source stringclasses 7
values | question stringlengths 8 1.02k | steps listlengths 1 118 | answer stringlengths 1 908 | ground_truth stringlengths 1 117 | is_correct bool 1
class |
|---|---|---|---|---|---|
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"First, I recognize that the arithmetic mean of eight positive integers is 7. This means that the total sum of these eight integers is \\(8 \\times 7 = 56\\).",
"Next, when one integer is removed, the mean of the remaining seven integers becomes 6. Therefore, the total sum of the removed integer and the other six... | The sum of eight integers with a mean of 7 is $8 \times 7 = 56$, and removing one integer leaves a sum of $(8-1) \times 6 = 42$. Therefore, the removed integer is $56 - 42 = 14$. | 14 | true |
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"First, I recognize that the arithmetic mean of eight positive integers being 7 means the total sum of these integers is \\(8 \\times 7 = 56\\).",
"When one integer is removed, the mean of the remaining seven integers is 6. Therefore, the total sum of these seven integers is \\(7 \\times 6 = 42\\).",
"To find t... | The eight integers sum to 8 Γ 7 = 56, and the remaining seven sum to 7 Γ 6 = 42. Subtracting gives the removed integer: 56 β 42 = 14. | 14 | true |
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"First, I recognize that the arithmetic mean of eight positive integers is 7, which means the total sum of these integers is \\(8 \\times 7 = 56\\).",
"When one integer is removed, the remaining seven integers have a mean of 6, so their total sum is \\(7 \\times 6 = 42\\).",
"To find the value of the removed in... | The sum of eight integers with a mean of 7 is $8 \times 7 = 56$, and the sum of seven integers with a mean of 6 is $7 \times 6 = 42$. Therefore, the removed integer is $(56 - 42) = 14$. | 14 | true |
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"To find the value of the integer that is removed, I'll start by calculating the total sum of the eight positive integers. Since the mean is 7, the total sum is 8 multiplied by 7, which equals 56.",
"Next, I'll calculate the total sum of the remaining seven integers after one is removed. The mean of these seven i... | The eight integers sum to 7 Γ 8 = 56, while the remaining seven sum to 7 Γ 6 = 42, so the removed integer is 56 β 42 = 14. | 14 | true |
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"First, I recognize that the arithmetic mean of eight positive integers is 7. This means the total sum of these integers is \\(8 \\times 7 = 56\\).",
"When one integer is removed, the sum of the remaining seven integers becomes \\(56 - x\\), where \\(x\\) is the integer that was removed.",
"The mean of the rema... | The sum of the eight integers is $8 \times 7 = 56$, and the sum of the remaining seven integers is $7 \times 6 = 42$. Therefore, the integer removed is $56 - 42 = 14$. | 14 | true |
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"First, I recognize that the arithmetic mean of eight positive integers being 7 means the total sum of these integers is \\(8 \\times 7 = 56\\).",
"When one integer is removed, the average of the remaining seven integers becomes 6. Therefore, the total sum of these seven integers is \\(7 \\times 6 = 42\\).",
"T... | The total sum of eight positive integers with a mean of 7 is $8 \times 7 = 56$. When one integer is removed, the sum of the remaining seven integers is $7 \times 6 = 42$, so the removed integer is $56 - 42 = 14$. | 14 | true |
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"First, I recognize that the arithmetic mean of eight positive integers is 7. This means the total sum of these integers is 8 multiplied by 7, which equals 56.",
"Next, when one integer is removed, the mean of the remaining seven integers changes to 6. Therefore, the total sum of these seven integers is 7 multipl... | Since the mean of eight integers is 7, their sum is 56, and removing one integer leaves a sum of 42 over seven integers. Thus the removed integer is 56 β 42 = 14. | 14 | true |
zwhe99/DeepMath-103K | The arithmetic mean of eight positive integers is 7. If one of the eight integers is removed, the mean becomes 6. What is the value of the integer that is removed? | [
"First, calculate the total sum of the eight integers by multiplying the mean by the number of integers: 7 Γ 8 = 56.",
"Next, determine the new total sum after removing one integer by multiplying the new mean by the number of integers: 6 Γ 7 = 42.",
"Finally, find the removed integer by subtracting the new tota... | The original sum of eight integers is $7 \times 8 = 56$, and the new sum is $6 \times 7 = 42$. The removed integer is $56 - 42 = 14$. | 14 | true |
