{-1}(2) = \frac{5}{2}$.
Step 4:
Now we can find $h(g(1))$ and $h(g^{-1}(2))$.
St... |
We are given that $g(x) = x^2 - 2x + 3$. So, to find $g(1)$, we substitute $x = 1$ into the expression for $g(x)$:
\[g(1) = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2.\]
Step 3:
Next, we need to find $g^{-1}(2)$. To do this, we set $g(x) = 2$ and solve for $x$:
\[2 = x^2 - 2x + 3.\]
Rearranging, we get the quadratic equation ... |
Given $g(x) = 2x - 3$, we can find $g(1)$ by substituting $x = 1$ into the equation: $g(1) = 2(1) - 3 = -1$.
Step 3:
To find $g^{-1}(2)$, we set $g(x) = 2$ and solve for $x$: $2 = 2x - 3$, so $2x = 5$ and $x = \frac{5}{2}$. Thus, $g^{-1}(2) = \frac{5}{2}$.
Step 4:
Now we can find $h(g(1))$ and $h(g^{-1}(2))$.
St... | -7.21875 | -3.09375 | -5.9375 | 4.875 | 5.65625 | 0.462891 | 2.296875 | 0.314453 | -0.542969 |
GSM_Rephrased | If Jeremy played a three-letter word on a triple word score to earn thirty points, and before the word score was tripled, the first and third letters were each worth one point, what was the value of the middle letter before the word score was tripled? | Let's think step by step.
Step 1: The word is a three-letter word, and the first and third letters were each worth one point, so the total value of the first and third letters before the word score was tripled is 1+1 = 2 points.
Step 2: | The word score was tripled to earn 30 points, so the original word score before tripling was 30/3 = 10 points. | The word score was tripled to earn 30 points. So, the value of the middle letter before the word score was tripled is 30 - 2 = 28 points.
Therefore, the value of the middle letter before the word score was tripled is 28 points.
The answer is: 28 | The word score was tripled to earn 30 points, so the original word score before tripling was 30/3 = 10 points.
Step 3: The total value of the first and third letters was 2 points, and the value of the middle letter was the remaining points, so the value of the middle letter before the word score was tripled is 10 - 2 ... | The word score was tripled to earn 30 points. So, the value of the middle letter before the word score was tripled is 30 - 2 = 28 points.
Therefore, the value of the middle letter before the word score was tripled is 28 points.
The answer is: 28 | 8 | The word score was tripled to earn 30 points, so the original word score before tripling was 30/3 = 10 points.
Step 3: The total value of the first and third letters was 2 points, and the value of the middle letter was the remaining points, so the value of the middle letter before the word score was tripled is 10 - 2 ... | The word score was tripled to earn 30 points. So, the value of the middle letter before the word score was tripled is 30 - 2 = 28 points.
Therefore, the value of the middle letter before the word score was tripled is 28 points.
The answer is: 8 | The word score was tripled to earn 30 points, so the original word score before tripling was 30/3 = 10 points.
Step 3: The total value of the first and third letters was 2 points, and the value of the middle letter was the remaining points, so the value of the middle letter before the word score was tripled is 10 - 2 ... | The word score was tripled to earn 30 points. So, the value of the middle letter before the word score was tripled is 30 - 2 = 28 points.
Therefore, the value of the middle letter before the word score was tripled is 28 points.
The answer is: 518 | -6.375 | -5.46875 | -4.03125 | -1.671875 | -4.1875 | -5.53125 | -4.125 | -5.6875 | -6.875 |
gpt-3.5-turbo-GSM | Let $A=2+i$, $B=3-2i$, $O=-4$, $P=-i$, $Q=5+3i$, $S=2+4i$, and $T=-3-6i$. Calculate $(A+B-O+P+Q+S)+T$. | Let's think step by step.
Step 1:
First, we need to find the sum of all the complex numbers in the problem.
We are given $A=2+i$, $B=3-2i$, $O=-4$, $P=-i$, $Q=5+3i$, $S=2+4i$, and $T=-3-6i$.
We need to find $(A+B-O+P+Q+S)+T$.
Step 2: |
Let's add all the complex numbers together.
First, add $A$ and $B$:
$$A+B = (2+i) + (3-2i) = 2+3+i-2i = 5-i.$$
Then, subtract $O$:
$$5-i - (-4) = 5-i+4 = 9-i.$$
Next, add $P$:
$$9-i + (-i) = 9-2i.$$
Then, add $Q$:
$$9-2i + (5+3i) = 9+5-2i+3i = 14+i.$$
Finally, add $S$:
$$14+i + (2+4i) = 14+2+i+4i = 16+5i.$$
Now, we a... |
First, let's add all the given complex numbers:
$A+B = (2+i) + (3-2i) = 5 - i$,
$-O = -(-4) = 4$,
$P = -i$,
$Q = 5+3i$,
$S = 2+4i$.
Now, let's add these results together:
$5 - i + 4 - i + 5 + 3i + 2 + 4i = 16 + i$. |
Let's add all the complex numbers together.
First, add $A$ and $B$:
$$A+B = (2+i) + (3-2i) = 2+3+i-2i = 5-i.$$
Then, subtract $O$:
$$5-i - (-4) = 5-i+4 = 9-i.$$
Next, add $P$:
$$9-i + (-i) = 9-2i.$$
Then, add $Q$:
$$9-2i + (5+3i) = 9+5-2i+3i = 14+i.$$
Finally, add $S$:
$$14+i + (2+4i) = 14+2+i+4i = 16+5i.$$
Now, we a... |
First, let's add all the given complex numbers:
$A+B = (2+i) + (3-2i) = 5 - i$,
$-O = -(-4) = 4$,
$P = -i$,
$Q = 5+3i$,
$S = 2+4i$.
Now, let's add these results together:
$5 - i + 4 - i + 5 + 3i + 2 + 4i = 16 + i$.
Step 3:
Now, let's add this result to $T = -3 - 6i$:
$16 + i + (-3 - 6i) = 16 - 3 + i - 6i = 13 -... | 13-i |
Let's add all the complex numbers together.
First, add $A$ and $B$:
$$A+B = (2+i) + (3-2i) = 2+3+i-2i = 5-i.$$
Then, subtract $O$:
$$5-i - (-4) = 5-i+4 = 9-i.$$
Next, add $P$:
$$9-i + (-i) = 9-2i.$$
Then, add $Q$:
$$9-2i + (5+3i) = 9+5-2i+3i = 14+i.$$
Finally, add $S$:
$$14+i + (2+4i) = 14+2+i+4i = 16+5i.$$
Now, we a... |
First, let's add all the given complex numbers:
$A+B = (2+i) + (3-2i) = 5 - i$,
$-O = -(-4) = 4$,
$P = -i$,
$Q = 5+3i$,
$S = 2+4i$.
Now, let's add these results together:
$5 - i + 4 - i + 5 + 3i + 2 + 4i = 16 + i$.
Step 3:
Now, let's add this result to $T = -3 - 6i$:
$16 + i + (-3 - 6i) = 16 - 3 + i - 6i = 13 -... |
Let's add all the complex numbers together.
First, add $A$ and $B$:
$$A+B = (2+i) + (3-2i) = 2+3+i-2i = 5-i.$$
Then, subtract $O$:
$$5-i - (-4) = 5-i+4 = 9-i.$$
Next, add $P$:
$$9-i + (-i) = 9-2i.$$
Then, add $Q$:
$$9-2i + (5+3i) = 9+5-2i+3i = 14+i.$$
Finally, add $S$:
$$14+i + (2+4i) = 14+2+i+4i = 16+5i.$$
Now, we a... |
First, let's add all the given complex numbers:
$A+B = (2+i) + (3-2i) = 5 - i$,
$-O = -(-4) = 4$,
$P = -i$,
$Q = 5+3i$,
$S = 2+4i$.
Now, let's add these results together:
$5 - i + 4 - i + 5 + 3i + 2 + 4i = 16 + i$.
Step 3:
Now, let's add this result to $T = -3 - 6i$:
$16 + i + (-3 - 6i) = 16 - 3 + i - 6i = 13 -... | -4.65625 | 1.265625 | -3.609375 | 1.265625 | 0.3125 | -1.609375 | -0.628906 | -2.71875 | -3.109375 |
GSM_Rephrased | Hannah aims to achieve the highest grade in the class on the upcoming math test. Due to her absence on the test day, she knows in advance the scores she needs to surpass. She discovers that one student scored 95% on the exam, while another student only got 3 questions wrong out of 40. How many questions does Hannah nee... | Let's think step by step.
Step 1: Determine the score for the student who got 95% on the exam.
Since 95% of 40 is 0.95 * 40 = 38, this student got 38 questions correct.
Step 2: Determine the score for the student who got 3 questions wrong out of 40.
If this student got 3 questions wrong, then he got 40 - 3 = 37 quest... | Determine how many questions Hannah needs to answer correctly to surpass both students.
To surpass the student who got 95%, Hannah needs to score higher than 38.
To surpass the student who got 3 questions wrong, Hannah needs to score higher than 37.
Since Hannah wants to achieve the highest score in the class, she nee... | Determine how many questions Hannah needs to answer correctly to achieve the highest score in the class.
Since the student who got 95% scored 38 questions correct and the student who got 3 questions wrong scored 37 questions correct, Hannah needs to answer at least 38 questions correctly to achieve the highest score i... | Determine how many questions Hannah needs to answer correctly to surpass both students.
To surpass the student who got 95%, Hannah needs to score higher than 38.
To surpass the student who got 3 questions wrong, Hannah needs to score higher than 37.
Since Hannah wants to achieve the highest score in the class, she nee... | Determine how many questions Hannah needs to answer correctly to achieve the highest score in the class.
Since the student who got 95% scored 38 questions correct and the student who got 3 questions wrong scored 37 questions correct, Hannah needs to answer at least 38 questions correctly to achieve the highest score i... | 39 | Determine how many questions Hannah needs to answer correctly to surpass both students.
To surpass the student who got 95%, Hannah needs to score higher than 38.
To surpass the student who got 3 questions wrong, Hannah needs to score higher than 37.
Since Hannah wants to achieve the highest score in the class, she nee... | Determine how many questions Hannah needs to answer correctly to achieve the highest score in the class.
Since the student who got 95% scored 38 questions correct and the student who got 3 questions wrong scored 37 questions correct, Hannah needs to answer at least 38 questions correctly to achieve the highest score i... | Determine how many questions Hannah needs to answer correctly to surpass both students.
To surpass the student who got 95%, Hannah needs to score higher than 38.
To surpass the student who got 3 questions wrong, Hannah needs to score higher than 37.
Since Hannah wants to achieve the highest score in the class, she nee... | Determine how many questions Hannah needs to answer correctly to achieve the highest score in the class.