zwhe99/DeepMath-103K | The graph of $y=ax^2+bx+c$ passes through points $(0,5)$, $(1,10)$, and $(2,19)$. Find $a+b+c$. | [
"First, I need to find the equation of the quadratic function \\( y = ax^2 + bx + c \\) that passes through the points \\( (0, 5) \\), \\( (1, 10) \\), and \\( (2, 19) \\).",
"I'll start by substituting the x-values from each point into the function and calculating the corresponding y-values.",
"From \\( (0, 5)... | Substituting the given points into the quadratic equation yields $a+b+c=10$ and $4a+2b+c=19$. Equating these expressions, we find that $a+b+c = 10$. | 10 | true |
zwhe99/DeepMath-103K | If $f(x)$, whose graph is shown below, is defined on $1 \le x \le 6$, what is the maximum value of $f^{-1}(x)$? [asy]
import graph; size(7.94cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.96,xmax=8.96,ymin=-2.66,ymax=4.38;
Label laxis; laxis.p=fontsize(10);
xaxis(... | [
"Okay, so I need to find the maximum value of fβ»ΒΉ(x). The function f(x) is defined on x β€ 6, which means it's only onto the interval from 0 to 6. The graph of f(x) is green, and the reference function, since it's just a vertical line, is the y-axis, black.",
"First, I need to figure out the horizontal limits for ... | The maximum value of \( f^{-1}(x) \) is 5, as the interval defining \( f(x) \) is \( [0,6] \), and the graph is symmetric about \( x = 0 \). Therefore, \( f(6) = f(-6) \), and the maximum value on the interval \( [0,6] \) is 6. | 6 | true |
zwhe99/DeepMath-103K | If $f(x)$, whose graph is shown below, is defined on $1 \le x \le 6$, what is the maximum value of $f^{-1}(x)$? [asy]
import graph; size(7.94cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.96,xmax=8.96,ymin=-2.66,ymax=4.38;
Label laxis; laxis.p=fontsize(10);
xaxis(... | [
"To find the maximum value of \\( f^{-1}(x) \\), we need to identify \\( x \\) such that \\( f(x) = 0 \\). This occurs when \\( x \\) is within the area covered by all points of the given graph.",
"First, we calculate the area covered by the graph on the \\( x \\)-axis. The graph starts at \\( x = 3 \\) and ends ... | The maximum value of \( f^{-1}(x) \) is 10, occurring at \( x = 5 \). This value is found by calculating the area covered by the graph from \( x=0 \) to \( x=6 \) and then doubling it. | 6 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor function of 14.6. The floor function returns the greatest integer less than or equal to a given number. Since 14.6 is between 14 and 15, the floor of 14.6 is 14.",
"Next, I need to evaluate the ceiling function of -14.6. The ceiling function returns the smallest integer greate... | Since β14.6β = 14 and ββ14.6β = β14, their difference is 14 β (β14) = 28. | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor of 14.6. The floor function returns the greatest integer less than or equal to the given number. Since 14.6 is between 14 and 15, the floor of 14.6 is 14.",
"Next, I need to evaluate the ceiling of -14.6. The ceiling function returns the smallest integer greater than or equal ... | The floor of 14.6 is 14 and the ceiling of -14.6 is -14, so \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil = 14 - (-14) = 28\). The final answer is 28. | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor of 14.6. The floor function returns the greatest integer less than or equal to the given number. For 14.6, the greatest integer less than or equal to 14.6 is 14. So, floor(14.6) is 14.",
"Next, I need to evaluate the ceiling of -14.6. The ceiling function returns the smallest ... | The floor of \(14.6\) is \(14\), and the ceiling of \(-14.6\) is \(-14\). Therefore, \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil = 14 - (-14) = 28\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"To evaluate the expression \\(\\lfloor 14.6 \\rfloor - \\lceil -14.6 \\rceil\\), I start by determining the floor of 14.6. The floor function returns the greatest integer less than or equal to the given number. For 14.6, the greatest integer less than or equal to it is 14.",
"Next, I calculate the ceiling of -14... | \(\lfloor 14.6 \rfloor = 14\) and \(\lceil -14.6 \rceil = -14\), so \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil = 14 - (-14) = 28\). Final answer: \(\boxed{28}\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"I need to evaluate the expression β14.6β β ββ14.6β.",
"First, I'll find the floor of 14.6. The floor function gives the greatest integer less than or equal to a number. For 14.6, the greatest integer less than or equal to it is 14.",
"Next, I'll determine the ceiling of β14.6. The ceiling function returns the ... | The floor of 14.6 is 14 and the ceiling of -14.6 is -14. Subtracting these values gives 14 - (-14) = 28. | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"To evaluate the expression \\(\\lfloor 14.6 \\rfloor - \\lceil -14.6 \\rceil\\), I need to determine the floor of 14.6 and the ceiling of -14.6 separately.",
"First, the floor function \\(\\lfloor x \\rfloor\\) returns the greatest integer less than or equal to \\(x\\). For 14.6, the greatest integer less than 1... | The expression \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil\) evaluates to \(14 - (-14)\). Therefore, the final result is \(28\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor function of 14.6. The floor function, denoted by \\( \\lfloor x \\rfloor \\), returns the greatest integer less than or equal to \\( x \\). Since 14.6 is between 14 and 15, the floor of 14.6 is 14.",
"Next, I need to evaluate the ceiling function of -14.6. The ceiling function... | The expression \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil\) evaluates to \(14 - (-14)\) where \(\lfloor 14.6 \rfloor = 14\) and \(\lceil -14.6 \rceil = -14\). Thus, the final answer is \(14 + 14 = 28\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor of 14.6. The floor function returns the greatest integer less than or equal to the given number. Since 14.6 is between 14 and 15, the floor of 14.6 is 14.",
"Next, I need to evaluate the ceiling of -14.6. The ceiling function returns the smallest integer greater than or equal ... | The floor of \(14.6\) is \(14\), and the ceiling of \(-14.6\) is \(-14\). Therefore, \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil = 14 - (-14) = 28\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor of 14.6. The floor function returns the greatest integer less than or equal to the given number. For 14.6, the greatest integer less than or equal to it is 14.",
"Next, I need to evaluate the ceiling of -14.6. The ceiling function returns the smallest integer greater than or e... | The floor of \(14.6\) is \(14\) and the ceiling of \(-14.6\) is \(-14\). Subtracting the ceiling from the floor gives \(14 - (-14) = 28\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor function of 14.6. The floor function returns the greatest integer less than or equal to the given number. Since 14.6 is between 14 and 15, the floor of 14.6 is 14.",
"Next, I need to evaluate the ceiling function of -14.6. The ceiling function returns the smallest integer grea... | The floor of 14.6 is 14 and the ceiling of -14.6 is -14. Subtracting these values: 14 - (-14) = 28. | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor of 14.6, which is the greatest integer less than or equal to 14.6. Since 14 is an integer, the floor of 14.6 is 14.",
"Next, I need to evaluate the ceiling of -14.6, which is the smallest integer greater than or equal to -14.6. Since -14.6 is a negative number, the ceiling of ... | The floor of 14.6 is 14, and the ceiling of -14.6 is -14. Therefore, \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil = 14 - (-14) = 28\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor function of 14.6. The floor function returns the greatest integer less than or equal to a given number. For 14.6, the greatest integer less than or equal to 14.6 is 14.",
"Next, I need to evaluate the ceiling function of -14.6. The ceiling function returns the smallest integer... | \(\lfloor 14.6 \rfloor = 14\) and \(\lceil -14.6 \rceil = -14\). Therefore, \(\lfloor 14.6 \rfloor - \lceil -14.6 \rceil = 14 - (-14) = 28\). | 28 | true |
zwhe99/DeepMath-103K | Evaluate $\lfloor 14.6 \rfloor-\lceil-14.6\rceil$. | [
"First, I need to evaluate the floor function of 14.6. The floor function returns the greatest integer less than or equal to the given number. For 14.6, the greatest integer less than or equal to it is 14. So, \\(\\lfloor 14.6 \\rfloor = 14\\).",
"Next, I need to evaluate the ceiling function of -14.6. The ceilin... | \(\lfloor 14.6 \rfloor = 14\) and \(\lceil -14.6 \rceil = -14\). Thus, \(14.6 - (-14.6) = 14 + 14 = 28\). | 28 | true |
zwhe99/DeepMath-103K | What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 2) and the midpoint of the segment with endpoints at (5, 0) and (6, 2)? Express your answer in simplest form. | [
"First, I need to find the midpoints of the two given line segments. For the segment with endpoints at (0, 0) and (2, 2), the midpoint can be calculated by averaging the x-coordinates and the y-coordinates separately. So, the x-coordinate of the midpoint is (0 + 2) divided by 2, which equals 1, and the y-coordinate... | The midpoints of the segments are \((1, 1)\) and \((5.6667, 1)\). The slope of the line connecting these midpoints is \(m = \frac{1 - 1}{1.6667 - 5} = 0\). | 0 | true |
zwhe99/DeepMath-103K | What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 2) and the midpoint of the segment with endpoints at (5, 0) and (6, 2)? Express your answer in simplest form. | [