Since the student who got 95% scored 38 questions correct and the student who got 3 questions wrong scored 37 questions correct, Hannah needs to answer at least 38 questions correctly to achieve the highest score i... | -5.59375 | -3.625 | -2.046875 | -3.625 | -2.046875 | -3.9375 | -5.53125 | -6.1875 | -6.59375 |
MATH_Rephrased | What is the monic polynomial, in terms of x, whose roots are 2 times the roots of the equation $x^3 - 3x^2 + 8 = 0$? | Let's think step by step.
Step 1: If $r_1, r_2, r_3$ are the roots of the equation $x^3 - 3x^2 + 8 = 0$, then the roots of the new equation are $2r_1, 2r_2, 2r_3$.
Step 2: The monic polynomial with roots $2r_1, 2r_2, 2r_3$ is given by $(x - 2r_1)(x - 2r_2)(x - 2r_3)$.
Step 3: Expanding this product, we get $(x - 2r_... | We know that $r_1 + r_2 + r_3 = 3$ (from the sum of the roots of the original equation), $r_1r_2 + r_2r_3 + r_3r_1 = 0$ (from the sum of the product of the roots taken two at a time), and $r_1r_2r_3 = -8$ (from the product of the roots of the original equation). | The coefficients of the polynomial $x^3 - 3x^2 + 8 = 0$ are 1, -3, and 8. Therefore, the sum of the roots $r_1 + r_2 + r_3 = 3$, the sum of the products of the roots taken two at a time $r_1r_2 + r_2r_3 + r_3r_1 = 0$, and the product of the roots $r_1r_2r_3 = 8$. | We know that $r_1 + r_2 + r_3 = 3$ (from the sum of the roots of the original equation), $r_1r_2 + r_2r_3 + r_3r_1 = 0$ (from the sum of the product of the roots taken two at a time), and $r_1r_2r_3 = -8$ (from the product of the roots of the original equation).
Step 5: Substituting these values into the expression in... | The coefficients of the polynomial $x^3 - 3x^2 + 8 = 0$ are 1, -3, and 8. Therefore, the sum of the roots $r_1 + r_2 + r_3 = 3$, the sum of the products of the roots taken two at a time $r_1r_2 + r_2r_3 + r_3r_1 = 0$, and the product of the roots $r_1r_2r_3 = 8$.
Step 5: Substituting these values into the expression ... | x^3-6x^2+64 | We know that $r_1 + r_2 + r_3 = 3$ (from the sum of the roots of the original equation), $r_1r_2 + r_2r_3 + r_3r_1 = 0$ (from the sum of the product of the roots taken two at a time), and $r_1r_2r_3 = -8$ (from the product of the roots of the original equation).
Step 5: Substituting these values into the expression in... | The coefficients of the polynomial $x^3 - 3x^2 + 8 = 0$ are 1, -3, and 8. Therefore, the sum of the roots $r_1 + r_2 + r_3 = 3$, the sum of the products of the roots taken two at a time $r_1r_2 + r_2r_3 + r_3r_1 = 0$, and the product of the roots $r_1r_2r_3 = 8$.
Step 5: Substituting these values into the expression ... | We know that $r_1 + r_2 + r_3 = 3$ (from the sum of the roots of the original equation), $r_1r_2 + r_2r_3 + r_3r_1 = 0$ (from the sum of the product of the roots taken two at a time), and $r_1r_2r_3 = -8$ (from the product of the roots of the original equation).
Step 5: Substituting these values into the expression in... | The coefficients of the polynomial $x^3 - 3x^2 + 8 = 0$ are 1, -3, and 8. Therefore, the sum of the roots $r_1 + r_2 + r_3 = 3$, the sum of the products of the roots taken two at a time $r_1r_2 + r_2r_3 + r_3r_1 = 0$, and the product of the roots $r_1r_2r_3 = 8$.
Step 5: Substituting these values into the expression ... | -2.171875 | -1.984375 | -2.34375 | 4.1875 | 3.703125 | 0.625 | 0.457031 | -0.859375 | -0.597656 |
MATH_AnsAug | Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}.\] | Let's think step by step.
Step 1: Expand the determinant using the first row.
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix} = \sin^2 A \begin{vmatrix} \cot B & 1 \\ \cot C & 1 \end{vmatrix} - \cot A \begin{vmatrix} \sin^2 B & 1 \\ \sin^2 C & 1 \end{vmatrix} + ... | Simplify the expression.
\[\sin^2 A \cot B - \sin^2 A \cot C - \cot A \sin^2 B + \cot A \sin^2 C + \sin^2 B \cot C - \sin^2 C \cot B = 0.\] | Simplify the expression.
\[\sin^2 A (\cot B - \cot C) - \cot A (\sin^2 B - \sin^2 C) + \sin^2 B \cot C - \sin^2 C \cot B.\]
We know that $\cot x = \frac{\cos x}{\sin x}$, and in a triangle, $\sin^2 A + \sin^2 B + \sin^2 C = 1$. Also, $\cos A + \cos B + \cos C = 1 + \cot A \sin A + \cot B \sin B + \cot C \sin C$.
Subst... | Simplify the expression.
\[\sin^2 A \cot B - \sin^2 A \cot C - \cot A \sin^2 B + \cot A \sin^2 C + \sin^2 B \cot C - \sin^2 C \cot B = 0.\]
Step 5: The determinant is 0.
So the value of the determinant is 0.
The answer is: 0 | Simplify the expression.
\[\sin^2 A (\cot B - \cot C) - \cot A (\sin^2 B - \sin^2 C) + \sin^2 B \cot C - \sin^2 C \cot B.\]
We know that $\cot x = \frac{\cos x}{\sin x}$, and in a triangle, $\sin^2 A + \sin^2 B + \sin^2 C = 1$. Also, $\cos A + \cos B + \cos C = 1 + \cot A \sin A + \cot B \sin B + \cot C \sin C$.
Su... | 0 | Simplify the expression.
\[\sin^2 A \cot B - \sin^2 A \cot C - \cot A \sin^2 B + \cot A \sin^2 C + \sin^2 B \cot C - \sin^2 C \cot B = 0.\]
Step 5: The determinant is 0.
So the value of the determinant is 0.
\[\cos B \sin C - \sin B \cos C - \frac{\ | Simplify the expression.
\[\sin^2 A (\cot B - \cot C) - \cot A (\sin^2 B - \sin^2 C) + \sin^2 B \cot C - \sin^2 C \cot B.\]
We know that $\cot x = \frac{\cos x}{\sin x}$, and in a triangle, $\sin^2 A + \sin^2 B + \sin^2 C = 1$. Also, $\cos A + \cos B + \cos C = 1 + \cot A \sin A + \cot B \sin B + \cot C \sin C$.
Su... | Simplify the expression.
\[\sin^2 A \cot B - \sin^2 A \cot C - \cot A \sin^2 B + \cot A \sin^2 C + \sin^2 B \cot C - \sin^2 C \cot B = 0.\]
Step 5: The determinant is 0.
So the value of the determinant is 0.
The answer is: -715 | Simplify the expression.
\[\sin^2 A (\cot B - \cot C) - \cot A (\sin^2 B - \sin^2 C) + \sin^2 B \cot C - \sin^2 C \cot B.\]
We know that $\cot x = \frac{\cos x}{\sin x}$, and in a triangle, $\sin^2 A + \sin^2 B + \sin^2 C = 1$. Also, $\cos A + \cos B + \cos C = 1 + \cot A \sin A + \cot B \sin B + \cot C \sin C$.
Su... | -0.984375 | 1.101563 | -2.015625 | 2.046875 | -2.359375 | 1.046875 | -2.4375 | -0.578125 | -0.211914 |
GSM_Rephrased | If Maryann can unlock a cheap pair of handcuffs in 6 minutes and an expensive pair in 8 minutes, how much time will it take her to rescue all three of her friends who have expensive handcuffs on their hands and cheap handcuffs on their ankles? | Let's think step by step.
Step 1: Maryann can unlock a cheap pair of handcuffs in 6 minutes.
Step 2: She can unlock an expensive pair of handcuffs in 8 minutes.
Step 3: | She needs to rescue three friends, and each friend has expensive handcuffs on their hands. So, she needs to unlock three expensive pairs of handcuffs. | She needs to rescue three friends. One friend has expensive handcuffs on their hands, and the other two friends have cheap handcuffs on their ankles. | She needs to rescue three friends, and each friend has expensive handcuffs on their hands. So, she needs to unlock three expensive pairs of handcuffs.
Step 4: The total time it will take her to unlock three expensive pairs of handcuffs is 3 * 8 = 24 minutes.
Step 5: Each friend also has cheap handcuffs on their ankles... | She needs to rescue three friends. One friend has expensive handcuffs on their hands, and the other two friends have cheap handcuffs on their ankles.
Step 4: To rescue the friend with expensive handcuffs on their hands, it will take her 8 minutes.
Step 5: To rescue the two friends with cheap handcuffs on their ankle... | 42 | She needs to rescue three friends, and each friend has expensive handcuffs on their hands. So, she needs to unlock three expensive pairs of handcuffs.
Step 4: The total time it will take her to unlock three expensive pairs of handcuffs is 3 * 8 = 24 minutes.
Step 5: Each friend also has cheap handcuffs on their ankles... | She needs to rescue three friends. One friend has expensive handcuffs on their hands, and the other two friends have cheap handcuffs on their ankles.
Step 4: To rescue the friend with expensive handcuffs on their hands, it will take her 8 minutes.
Step 5: To rescue the two friends with cheap handcuffs on their ankle... | She needs to rescue three friends, and each friend has expensive handcuffs on their hands. So, she needs to unlock three expensive pairs of handcuffs.
Step 4: The total time it will take her to unlock three expensive pairs of handcuffs is 3 * 8 = 24 minutes.
Step 5: Each friend also has cheap handcuffs on their ankles... | She needs to rescue three friends. One friend has expensive handcuffs on their hands, and the other two friends have cheap handcuffs on their ankles.
Step 4: To rescue the friend with expensive handcuffs on their hands, it will take her 8 minutes.
Step 5: To rescue the two friends with cheap handcuffs on their ankle... | -10.5 | -9.875 | -9.5625 | -2.84375 | -4.71875 | -6.875 | -6.625 | -6.8125 | -6.6875 |
gpt-3.5-turbo-GSM | Find the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505} - x^{505}$. | Let's think step by step.
Step 1:
The Binomial Theorem tells us that the coefficient of $x^k$ in the expansion of $(2 + x)^{505}$ is $\binom{505}{k} \cdot 2^{505-k}$.
Step 2:
So the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505}$ is $\binom{505}{504} \cdot 2^{505-504} = 505 \cdot 2 = 1010$.
Step 3: |
The coefficient of $x^{504}$ in the expansion of $x^{505}$ is simply 0 because the power of $x$ is not 0. |
The coefficient of $x^{504}$ in the expansion of $-x^{505}$ is simply $-1$. |
The coefficient of $x^{504}$ in the expansion of $x^{505}$ is simply 0 because the power of $x$ is not 0.
Step 4:
Therefore, the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505} - x^{505}$ is $1010 - 0 = 1010$.
The answer is: 1010 |
The coefficient of $x^{504}$ in the expansion of $-x^{505}$ is simply $-1$.
Step 4:
To find the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505} - x^{505}$, we subtract the coefficient from the expansion of $-x^{505}$ from the coefficient from the expansion of $(2 + x)^{505}$.
Step 5:
So the coefficien... | 1010 |
The coefficient of $x^{504}$ in the expansion of $x^{505}$ is simply 0 because the power of $x$ is not 0.
Step 4:
Therefore, the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505} - x^{505}$ is $1010 - 0 = 1010$.
The answer is: 1009 |
The coefficient of $x^{504}$ in the expansion of $-x^{505}$ is simply $-1$.
Step 4:
To find the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505} - x^{505}$, we subtract the coefficient from the expansion of $-x^{505}$ from the coefficient from the expansion of $(2 + x)^{505}$.
Step 5:
So the coefficien... |
The coefficient of $x^{504}$ in the expansion of $x^{505}$ is simply 0 because the power of $x$ is not 0.
Step 4:
Therefore, the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505} - x^{505}$ is $1010 - 0 = 1010$.
The answer is: -791 |
The coefficient of $x^{504}$ in the expansion of $-x^{505}$ is simply $-1$.
Step 4:
To find the coefficient of $x^{504}$ in the expansion of $(2 + x)^{505} - x^{505}$, we subtract the coefficient from the expansion of $-x^{505}$ from the coefficient from the expansion of $(2 + x)^{505}$.
Step 5:
So the coefficien... | -2.375 | -1.546875 | -0.949219 | 4.5 | 3.703125 | 1.328125 | -0.097168 | -0.953125 | -0.773438 |
gpt-3.5-turbo-GSM | For the real number $z$, it holds that $z^2 - 6z + 8 < 0.$ Determine the interval for $z^2 + 6z + 9.$ | Let's think step by step.
Step 1:
First, we need to solve the inequality $z^2 - 6z + 8 < 0$. This is a quadratic inequality. We can rewrite it as $(z-4)(z-2) < 0$. The solutions to the equation $(z-4)(z-2) = 0$ are $z=4$ and $z=2$.
The sign of $(z-4)(z-2)$ changes at $z=2$ and $z=4$. When $z<2$, both $(z-4)$ and $(z... |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (that is, it's always greater than or equal to 0) for all real $z$, the interval for $z^2 + 6z + 9$ is $[0, \infty)$.
However, we know that $z$ must be in the interval $(2, 4)$ from the first p... |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (because the square of any real number is nonnegative), and we know from step 1 that $2 < z < 4$, we can conclude that the minimum value of $(z+3)^2$ is achieved when $z=2$, and the maximum valu... |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (that is, it's always greater than or equal to 0) for all real $z$, the interval for $z^2 + 6z + 9$ is $[0, \infty)$.
However, we know that $z$ must be in the interval $(2, 4)$ from the first p... |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (because the square of any real number is nonnegative), and we know from step 1 that $2 < z < 4$, we can conclude that the minimum value of $(z+3)^2$ is achieved when $z=2$, and the maximum valu... | (25,49) |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (that is, it's always greater than or equal to 0) for all real $z$, the interval for $z^2 + 6z + 9$ is $[0, \infty)$.
However, we know that $z$ must be in the interval $(2, 4)$ from the first p... |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (because the square of any real number is nonnegative), and we know from step 1 that $2 < z < 4$, we can conclude that the minimum value of $(z+3)^2$ is achieved when $z=2$, and the maximum valu... |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (that is, it's always greater than or equal to 0) for all real $z$, the interval for $z^2 + 6z + 9$ is $[0, \infty)$.
However, we know that $z$ must be in the interval $(2, 4)$ from the first p... |
Now we want to find the interval for $z^2 + 6z + 9$. We can rewrite this as $(z+3)^2$. Since $(z+3)^2$ is always nonnegative (because the square of any real number is nonnegative), and we know from step 1 that $2 < z < 4$, we can conclude that the minimum value of $(z+3)^2$ is achieved when $z=2$, and the maximum valu... | -1.226563 | 1.695313 | 2.640625 | 1.695313 | 2.765625 | 1.789063 | 1.367188 | -0.351563 | 0.486328 |
gpt-3.5-turbo-GSM | A bag contains 50 balls numbered from 1 to 50, and a box contains 60 balls numbered from 41 to 100. A ball is randomly selected from each. What is the probability that the ball from the bag is divisible by 4 and the ball from the box is either prime or less than 60? Express your answer as a common fraction. | Let's think step by step.
Step 1:
First, we need to find the balls that are divisible by 4 from the bag. The numbers on the balls in the bag are from 1 to 50. The divisible by 4 numbers in this range are 4, 8, 12, ..., 50. We can see that these are all the multiples of 4 in this range. So, we have 12 numbers that are... |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. There are 13 prime numbers.
In addition to the prime numbers, the numbers less than 60 in ... |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. We also need to include the numbers less than 60 in this range, which are all the numbers ... |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. There are 13 prime numbers.
In addition to the prime numbers, the numbers less than 60 in ... |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. We also need to include the numbers less than 60 in this range, which are all the numbers ... | \frac{16}{125} |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. There are 13 prime numbers.
In addition to the prime numbers, the numbers less than 60 in ... |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. We also need to include the numbers less than 60 in this range, which are all the numbers ... |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. There are 13 prime numbers.
In addition to the prime numbers, the numbers less than 60 in ... |
Next, we need to find the balls that are either prime or less than 60 from the box. The numbers on the balls in the box are from 41 to 100. The prime numbers in this range are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. We also need to include the numbers less than 60 in this range, which are all the numbers ... | -3.703125 | -0.228516 | -0.871094 | 2.8125 | 0.816406 | -1.109375 | 0.199219 | 0.010986 | -1.820313 |
GSM_Rephrased | If Johnny purchases 15 packs of colored pencils for his class, with each pack containing a red, yellow, and green pencil, but 3 of the packs have two extra red pencils, how many red colored pencils did Johnny buy in total? | Let's think step by step.
Step 1: | Calculate the number of red pencils in the regular packs.
Each pack contains a red, yellow, and green pencil, so each pack has 1 red pencil.
Johnny bought 15 packs of colored pencils, so the number of red pencils in the regular packs is 15 * 1 = 15. |
Johnny purchased 15 packs of colored pencils. Each pack contains a red, yellow, and green pencil. So in each pack, there are 3 pencils. So in total, he bought 15 * 3 = 45 pencils. | Calculate the number of red pencils in the regular packs.
Each pack contains a red, yellow, and green pencil, so each pack has 1 red pencil.
Johnny bought 15 packs of colored pencils, so the number of red pencils in the regular packs is 15 * 1 = 15.
Step 2: Calculate the number of extra red pencils.
3 of the packs hav... |
Johnny purchased 15 packs of colored pencils. Each pack contains a red, yellow, and green pencil. So in each pack, there are 3 pencils. So in total, he bought 15 * 3 = 45 pencils.
Step 2:
However, 3 of the packs had two extra red pencils. Each pack has an extra red pencil for 3 packs, so that's 3 * 2 = 6 extra red ... | 21 | Calculate the number of red pencils in the regular packs.
Each pack contains a red, yellow, and green pencil, so each pack has 1 red pencil.
Johnny bought 15 packs of colored pencils, so the number of red pencils in the regular packs is 15 * 1 = 15.
Step 2: Calculate the number of extra red pencils.
3 of the packs hav... |
Johnny purchased 15 packs of colored pencils. Each pack contains a red, yellow, and green pencil. So in each pack, there are 3 pencils. So in total, he bought 15 * 3 = 45 pencils.
Step 2:
However, 3 of the packs had two extra red pencils. Each pack has an extra red pencil for 3 packs, so that's 3 * 2 = 6 extra red ... | Calculate the number of red pencils in the regular packs.
Each pack contains a red, yellow, and green pencil, so each pack has 1 red pencil.
Johnny bought 15 packs of colored pencils, so the number of red pencils in the regular packs is 15 * 1 = 15.
Step 2: Calculate the number of extra red pencils.
3 of the packs hav... |
Johnny purchased 15 packs of colored pencils. Each pack contains a red, yellow, and green pencil. So in each pack, there are 3 pencils. So in total, he bought 15 * 3 = 45 pencils.
Step 2:
However, 3 of the packs had two extra red pencils. Each pack has an extra red pencil for 3 packs, so that's 3 * 2 = 6 extra red ... | -9.375 | -6.96875 | -8.8125 | -0.730469 | -5.28125 | -5.8125 | -5.125 | -5.875 | -6.6875 |
GSM_Rephrased | After accidentally dropping and losing half of the 12 marbles Carl took out to play his game, he takes out an additional 10 marbles. His mother then brings him a bag with 25 marbles. If Carl cannot find his lost marbles, how many marbles will he have to put back in the jar after playing his game, considering both his o... | Let's think step by step.
Step 1:
Carl initially had 12 marbles. He lost half of them, so he lost 12/2 = 6 marbles.
So, he was left with 12 - 6 = 6 marbles.
Step 2:
He then took out an additional 10 marbles, so he now has 6 + 10 = 16 marbles.
Step 3:
His mother brings him a bag with 25 marbles, so he now has 16 + ... |
If the game requires him to return any marbles he does not use back into the jar, and if he cannot find his lost marbles, he will have to return all the marbles he currently has back into the jar. So, he will have to return 41 marbles back into the jar.
The answer is: 41 |
If Carl cannot find his lost marbles, he will have to put back in the jar the number of marbles he lost. Since he lost 6 marbles, he will have to put back 6 marbles.
So the answer is $\boxed{6}$.
The answer is: 6 |
If the game requires him to return any marbles he does not use back into the jar, and if he cannot find his lost marbles, he will have to return all the marbles he currently has back into the jar. So, he will have to return 41 marbles back into the jar.
The answer is: 41 |
If Carl cannot find his lost marbles, he will have to put back in the jar the number of marbles he lost. Since he lost 6 marbles, he will have to put back 6 marbles.
So the answer is $\boxed{6}$.
The answer is: 6 | 41 |
If the game requires him to return any marbles he does not use back into the jar, and if he cannot find his lost marbles, he will have to return all the marbles he currently has back into the jar. So, he will have to return 41 marbles back into the jar.
The answer is: 6 |
If Carl cannot find his lost marbles, he will have to put back in the jar the number of marbles he lost. Since he lost 6 marbles, he will have to put back 6 marbles.
So the answer is $\boxed{6}$.
The answer is: 41 |
If the game requires him to return any marbles he does not use back into the jar, and if he cannot find his lost marbles, he will have to return all the marbles he currently has back into the jar. So, he will have to return 41 marbles back into the jar.
The answer is: 116 |
If Carl cannot find his lost marbles, he will have to put back in the jar the number of marbles he lost. Since he lost 6 marbles, he will have to put back 6 marbles.
So the answer is $\boxed{6}$.
The answer is: 116 | -5.375 | -1.351563 | -3.078125 | -1.351563 | -3.171875 | -3.75 | -6.15625 | -5.03125 | -4.84375 |
MATH_Rephrased | Paul and Yuri are planning a World War III preenactment and they want it to be sunny for exactly one day during the four-day Labor Day weekend. The Boston weatherman predicts a 75 percent chance of rain each day. If it doesn't rain, then the weather will be sunny. What is the probability that Paul and Yuri get the weat... | Let's think step by step.
Step 1:
Paul and Yuri want exactly one day to be sunny during the four-day weekend. This means that three days will be rainy.
Step 2:
The probability of rain each day is 75% or 0.75. So, the probability of no rain (sunny) each day is $1 - 0.75 = 0.25$.
Step 3:
The number of ways to choo... |
The probability of getting exactly one sunny day is the number of ways to choose one sunny day times the probability of that happening. The probability of getting three rainy days in a row is $0.75^3$, and the probability of getting one sunny day is $0.25$. So the probability of getting exactly one sunny day is $4 \t... |
The probability of having exactly one sunny day is the number of ways to choose one sunny day times the probability of that day being sunny and the other three days being rainy. The probability of a sunny day is $0.25$ and the probability of a rainy day is $0.75$. So the probability of exactly one sunny day is:
\[4 \... |
The probability of getting exactly one sunny day is the number of ways to choose one sunny day times the probability of that happening. The probability of getting three rainy days in a row is $0.75^3$, and the probability of getting one sunny day is $0.25$. So the probability of getting exactly one sunny day is $4 \t... |
The probability of having exactly one sunny day is the number of ways to choose one sunny day times the probability of that day being sunny and the other three days being rainy. The probability of a sunny day is $0.25$ and the probability of a rainy day is $0.75$. So the probability of exactly one sunny day is:
\[4 ... | \frac{27}{64} |
The probability of getting exactly one sunny day is the number of ways to choose one sunny day times the probability of that happening. The probability of getting three rainy days in a row is $0.75^3$, and the probability of getting one sunny day is $0.25$. So the probability of getting exactly one sunny day is $4 \t... |
The probability of having exactly one sunny day is the number of ways to choose one sunny day times the probability of that day being sunny and the other three days being rainy. The probability of a sunny day is $0.25$ and the probability of a rainy day is $0.75$. So the probability of exactly one sunny day is:
\[4 ... |
The probability of getting exactly one sunny day is the number of ways to choose one sunny day times the probability of that happening. The probability of getting three rainy days in a row is $0.75^3$, and the probability of getting one sunny day is $0.25$. So the probability of getting exactly one sunny day is $4 \t... |
The probability of having exactly one sunny day is the number of ways to choose one sunny day times the probability of that day being sunny and the other three days being rainy. The probability of a sunny day is $0.25$ and the probability of a rainy day is $0.75$. So the probability of exactly one sunny day is:
\[4 ... | -4.25 | 1.171875 | 1.109375 | 1.171875 | 0.570313 | -0.863281 | -2.46875 | -4.5 | -2.984375 |
gpt-3.5-turbo-GSM | Using the notation $\{x\}$ for the fractional part of $x$, which is $x - [x]$, determine $\{\sqrt{2}\}$ | Let's think step by step.
Step 1:
First, we need to find the square root of 2. The square root of 2 is approximately 1.4142135623730951.
Step 2:
Now, we need to find the fractional part of this number. The fractional part of a number is the number minus its integer part. So, we first need to find the integer part o... |
So, the fractional part of the square root of 2 is approximately 0.4142135623730951.
Therefore, $\{\sqrt{2}\} \approx 0.4142135623730951$.
However, it is known that the square root of 2 is an irrational number, which means that its decimal representation never ends and never repeats. Therefore, the exact value of the... |
To express this decimal as a fraction, we can find the closest fraction to it. The fraction 1/2 is 0.5, which is too large, and the fraction 1/3 is 0.3333333333333333, which is too small. The fraction 2/5 is 0.4, which is close, but still too small, and the fraction 3/7 is 0.42857142857142855, which is very close. So... |
So, the fractional part of the square root of 2 is approximately 0.4142135623730951.
Therefore, $\{\sqrt{2}\} \approx 0.4142135623730951$.
However, it is known that the square root of 2 is an irrational number, which means that its decimal representation never ends and never repeats. Therefore, the exact value of the... |
To express this decimal as a fraction, we can find the closest fraction to it. The fraction 1/2 is 0.5, which is too large, and the fraction 1/3 is 0.3333333333333333, which is too small. The fraction 2/5 is 0.4, which is close, but still too small, and the fraction 3/7 is 0.42857142857142855, which is very close. So... | \sqrt{2}-1 |
So, the fractional part of the square root of 2 is approximately 0.4142135623730951.
Therefore, $\{\sqrt{2}\} \approx 0.4142135623730951$.
However, it is known that the square root of 2 is an irrational number, which means that its decimal representation never ends and never repeats. Therefore, the exact value of the... |
To express this decimal as a fraction, we can find the closest fraction to it. The fraction 1/2 is 0.5, which is too large, and the fraction 1/3 is 0.3333333333333333, which is too small. The fraction 2/5 is 0.4, which is close, but still too small, and the fraction 3/7 is 0.42857142857142855, which is very close. So... |
So, the fractional part of the square root of 2 is approximately 0.4142135623730951.
Therefore, $\{\sqrt{2}\} \approx 0.4142135623730951$.
However, it is known that the square root of 2 is an irrational number, which means that its decimal representation never ends and never repeats. Therefore, the exact value of the... |
To express this decimal as a fraction, we can find the closest fraction to it. The fraction 1/2 is 0.5, which is too large, and the fraction 1/3 is 0.3333333333333333, which is too small. The fraction 2/5 is 0.4, which is close, but still too small, and the fraction 3/7 is 0.42857142857142855, which is very close. So... | -7.625 | -3.921875 | -7.28125 | -3.921875 | -8.6875 | -6.0625 | -6.65625 | -7.0625 | -8.9375 |
GSM_AnsAug | Johnny buys 15 packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. How many red colored pencils did Johnny buy? | Let's think step by step.
Step 1: | First, we know that each pack has one red, one yellow, and one green pencil. So in 15 packs, there should be 15 packs * 1 red pencil/pack = 15 red pencils. |
First, we know that Johnny bought 15 packs of colored pencils. Each pack has 3 colors: red, yellow, and green. So, if there were no extra pencils, he would have bought 15 packs x 3 pencils per pack = 45 red pencils. | First, we know that each pack has one red, one yellow, and one green pencil. So in 15 packs, there should be 15 packs * 1 red pencil/pack = 15 red pencils.
Step 2: However, 3 packs have two extra red pencils each. So the extra red pencils from those 3 packs is 3 packs * 2 red pencils/pack = 6 red pencils.
Step 3: To f... |
First, we know that Johnny bought 15 packs of colored pencils. Each pack has 3 colors: red, yellow, and green. So, if there were no extra pencils, he would have bought 15 packs x 3 pencils per pack = 45 red pencils.
Step 2:
However, he noticed that 3 packs had two extra red pencils each. So, these 3 packs added 3 x ... | 21 | First, we know that each pack has one red, one yellow, and one green pencil. So in 15 packs, there should be 15 packs * 1 red pencil/pack = 15 red pencils.
Step 2: However, 3 packs have two extra red pencils each. So the extra red pencils from those 3 packs is 3 packs * 2 red pencils/pack = 6 red pencils.
Step 3: To f... |
First, we know that Johnny bought 15 packs of colored pencils. Each pack has 3 colors: red, yellow, and green. So, if there were no extra pencils, he would have bought 15 packs x 3 pencils per pack = 45 red pencils.
Step 2:
However, he noticed that 3 packs had two extra red pencils each. So, these 3 packs added 3 x ... | First, we know that each pack has one red, one yellow, and one green pencil. So in 15 packs, there should be 15 packs * 1 red pencil/pack = 15 red pencils.
Step 2: However, 3 packs have two extra red pencils each. So the extra red pencils from those 3 packs is 3 packs * 2 red pencils/pack = 6 red pencils.
Step 3: To f... |
First, we know that Johnny bought 15 packs of colored pencils. Each pack has 3 colors: red, yellow, and green. So, if there were no extra pencils, he would have bought 15 packs x 3 pencils per pack = 45 red pencils.
Step 2:
However, he noticed that 3 packs had two extra red pencils each. So, these 3 packs added 3 x ... | -9.125 | -7.9375 | -7.125 | -0.249023 | -0.761719 | -5.53125 | -4.96875 | -6.46875 | -6.125 |
gpt-3.5-turbo-GSM | If $e^{3i \gamma} = \frac{4 - i \sqrt{7}}{5},$ then find $\sin 6 \gamma.$ | Let's think step by step.
Step 1: | We know that $e^{i\theta} = \cos \theta + i \sin \theta$. So, we can write $e^{3i \gamma} = \cos 3\gamma + i \sin 3\gamma$. |
We start by writing $e^{3i \gamma}$ in the form $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude and $\theta$ is the argument of the complex number.
We know that $r = \sqrt{a^2 + b^2} = \sqrt{(4)^2 + (-\sqrt{7})^2} = \sqrt{16 + 7} = \sqrt{23}$.
And $\theta = \arctan \frac{b}{a} = \arctan \frac{-\sqrt{7}}{... | We know that $e^{i\theta} = \cos \theta + i \sin \theta$. So, we can write $e^{3i \gamma} = \cos 3\gamma + i \sin 3\gamma$.
Step 2: We are given that $e^{3i \gamma} = \frac{4 - i \sqrt{7}}{5}$. So, we have $\cos 3\gamma + i \sin 3\gamma = \frac{4 - i \sqrt{7}}{5}$.
Step 3: Comparing the real and imaginary parts on bot... |
We start by writing $e^{3i \gamma}$ in the form $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude and $\theta$ is the argument of the complex number.
We know that $r = \sqrt{a^2 + b^2} = \sqrt{(4)^2 + (-\sqrt{7})^2} = \sqrt{16 + 7} = \sqrt{23}$.
And $\theta = \arctan \frac{b}{a} = \arctan \frac{-\sqrt{7}}{... | -\frac{8\sqrt{7}}{25} | We know that $e^{i\theta} = \cos \theta + i \sin \theta$. So, we can write $e^{3i \gamma} = \cos 3\gamma + i \sin 3\gamma$.
Step 2: We are given that $e^{3i \gamma} = \frac{4 - i \sqrt{7}}{5}$. So, we have $\cos 3\gamma + i \sin 3\gamma = \frac{4 - i \sqrt{7}}{5}$.
Step 3: Comparing the real and imaginary parts on bot... |
We start by writing $e^{3i \gamma}$ in the form $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude and $\theta$ is the argument of the complex number.
We know that $r = \sqrt{a^2 + b^2} = \sqrt{(4)^2 + (-\sqrt{7})^2} = \sqrt{16 + 7} = \sqrt{23}$.
And $\theta = \arctan \frac{b}{a} = \arctan \frac{-\sqrt{7}}{... | We know that $e^{i\theta} = \cos \theta + i \sin \theta$. So, we can write $e^{3i \gamma} = \cos 3\gamma + i \sin 3\gamma$.
Step 2: We are given that $e^{3i \gamma} = \frac{4 - i \sqrt{7}}{5}$. So, we have $\cos 3\gamma + i \sin 3\gamma = \frac{4 - i \sqrt{7}}{5}$.
Step 3: Comparing the real and imaginary parts on bot... |
We start by writing $e^{3i \gamma}$ in the form $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude and $\theta$ is the argument of the complex number.
We know that $r = \sqrt{a^2 + b^2} = \sqrt{(4)^2 + (-\sqrt{7})^2} = \sqrt{16 + 7} = \sqrt{23}$.
And $\theta = \arctan \frac{b}{a} = \arctan \frac{-\sqrt{7}}{... | -5 | -1.65625 | -2.640625 | 8.3125 | 4.03125 | 3.84375 | 4.90625 | 3.515625 | 1.125 |
GSM_Rephrased | What is the average rainfall total for the duration of the storm if it dropped 5 inches of rain in the first 30 minutes, half that amount in the next 30 minutes, and 1/2 inch of rain for the following hour? | Let's think step by step.
Step 1:
In the first 30 minutes, it dropped 5 inches of rain.
Step 2:
In the next 30 minutes, it dropped half of the amount from the first 30 minutes, which is 5/2 = 2.5 inches of rain.
Step 3:
For the following hour, it dropped 1/2 inch of rain.
Step 4:
To find the total rainfall, we ... |
The storm lasted for a total of 1 hour (or 60 minutes) + 30 minutes + 30 minutes = 2 hours. |
To find the average rainfall per time period, we divide the total rainfall by the total time: 8 inches / 1 hour (or 60 minutes) = 8/60 = 4/30 = 2/15 inches per minute.
However, the question asks for the average rainfall total for the duration of the storm, not the average rainfall per minute. |
The storm lasted for a total of 1 hour (or 60 minutes) + 30 minutes + 30 minutes = 2 hours.
Step 6:
To find the average rainfall, we divide the total rainfall by the total time: 8 inches / 2 hours = 4 inches per hour.
The answer is: 4 |
To find the average rainfall per time period, we divide the total rainfall by the total time: 8 inches / 1 hour (or 60 minutes) = 8/60 = 4/30 = 2/15 inches per minute.
However, the question asks for the average rainfall total for the duration of the storm, not the average rainfall per minute.
Step 6:
To find the a... | 4 |
The storm lasted for a total of 1 hour (or 60 minutes) + 30 minutes + 30 minutes = 2 hours.
Step 6:
To find the average rainfall, we divide the total rainfall by the total time: 8 inches / 2 hours = 4 inches per hour.
The answer is: 8 |
To find the average rainfall per time period, we divide the total rainfall by the total time: 8 inches / 1 hour (or 60 minutes) = 8/60 = 4/30 = 2/15 inches per minute.
However, the question asks for the average rainfall total for the duration of the storm, not the average rainfall per minute.
Step 6:
To find the a... |
The storm lasted for a total of 1 hour (or 60 minutes) + 30 minutes + 30 minutes = 2 hours.
Step 6:
To find the average rainfall, we divide the total rainfall by the total time: 8 inches / 2 hours = 4 inches per hour.
The answer is: -939 |
To find the average rainfall per time period, we divide the total rainfall by the total time: 8 inches / 1 hour (or 60 minutes) = 8/60 = 4/30 = 2/15 inches per minute.
However, the question asks for the average rainfall total for the duration of the storm, not the average rainfall per minute.
Step 6:
To find the a... | -6.5625 | -8.25 | -6.625 | -2.40625 | -5 | -7.28125 | -4.5 | -6.84375 | -8.0625 |
GSM_Rephrased | Jenna is creating a costume for her role in Oliver Twist, which includes an overskirt, two petticoats, and a bodice. The overskirt and petticoats require rectangles of material measuring 12 feet by 4 feet each. The bodice requires 2 square feet of material for the shirt and 5 square feet of fabric for each sleeve. If t... | Let's think step by step.
Step 1: | Calculate the area of the overskirt and petticoats. Each of them is a rectangle with dimensions 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area of the petticoats is 2*48 = 96 square feet. The overskirt itself is also 48 square feet. So the total area of t... |
First, we need to find the area of each piece of material. The area of a rectangle is calculated by multiplying its length by its width.
The overskirt and each petticoat each require a rectangle measuring 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area ... | Calculate the area of the overskirt and petticoats. Each of them is a rectangle with dimensions 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area of the petticoats is 2*48 = 96 square feet. The overskirt itself is also 48 square feet. So the total area of t... |
First, we need to find the area of each piece of material. The area of a rectangle is calculated by multiplying its length by its width.
The overskirt and each petticoat each require a rectangle measuring 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area... | 468 | Calculate the area of the overskirt and petticoats. Each of them is a rectangle with dimensions 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area of the petticoats is 2*48 = 96 square feet. The overskirt itself is also 48 square feet. So the total area of t... |
First, we need to find the area of each piece of material. The area of a rectangle is calculated by multiplying its length by its width.
The overskirt and each petticoat each require a rectangle measuring 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area... | Calculate the area of the overskirt and petticoats. Each of them is a rectangle with dimensions 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area of the petticoats is 2*48 = 96 square feet. The overskirt itself is also 48 square feet. So the total area of t... |
First, we need to find the area of each piece of material. The area of a rectangle is calculated by multiplying its length by its width.
The overskirt and each petticoat each require a rectangle measuring 12 feet by 4 feet, so the area of each is 12*4 = 48 square feet. Since there are two petticoats, the total area... | -9.5 | -5.71875 | -6.90625 | -0.034424 | -0.601563 | -3.765625 | -3.15625 | -4.3125 | -5.34375 |
MATH_Rephrased | In how many ways can non-similar triangles be formed, where the degree measures of their angles are distinct positive integers in an arithmetic progression? | Let's think step by step.
Step 1:
The sum of the degree measures of the angles in a triangle is always 180 degrees.
Step 2:
If the degree measures of the angles are in an arithmetic progression, let's say the first term is $a$ and the common difference is $d$. Then the three angles are $a$, $a+d$, and $a+2d$.
Ste... |
Now we need to find the number of pairs of positive integers $(a, d)$ that satisfy $a+d = 60$. This is simply the number of ways to choose $a$ and $d$ from the numbers 1 through 59, since $a$ and $d$ must be positive integers. |
Now we need to find all possible pairs $(a,d)$ of positive integers that satisfy this equation. We can start by setting $a=1$ and find the corresponding value of $d$. If $a=1$, then $1+d=60$, which gives $d=59$. So one possible set of angles is $\{1, 59, 101\}$. |
Now we need to find the number of pairs of positive integers $(a, d)$ that satisfy $a+d = 60$. This is simply the number of ways to choose $a$ and $d$ from the numbers 1 through 59, since $a$ and $d$ must be positive integers.
Step 5:
The number of ways to choose $a$ and $d$ is the number of pairs $(a, 60-a)$, wher... |
Now we need to find all possible pairs $(a,d)$ of positive integers that satisfy this equation. We can start by setting $a=1$ and find the corresponding value of $d$. If $a=1$, then $1+d=60$, which gives $d=59$. So one possible set of angles is $\{1, 59, 101\}$.
Step 5:
We can continue this process for increasing v... | 59 |
Now we need to find the number of pairs of positive integers $(a, d)$ that satisfy $a+d = 60$. This is simply the number of ways to choose $a$ and $d$ from the numbers 1 through 59, since $a$ and $d$ must be positive integers.
Step 5:
The number of ways to choose $a$ and $d$ is the number of pairs $(a, 60-a)$, wher... |
Now we need to find all possible pairs $(a,d)$ of positive integers that satisfy this equation. We can start by setting $a=1$ and find the corresponding value of $d$. If $a=1$, then $1+d=60$, which gives $d=59$. So one possible set of angles is $\{1, 59, 101\}$.
Step 5:
We can continue this process for increasing v... |
Now we need to find the number of pairs of positive integers $(a, d)$ that satisfy $a+d = 60$. This is simply the number of ways to choose $a$ and $d$ from the numbers 1 through 59, since $a$ and $d$ must be positive integers.
Step 5:
The number of ways to choose $a$ and $d$ is the number of pairs $(a, 60-a)$, wher... |
Now we need to find all possible pairs $(a,d)$ of positive integers that satisfy this equation. We can start by setting $a=1$ and find the corresponding value of $d$. If $a=1$, then $1+d=60$, which gives $d=59$. So one possible set of angles is $\{1, 59, 101\}$.
Step 5:
We can continue this process for increasing v... | -5.71875 | -6.15625 | -6.875 | 0.613281 | -3.578125 | -2.796875 | -3.6875 | -4.03125 | -3.84375 |
gpt-3.5-turbo-GSM | Determine the result of \(\sin(\arcsin(\frac{3}{5}) + \arctan(2))\). | Let's think step by step.
Step 1: First, we find the value of $\arcsin(\frac{3}{5})$.
If we let $\theta = \arcsin(\frac{3}{5})$, then $\sin(\theta) = \frac{3}{5}$.
We know that $\sin^2(\theta) + \cos^2(\theta) = 1$, so $\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \frac{9}{25}} = \frac{4}{5}$.
Step 2: | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can solve for $\sin(\phi)$ and $\cos(\phi)$ using the Pythagorean identity $\sin^2(\phi) + \cos^2(\phi) = 1$.
From $\frac{... | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can square both sides to get $\frac{\sin^2(\phi)}{\cos^2(\phi)} = 4$, and then use the identity $\sin^2(\phi) + \cos^2(\ph... | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can solve for $\sin(\phi)$ and $\cos(\phi)$ using the Pythagorean identity $\sin^2(\phi) + \cos^2(\phi) = 1$.
From $\frac{... | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can square both sides to get $\frac{\sin^2(\phi)}{\cos^2(\phi)} = 4$, and then use the identity $\sin^2(\phi) + \cos^2(\ph... | \frac{11\sqrt{5}}{25} | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can solve for $\sin(\phi)$ and $\cos(\phi)$ using the Pythagorean identity $\sin^2(\phi) + \cos^2(\phi) = 1$.
From $\frac{... | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can square both sides to get $\frac{\sin^2(\phi)}{\cos^2(\phi)} = 4$, and then use the identity $\sin^2(\phi) + \cos^2(\ph... | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can solve for $\sin(\phi)$ and $\cos(\phi)$ using the Pythagorean identity $\sin^2(\phi) + \cos^2(\phi) = 1$.
From $\frac{... | Next, we find the value of $\arctan(2)$.
If we let $\phi = \arctan(2)$, then $\tan(\phi) = 2$.
We know that $\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}$, so $\frac{\sin(\phi)}{\cos(\phi)} = 2$.
We can square both sides to get $\frac{\sin^2(\phi)}{\cos^2(\phi)} = 4$, and then use the identity $\sin^2(\phi) + \cos^2(\ph... | -2.65625 | -2.234375 | -3.265625 | 5.875 | 5.75 | 0.601563 | 1.4375 | -0.170898 | -0.632813 |
GSM_Rephrased | What is the average temperature for July 4th in Washington, DC over the past five years, considering the following high temperatures: 90 degrees in 2020, 90 degrees in 2019, 90 degrees in 2018, 79 degrees in 2017, and 71 degrees in 2016? | Let's think step by step.
Step 1: | Add up all the high temperatures for July 4th over the past five years.
The high temperatures for 2020, 2019, and 2018 were all 90 degrees, so 90 + 90 + 90 = 270 degrees.
The high temperature for 2017 was 79 degrees, and for 2016 it was 71 degrees, so 79 + 71 = 150 degrees.
Adding these together, the total high temper... | Add up all the high temperatures for July 4th over the past five years.
The high temperatures were: 90 degrees in 2020, 90 degrees in 2019, 90 degrees in 2018, 79 degrees in 2017, and 71 degrees in 2016.
So, we add these up: 90 + 90 + 90 + 79 + 71 = 410 degrees. | Add up all the high temperatures for July 4th over the past five years.
The high temperatures for 2020, 2019, and 2018 were all 90 degrees, so 90 + 90 + 90 = 270 degrees.
The high temperature for 2017 was 79 degrees, and for 2016 it was 71 degrees, so 79 + 71 = 150 degrees.
Adding these together, the total high temper... | Add up all the high temperatures for July 4th over the past five years.
The high temperatures were: 90 degrees in 2020, 90 degrees in 2019, 90 degrees in 2018, 79 degrees in 2017, and 71 degrees in 2016.
So, we add these up: 90 + 90 + 90 + 79 + 71 = 410 degrees.
Step 2: To find the average, we divide the total by... | 84 | Add up all the high temperatures for July 4th over the past five years.
The high temperatures for 2020, 2019, and 2018 were all 90 degrees, so 90 + 90 + 90 = 270 degrees.
The high temperature for 2017 was 79 degrees, and for 2016 it was 71 degrees, so 79 + 71 = 150 degrees.
Adding these together, the total high temper... | Add up all the high temperatures for July 4th over the past five years.
The high temperatures were: 90 degrees in 2020, 90 degrees in 2019, 90 degrees in 2018, 79 degrees in 2017, and 71 degrees in 2016.
So, we add these up: 90 + 90 + 90 + 79 + 71 = 410 degrees.
Step 2: To find the average, we divide the total by... | Add up all the high temperatures for July 4th over the past five years.
The high temperatures for 2020, 2019, and 2018 were all 90 degrees, so 90 + 90 + 90 = 270 degrees.
The high temperature for 2017 was 79 degrees, and for 2016 it was 71 degrees, so 79 + 71 = 150 degrees.
Adding these together, the total high temper... | Add up all the high temperatures for July 4th over the past five years.
The high temperatures were: 90 degrees in 2020, 90 degrees in 2019, 90 degrees in 2018, 79 degrees in 2017, and 71 degrees in 2016.
So, we add these up: 90 + 90 + 90 + 79 + 71 = 410 degrees.
Step 2: To find the average, we divide the total by... | -9.0625 | -8.8125 | -6.40625 | -2.421875 | -1.820313 | -4.875 | -4.75 | -6.15625 | -7.125 |
GSM_Rephrased | If Grace charges 300 dollars per week and her client pays her every 2 weeks, how many weeks will it take for Grace to earn a total of 1800 dollars? | Let's think step by step.
Step 1:
First, we need to figure out how much Grace earns in 2 weeks. Since she charges 300 dollars per week, in 2 weeks she earns $300 \times 2 = 600$ dollars.
Step 2:
Now, we want to find out how many 2-week periods it takes for Grace to earn 1800 dollars. To do this, we divide the total ... |
Doing the division, we find that $\frac{1800}{600} = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars. Since each period is 2 weeks long, it will take her $3 \times 2 = 6$ weeks to earn a total of 1800 dollars.
The answer is: 6 |
Doing the division, we find that $1800 \div 600 = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars.
The answer is: 3 |
Doing the division, we find that $\frac{1800}{600} = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars. Since each period is 2 weeks long, it will take her $3 \times 2 = 6$ weeks to earn a total of 1800 dollars.
The answer is: 6 |
Doing the division, we find that $1800 \div 600 = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars.
The answer is: 3 | 6 |
Doing the division, we find that $\frac{1800}{600} = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars. Since each period is 2 weeks long, it will take her $3 \times 2 = 6$ weeks to earn a total of 1800 dollars.
The answer is: 3 |
Doing the division, we find that $1800 \div 600 = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars.
The answer is: 6 |
Doing the division, we find that $\frac{1800}{600} = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars. Since each period is 2 weeks long, it will take her $3 \times 2 = 6$ weeks to earn a total of 1800 dollars.
The answer is: 232 |
Doing the division, we find that $1800 \div 600 = 3$.
So, it will take Grace 3 two-week periods to earn a total of 1800 dollars.
The answer is: 232 | -5.8125 | -1.328125 | -1.765625 | -1.328125 | -1.859375 | -3.296875 | -3.703125 | -6.8125 | -7.15625 |
GSM_Rephrased | If Apollo needs to replace his chariot wheels once a month and trades golden apples with Hephaestus for the wheels, and Hephaestus raised his rates halfway through the year, now demanding twice as many golden apples as before, how many golden apples does Apollo have to pay for a full year of chariot wheels, considering... | Let's think step by step.
Step 1:
Hephaestus charges three golden apples for the first six months. Since there are 12 months in a year and the first six months are already accounted for, there are 12 - 6 = 6 months left in the year.
Step 2:
Before the rate increase, Hephaestus charged three golden apples for each mo... |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for each of the remaining six months, Apollo would have to pay 2 * 3 = 6 golden apples. |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for the remaining six months, Apollo would have to pay 2 * 6 = 12 golden apples. |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for each of the remaining six months, Apollo would have to pay 2 * 3 = 6 golden apples.
Step 4:
To find the total number of golden apples Apollo has to pay for a full year of chariot wheels, we add the golden apples for the fir... |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for the remaining six months, Apollo would have to pay 2 * 6 = 12 golden apples.
Step 4:
To find the total number of golden apples Apollo has to pay for a full year of chariot wheels, we add the number of golden apples for the ... | 54 |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for each of the remaining six months, Apollo would have to pay 2 * 3 = 6 golden apples.
Step 4:
To find the total number of golden apples Apollo has to pay for a full year of chariot wheels, we add the golden apples for the fir... |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for the remaining six months, Apollo would have to pay 2 * 6 = 12 golden apples.
Step 4:
To find the total number of golden apples Apollo has to pay for a full year of chariot wheels, we add the number of golden apples for the ... |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for each of the remaining six months, Apollo would have to pay 2 * 3 = 6 golden apples.
Step 4:
To find the total number of golden apples Apollo has to pay for a full year of chariot wheels, we add the golden apples for the fir... |
After the rate increase, Hephaestus now demands twice as many golden apples as before. So for the remaining six months, Apollo would have to pay 2 * 6 = 12 golden apples.
Step 4:
To find the total number of golden apples Apollo has to pay for a full year of chariot wheels, we add the number of golden apples for the ... | -5.46875 | -7.75 | -9.0625 | 1.023438 | -0.761719 | -4.53125 | -4.875 | -5.40625 | -6.21875 |
GSM_AnsAug | 3 years ago James turned 27. In 5 years Matt will be twice James age. How old is Matt now? | Let's think step by step.
Step 1: 3 years ago, James was 27, so his current age is 27 + 3 = 30 years old.
Step 2: | In 5 years, James will be 30 + 5 = 35 years old. | In 5 years, Matt will be twice James' age. This means Matt's age in 5 years will be 2 * 30 = 60 years old. | In 5 years, James will be 30 + 5 = 35 years old.
Step 3: According to the problem, in 5 years Matt will be twice James' age, so in 5 years, Matt will be 2 * 35 = 70 years old.
Step 4: To find Matt's current age, we need to subtract 5 years from his age in 5 years, so Matt is currently 70 - 5 = 65 years old.
So the ans... | In 5 years, Matt will be twice James' age. This means Matt's age in 5 years will be 2 * 30 = 60 years old.
Step 3: To find out how old Matt is now, we need to subtract 5 years from his age in 5 years. So, Matt's current age is 60 - 5 = 55 years old.
So, the answer is $\boxed{55}$.
The answer is: 55 | 65 | In 5 years, James will be 30 + 5 = 35 years old.
Step 3: According to the problem, in 5 years Matt will be twice James' age, so in 5 years, Matt will be 2 * 35 = 70 years old.
Step 4: To find Matt's current age, we need to subtract 5 years from his age in 5 years, so Matt is currently 70 - 5 = 65 years old.
So the ans... | In 5 years, Matt will be twice James' age. This means Matt's age in 5 years will be 2 * 30 = 60 years old.
Step 3: To find out how old Matt is now, we need to subtract 5 years from his age in 5 years. So, Matt's current age is 60 - 5 = 55 years old.
So, the answer is $\boxed{55}$.
The answer is: 65 | In 5 years, James will be 30 + 5 = 35 years old.
Step 3: According to the problem, in 5 years Matt will be twice James' age, so in 5 years, Matt will be 2 * 35 = 70 years old.
Step 4: To find Matt's current age, we need to subtract 5 years from his age in 5 years, so Matt is currently 70 - 5 = 65 years old.
So the ans... | In 5 years, Matt will be twice James' age. This means Matt's age in 5 years will be 2 * 30 = 60 years old.
Step 3: To find out how old Matt is now, we need to subtract 5 years from his age in 5 years. So, Matt's current age is 60 - 5 = 55 years old.
So, the answer is $\boxed{55}$.
The answer is: 149 | -6.5 | -7.28125 | -5.78125 | -0.707031 | -0.396484 | -3.265625 | -2.5 | -3.71875 | -3.828125 |
GSM_Rephrased | If Erika and her brother Rick want to purchase a gift for their mother worth $250, and Erika has saved $155 while her brother has saved only half the cost of the gift, and they also plan to surprise their mother with a $25 birthday cake, how much money will they have remaining after purchasing the gift and cake? | Let's think step by step.
Step 1:
First, we need to find out how much money Rick has saved. Since Rick has saved only half the cost of the gift, and the gift is worth $250, Rick has saved $250/2 = $125.
Step 2:
Now, let's add up the money that Erika and Rick have saved. Erika has saved $155, and Rick has saved $125,... |
They also plan to buy a $25 birthday cake. So, the total cost of the gift and cake will be $250 + $25 = $275. |
They also plan to buy a $25 birthday cake. So, we need to add this cost to the total amount of money they have saved. $280 + $25 = $305. |
They also plan to buy a $25 birthday cake. So, the total cost of the gift and cake will be $250 + $25 = $275.
Step 4:
To find out how much money they will have remaining, we need to subtract the total cost of the gift and cake from the total amount of money they have saved. So, $280 - $275 = $5.
So, they will have $5 ... |
They also plan to buy a $25 birthday cake. So, we need to add this cost to the total amount of money they have saved. $280 + $25 = $305.
Step 4:
The total cost of the gift and cake is $305. Since they want to find out how much money they will have remaining, we need to subtract this total from the amount of money the... | 5 |
They also plan to buy a $25 birthday cake. So, the total cost of the gift and cake will be $250 + $25 = $275.
Step 4:
To find out how much money they will have remaining, we need to subtract the total cost of the gift and cake from the total amount of money they have saved. So, $280 - $275 = $5.
So, they will have $5 ... |
They also plan to buy a $25 birthday cake. So, we need to add this cost to the total amount of money they have saved. $280 + $25 = $305.
Step 4:
The total cost of the gift and cake is $305. Since they want to find out how much money they will have remaining, we need to subtract this total from the amount of money the... |
They also plan to buy a $25 birthday cake. So, the total cost of the gift and cake will be $250 + $25 = $275.
Step 4:
To find out how much money they will have remaining, we need to subtract the total cost of the gift and cake from the total amount of money they have saved. So, $280 - $275 = $5.
So, they will have $5 ... |
They also plan to buy a $25 birthday cake. So, we need to add this cost to the total amount of money they have saved. $280 + $25 = $305.
Step 4:
The total cost of the gift and cake is $305. Since they want to find out how much money they will have remaining, we need to subtract this total from the amount of money the... | -5.875 | -5.96875 | -5.84375 | -1.070313 | -3.875 | -5.09375 | -4.125 | -6.15625 | -7.03125 |
gpt-3.5-turbo-GSM | Let $0 \le a,$ $b,$ $c,$ $d,$ $e,$ $f,$ $g,$ $h \le 1.$ Find the maximum value of the expression
\[\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - e)^2} + \sqrt{e^2 + (1 - f)^2} + \sqrt{f^2 + (1 - g)^2} + \sqrt{g^2 + (1 - h)^2} + \sqrt{h^2 + (1 - a)^2}.\] | Let's think step by step.
Step 1: | Consider the expression
\[\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - e)^2} + \sqrt{e^2 + (1 - f)^2} + \sqrt{f^2 + (1 - g)^2} + \sqrt{g^2 + (1 - h)^2} + \sqrt{h^2 + (1 - a)^2}.\]
By the triangle inequality, we have
\[\sqrt{a^2 + (1 - b)^2} \le a + (1 - b),\]
\[\sqrt{b^2 ... | We know that the shortest distance between two points is a straight line. So we can think of each term in the expression as the distance between two points in a circle. The first point is $(a, b)$, the second point is $(1, 0)$. The distance between these two points is $\sqrt{a^2 + (1 - b)^2}$. Similarly, the distance ... | Consider the expression
\[\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - e)^2} + \sqrt{e^2 + (1 - f)^2} + \sqrt{f^2 + (1 - g)^2} + \sqrt{g^2 + (1 - h)^2} + \sqrt{h^2 + (1 - a)^2}.\]
By the triangle inequality, we have
\[\sqrt{a^2 + (1 - b)^2} \le a + (1 - b),\]
\[\sqrt{b^2 ... | We know that the shortest distance between two points is a straight line. So we can think of each term in the expression as the distance between two points in a circle. The first point is $(a, b)$, the second point is $(1, 0)$. The distance between these two points is $\sqrt{a^2 + (1 - b)^2}$. Similarly, the distance ... | 8 | Consider the expression
\[\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - e)^2} + \sqrt{e^2 + (1 - f)^2} + \sqrt{f^2 + (1 - g)^2} + \sqrt{g^2 + (1 - h)^2} + \sqrt{h^2 + (1 - a)^2}.\]
By the triangle inequality, we have
\[\sqrt{a^2 + (1 - b)^2} \le a + (1 - b),\]
\[\sqrt{b^2 ... | We know that the shortest distance between two points is a straight line. So we can think of each term in the expression as the distance between two points in a circle. The first point is $(a, b)$, the second point is $(1, 0)$. The distance between these two points is $\sqrt{a^2 + (1 - b)^2}$. Similarly, the distance ... | Consider the expression
\[\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - e)^2} + \sqrt{e^2 + (1 - f)^2} + \sqrt{f^2 + (1 - g)^2} + \sqrt{g^2 + (1 - h)^2} + \sqrt{h^2 + (1 - a)^2}.\]
By the triangle inequality, we have
\[\sqrt{a^2 + (1 - b)^2} \le a + (1 - b),\]
\[\sqrt{b^2 ... | We know that the shortest distance between two points is a straight line. So we can think of each term in the expression as the distance between two points in a circle. The first point is $(a, b)$, the second point is $(1, 0)$. The distance between these two points is $\sqrt{a^2 + (1 - b)^2}$. Similarly, the distance ... | -5.28125 | 1.546875 | -5.90625 | 1.546875 | 0.194336 | -0.416016 | -0.447266 | -0.859375 | -1.445313 |
gpt-3.5-turbo-GSM | Inside triangle ABC, there is a circle, named circle O, which happens to be the inscribed circle of the triangle and the circumscribed circle of another triangle XYZ. Point X lies on side BC, point Y on side AB, and point Z on side AC. Given that the angles of triangle ABC are such that $\angle A$ equals $40^\circ$, $\... | Let's think step by step.
Step 1: | Determine the measures of angles of triangle XYZ.
Since circle O is the inscribed circle of triangle XYZ, we can use the property that the sum of the measures of the angles opposite the sides of a triangle is equal to the measure of the angle at the center of the circle subtended by those sides.
So, $\angle Y + \angle... |
The inscribed circle of triangle ABC touches the sides of the triangle at points X, Y, and Z.
Therefore, the angles of triangle XYZ are equal to the corresponding angles of triangle ABC. So, $\angle X = \angle A = 40^\circ$, $\angle Y = \angle B = 60^\circ$, and $\angle Z = \angle C = 80^\circ$. | Determine the measures of angles of triangle XYZ.
Since circle O is the inscribed circle of triangle XYZ, we can use the property that the sum of the measures of the angles opposite the sides of a triangle is equal to the measure of the angle at the center of the circle subtended by those sides.
So, $\angle Y + \angle... |
The inscribed circle of triangle ABC touches the sides of the triangle at points X, Y, and Z.
Therefore, the angles of triangle XYZ are equal to the corresponding angles of triangle ABC. So, $\angle X = \angle A = 40^\circ$, $\angle Y = \angle B = 60^\circ$, and $\angle Z = \angle C = 80^\circ$.
Step 2:
To find $... | 120 | Determine the measures of angles of triangle XYZ.
Since circle O is the inscribed circle of triangle XYZ, we can use the property that the sum of the measures of the angles opposite the sides of a triangle is equal to the measure of the angle at the center of the circle subtended by those sides.
So, $\angle Y + \angle... |
The inscribed circle of triangle ABC touches the sides of the triangle at points X, Y, and Z.
Therefore, the angles of triangle XYZ are equal to the corresponding angles of triangle ABC. So, $\angle X = \angle A = 40^\circ$, $\angle Y = \angle B = 60^\circ$, and $\angle Z = \angle C = 80^\circ$.
Step 2:
To find $... | Determine the measures of angles of triangle XYZ.
Since circle O is the inscribed circle of triangle XYZ, we can use the property that the sum of the measures of the angles opposite the sides of a triangle is equal to the measure of the angle at the center of the circle subtended by those sides.
So, $\angle Y + \angle... |
The inscribed circle of triangle ABC touches the sides of the triangle at points X, Y, and Z.
Therefore, the angles of triangle XYZ are equal to the corresponding angles of triangle ABC. So, $\angle X = \angle A = 40^\circ$, $\angle Y = \angle B = 60^\circ$, and $\angle Z = \angle C = 80^\circ$.
Step 2:
To find $... | -5.53125 | -0.408203 | -3.328125 | 4.375 | 4.59375 | 0.949219 | 1.640625 | 1.234375 | 0.201172 |
GSM_Rephrased | If Beatrice purchased 20 books at the supermarket, paying $20 for each of the first 5 books and receiving a $2 discount for each additional book, what is the total amount she paid at the supermarket? | Let's think step by step.
Step 1:
For the first 5 books, Beatrice paid $20 for each book. So, the total cost for these books is 5 * $20 = $100.
Step 2: |
For the remaining 15 books, she received a $2 discount for each book. So, the price for each of these books is $20 - $2 = $18. |
For the remaining 15 books (20 total books - 5 books), Beatrice received a $2 discount for each book. So, the discount for each book is $2. Therefore, the total discount for these books is 15 * $2 = $30. |
For the remaining 15 books, she received a $2 discount for each book. So, the price for each of these books is $20 - $2 = $18.
Step 3:
The total cost for the remaining 15 books is 15 * $18 = $270.
Step 4:
To find the total amount Beatrice paid at the supermarket, we add the cost of the first 5 books and the cost of th... |
For the remaining 15 books (20 total books - 5 books), Beatrice received a $2 discount for each book. So, the discount for each book is $2. Therefore, the total discount for these books is 15 * $2 = $30.
Step 3:
To find the total amount Beatrice paid at the supermarket, we need to subtract the discount from the total... | 370 |
For the remaining 15 books, she received a $2 discount for each book. So, the price for each of these books is $20 - $2 = $18.
Step 3:
The total cost for the remaining 15 books is 15 * $18 = $270.
Step 4:
To find the total amount Beatrice paid at the supermarket, we add the cost of the first 5 books and the cost of th... |
For the remaining 15 books (20 total books - 5 books), Beatrice received a $2 discount for each book. So, the discount for each book is $2. Therefore, the total discount for these books is 15 * $2 = $30.
Step 3:
To find the total amount Beatrice paid at the supermarket, we need to subtract the discount from the total... |
For the remaining 15 books, she received a $2 discount for each book. So, the price for each of these books is $20 - $2 = $18.
Step 3:
The total cost for the remaining 15 books is 15 * $18 = $270.
Step 4:
To find the total amount Beatrice paid at the supermarket, we add the cost of the first 5 books and the cost of th... |
For the remaining 15 books (20 total books - 5 books), Beatrice received a $2 discount for each book. So, the discount for each book is $2. Therefore, the total discount for these books is 15 * $2 = $30.
Step 3:
To find the total amount Beatrice paid at the supermarket, we need to subtract the discount from the total... | -5.71875 | -6.78125 | -8.6875 | -0.026489 | -5.4375 | -5.90625 | -4.625 | -4.53125 | -6.875 |
GSM_Rephrased | If each pie costs $4 per piece and there are 3 pieces in each pie, and the bakery can produce 12 pies per hour, with a production cost of $0.5 per pie, how much money would the bakery make if all pie pieces were sold? | Let's think step by step.
Step 1:
The bakery can produce 12 pies per hour, and each pie has 3 pieces, so the total number of pieces produced per hour is 12 * 3 = 36 pieces.
Step 2: |
Each piece of pie costs $4, so if all the pieces were sold, the total revenue would be 36 * $4 = $144. |
Each piece of pie costs $4 to make, so the total cost of making all the pieces per hour is 36 * $4 = $144. |
Each piece of pie costs $4, so if all the pieces were sold, the total revenue would be 36 * $4 = $144.
Step 3:
The production cost per pie is $0.5, and the bakery produces 12 pies per hour, so the total production cost is 12 * $0.5 = $6.
Step 4:
To find the profit, we need to subtract the production cost from the t... |
Each piece of pie costs $4 to make, so the total cost of making all the pieces per hour is 36 * $4 = $144.
Step 3:
The production cost per pie is $0.5, and the bakery produces 12 pies per hour, so the total production cost per hour is 12 * $0.5 = $6.
Step 4:
The total revenue per hour is the total number of piece... | 138 |
Each piece of pie costs $4, so if all the pieces were sold, the total revenue would be 36 * $4 = $144.
Step 3:
The production cost per pie is $0.5, and the bakery produces 12 pies per hour, so the total production cost is 12 * $0.5 = $6.
Step 4:
To find the profit, we need to subtract the production cost from the t... |
Each piece of pie costs $4 to make, so the total cost of making all the pieces per hour is 36 * $4 = $144.
Step 3:
The production cost per pie is $0.5, and the bakery produces 12 pies per hour, so the total production cost per hour is 12 * $0.5 = $6.
Step 4:
The total revenue per hour is the total number of piece... |
Each piece of pie costs $4, so if all the pieces were sold, the total revenue would be 36 * $4 = $144.
Step 3:
The production cost per pie is $0.5, and the bakery produces 12 pies per hour, so the total production cost is 12 * $0.5 = $6.
Step 4:
To find the profit, we need to subtract the production cost from the t... |
Each piece of pie costs $4 to make, so the total cost of making all the pieces per hour is 36 * $4 = $144.
Step 3:
The production cost per pie is $0.5, and the bakery produces 12 pies per hour, so the total production cost per hour is 12 * $0.5 = $6.
Step 4:
The total revenue per hour is the total number of piece... | -6.03125 | -3.578125 | -10.875 | -0.205078 | -5.53125 | -6.5 | -4.875 | -4.84375 | -6.78125 |
gpt-3.5-turbo-GSM | Determine the least positive integer $x$ for which the sum of $x$ and 4609 gives a remainder of 2104 when divided by 12. | Let's think step by step.
Step 1:
We know that $x + 4609 \equiv 2104 \pmod{12}$.
Step 2:
We can rewrite the given congruence as $x \equiv 2104 - 4609 \pmod{12}$.
Step 3:
Calculating $2104 - 4609 = -2505$.
Step 4: |
Now we need to find the equivalent positive integer for $-2505 \pmod{12}$.
To do this, we divide $-2505$ by $12$ and find the remainder.
$-2505 \div 12 = -208 \text{ remainder } -9$.
However, we want a positive remainder, so we add $12$ to $-9$ to get a positive remainder.
$-9 + 12 = 3$.
So, $-2505 \equiv 3 \pmod{12}$... |
We want to find the positive integer $x$, so we add $12$ until we get a positive number.
$-2505 + 12 \cdot 210 = 30$. |
Now we need to find the equivalent positive integer for $-2505 \pmod{12}$.
To do this, we divide $-2505$ by $12$ and find the remainder.
$-2505 \div 12 = -208 \text{ remainder } -9$.
However, we want a positive remainder, so we add $12$ to $-9$ to get a positive remainder.
$-9 + 12 = 3$.
So, $-2505 \equiv 3 \pmod{12}$... |
We want to find the positive integer $x$, so we add $12$ until we get a positive number.
$-2505 + 12 \cdot 210 = 30$.
Step 5:
So $x \equiv 30 \pmod{12}$.
Step 6:
The least positive integer $x$ that satisfies this congruence is $x = 6$.
The answer is: 6 | 3 |
Now we need to find the equivalent positive integer for $-2505 \pmod{12}$.
To do this, we divide $-2505$ by $12$ and find the remainder.
$-2505 \div 12 = -208 \text{ remainder } -9$.
However, we want a positive remainder, so we add $12$ to $-9$ to get a positive remainder.
$-9 + 12 = 3$.
So, $-2505 \equiv 3 \pmod{12}$... |
We want to find the positive integer $x$, so we add $12$ until we get a positive number.
$-2505 + 12 \cdot 210 = 30$.
Step 5:
So $x \equiv 30 \pmod{12}$.
Step 6:
The least positive integer $x$ that satisfies this congruence is $x = 6$.
The answer is: 3 |
Now we need to find the equivalent positive integer for $-2505 \pmod{12}$.
To do this, we divide $-2505$ by $12$ and find the remainder.
$-2505 \div 12 = -208 \text{ remainder } -9$.
However, we want a positive remainder, so we add $12$ to $-9$ to get a positive remainder.
$-9 + 12 = 3$.
So, $-2505 \equiv 3 \pmod{12}$... |
We want to find the positive integer $x$, so we add $12$ until we get a positive number.
$-2505 + 12 \cdot 210 = 30$.
Step 5:
So $x \equiv 30 \pmod{12}$.
Step 6:
The least positive integer $x$ that satisfies this congruence is $x = 6$.
The answer is: -141 | -2.171875 | -0.302734 | -1.554688 | 2.015625 | 0.345703 | -1.835938 | -1.289063 | -2.109375 | -3.828125 |
GSM_AnsAug | Seven parrots and some crows are perched on a tree branch. There was a noise and the same number of parrots and crows flew away. If only 2 parrots and 1 crow are left on the tree branch now, how many birds were perched on the branch to begin with? | Let's think step by step.
Step 1:
We know that the number of parrots that flew away is equal to the number of crows that flew away. Let's denote the number of parrots that flew away as $P$ and the number of crows that flew away as $C$.
Step 2:
We know that initially, there were 7 parrots on the branch. After some fl... |
Let's denote the initial number of crows as $N$. After some flew away, there is 1 crow left. So, we can write the equation: $N - C = 1$, which simplifies to $N - 5 = 1$, and then $N = 6$. This means there were initially 6 crows on the branch. |
We also know that initially, there were some crows on the branch. After some flew away, there is 1 crow left. So, we can write the equation: $C - C = 1$, which simplifies to $C = 5$. This means there were 5 crows initially on the branch. |
Let's denote the initial number of crows as $N$. After some flew away, there is 1 crow left. So, we can write the equation: $N - C = 1$, which simplifies to $N - 5 = 1$, and then $N = 6$. This means there were initially 6 crows on the branch.
Step 5:
To find the total number of birds on the branch initially, we add th... |
We also know that initially, there were some crows on the branch. After some flew away, there is 1 crow left. So, we can write the equation: $C - C = 1$, which simplifies to $C = 5$. This means there were 5 crows initially on the branch.
Step 5:
To find out how many birds were perched on the branch to begin with, we ... | 13 |
Let's denote the initial number of crows as $N$. After some flew away, there is 1 crow left. So, we can write the equation: $N - C = 1$, which simplifies to $N - 5 = 1$, and then $N = 6$. This means there were initially 6 crows on the branch.
Step 5:
To find the total number of birds on the branch initially, we add th... |
We also know that initially, there were some crows on the branch. After some flew away, there is 1 crow left. So, we can write the equation: $C - C = 1$, which simplifies to $C = 5$. This means there were 5 crows initially on the branch.
Step 5:
To find out how many birds were perched on the branch to begin with, we ... |
Let's denote the initial number of crows as $N$. After some flew away, there is 1 crow left. So, we can write the equation: $N - C = 1$, which simplifies to $N - 5 = 1$, and then $N = 6$. This means there were initially 6 crows on the branch.
Step 5:
To find the total number of birds on the branch initially, we add th... |
We also know that initially, there were some crows on the branch. After some flew away, there is 1 crow left. So, we can write the equation: $C - C = 1$, which simplifies to $C = 5$. This means there were 5 crows initially on the branch.
Step 5:
To find out how many birds were perched on the branch to begin with, we ... | -5.03125 | -1.9375 | -7.0625 | 1.671875 | -3.28125 | -5.125 | -3.203125 | -3.703125 | -5.78125 |
MATH_Rephrased | Let $m$ be the smallest positive integer with only two positive divisors, and let $n$ be the largest integer less than $100$ with exactly three positive divisors. What is the sum of $m$ and $n$? | Let's think step by step.
Step 1: To find the smallest positive integer $m$ with only two positive divisors, we need to find a prime number. The smallest prime number is $2$. The only divisors of $2$ are $1$ and $2$ itself, so $m=2$.
Step 2: | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a square of a prime number. The largest prime number less than $10$ is $7$, and $7^2 = 49$. The positive divisors of $49$ are $1$, $7$, and $49$ itself, so $n=49$. | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a perfect square. The largest perfect square less than $100$ is $81$, which is $9^2$. The positive divisors of $81$ are $1$, $3$, $9$, and $27$, so $n=81$. | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a square of a prime number. The largest prime number less than $10$ is $7$, and $7^2 = 49$. The positive divisors of $49$ are $1$, $7$, and $49$ itself, so $n=49$.
Step 3: To find the sum of $m$ and $n$, we simply add... | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a perfect square. The largest perfect square less than $100$ is $81$, which is $9^2$. The positive divisors of $81$ are $1$, $3$, $9$, and $27$, so $n=81$.
Step 3: To find the sum of $m$ and $n$, we simply add them t... | 51 | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a square of a prime number. The largest prime number less than $10$ is $7$, and $7^2 = 49$. The positive divisors of $49$ are $1$, $7$, and $49$ itself, so $n=49$.
Step 3: To find the sum of $m$ and $n$, we simply add... | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a perfect square. The largest perfect square less than $100$ is $81$, which is $9^2$. The positive divisors of $81$ are $1$, $3$, $9$, and $27$, so $n=81$.
Step 3: To find the sum of $m$ and $n$, we simply add them t... | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a square of a prime number. The largest prime number less than $10$ is $7$, and $7^2 = 49$. The positive divisors of $49$ are $1$, $7$, and $49$ itself, so $n=49$.
Step 3: To find the sum of $m$ and $n$, we simply add... | To find the largest integer $n$ less than $100$ with exactly three positive divisors, we need to find a perfect square. The largest perfect square less than $100$ is $81$, which is $9^2$. The positive divisors of $81$ are $1$, $3$, $9$, and $27$, so $n=81$.
Step 3: To find the sum of $m$ and $n$, we simply add them t... | -4.5 | -0.279297 | -2.3125 | 1.476563 | -0.503906 | -2.421875 | -1.742188 | -2.625 | -3.0625 |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 3