"To find the slope of the line segment connecting the two midpoints, I'll start by calculating the midpoint of the segment with endpoints (0, 0) and (2, 2). The midpoint formula is (((0 + 2) divided by 2, (0 + 2) divided by 2)). This gives me the midpoint (1, 1).",
"Next, I'll calculate the midpoint of the segmen... | The midpoints are (1, 1) and (5, 1), so the slope of the line joining them is (1 β 1)/(5 β 1) = 0. The slope is **0**. | 0 | true |
zwhe99/DeepMath-103K | What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 2) and the midpoint of the segment with endpoints at (5, 0) and (6, 2)? Express your answer in simplest form. | [
"Okay, so I have this problem here: I need to find the slope of the line that connects the midpoints of two given segments. The endpoints of the first segment are (0, 0) and (2, 2), and the endpoints of the second segment are (5, 0) and (6, 2). Hmm, how do I approach this?",
"First, I remember that the midpoint o... | The slope of the line containing the midpoints of the segment with endpoints at \((0, 0)\) and \((2, 2)\) and the segment with endpoints at \((5, 0)\) and \((6, 2)\) is 0. | 0 | true |
zwhe99/DeepMath-103K | What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 2) and the midpoint of the segment with endpoints at (5, 0) and (6, 2)? Express your answer in simplest form. | [
"First, I need to find the midpoints of the two given line segments. For the segment with endpoints at (0, 0) and (2, 2), the midpoint is calculated by averaging the x-coordinates and the y-coordinates of the endpoints. This gives me the midpoint (1, 1).",
"Next, I'll calculate the midpoint of the segment with en... | The slope of the line containing the midpoints \((1, 1)\) and \((5.5, 1)\) is calculated using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) as \(m = \frac{1 - 1}{5.5 - 1} = \frac{0}{4.5} = 0\). The final answer is 0. | 0 | true |
zwhe99/DeepMath-103K | Let $a$ and $b$ be real numbers. The function $h(x)=ax+b$ satisfies $h(1)=5$ and $h(-1)=1$. What is $h(6)$? | [
"First, I recognize that the function \\( h(x) = ax + b \\) is a linear function. This means that for any real numbers \\( x \\) and \\( a \\), the value of \\( h(x) \\) is solely determined by the coefficients \\( a \\) and \\( b \\).",
"Given that \\( h(1) = 5 \\), I can substitute \\( x = 1 \\) into the functi... | The function is \( h(x) = 2x + 3 \). Therefore, \( h(6) = 2(6) + 3 = 15 \). | 15 | true |
zwhe99/DeepMath-103K | Let $a$ and $b$ be real numbers. The function $h(x)=ax+b$ satisfies $h(1)=5$ and $h(-1)=1$. What is $h(6)$? | [
"First, I need to determine the values of \\( a \\) and \\( b \\) in the function \\( h(x) = ax + b \\).",
"Given that \\( h(1) = 5 \\), I can substitute \\( x = 1 \\) into the equation:",
"\\[",
"a(1) + b = 5",
"\\]",
"This simplifies to:",
"\\[",
"a + b = 5",
"\\]",
"Next, substituting \\( x = -... | Given \( h(x) = ax + b \), we found \( a=2 \) and \( b=3 \), so \( h(x) = 2x + 3 \). Therefore, \( h(6) = 2(6) + 3 = 15 \). | 15 | true |
zwhe99/DeepMath-103K | How many different triangles can be formed using three vertices of a hexagon as vertices of a triangle? [asy]size(75);
draw(dir(0)--dir(30)--dir(110)--dir(175)--dir(250)--dir(300)--cycle);
[/asy] | [
"First, I need to determine the total number of triangles that can be formed by selecting three vertices out of a hexagon. I can use the combination formula to calculate this.",
"The combination formula for selecting 3 vertices out of 6 (since a hexagon has 6 vertices) is given by:",
"\\[",
"\\binom{6}{3} = \... | The number of different triangles that can be formed by selecting three vertices of a hexagon is given by the combination formula $\binom{6}{3}$. This calculation results in 15 distinct triangles. | 20 | true |
zwhe99/DeepMath-103K | How many different triangles can be formed using three vertices of a hexagon as vertices of a triangle? [asy]size(75);
draw(dir(0)--dir(30)--dir(110)--dir(175)--dir(250)--dir(300)--cycle);
[/asy] | [
"To find the number of different triangles that can be formed using three vertices of a hexagon, I start by determining the total number of possible triangles that can be formed by choosing any three vertices from the six vertices of a hexagon. This is calculated using the combination formula C(6, 3), which equals ... | The total number of triangles is $\binom{6}{3} = 20$. Subtracting the 4 triangles formed by duplicate vertices on the y-axis leaves 16 unique triangles. | 20 | true |
zwhe99/DeepMath-103K | How many different triangles can be formed using three vertices of a hexagon as vertices of a triangle? [asy]size(75);
draw(dir(0)--dir(30)--dir(110)--dir(175)--dir(250)--dir(300)--cycle);
[/asy] | [
"To determine how many different triangles can be formed using three vertices of a hexagon, I start by identifying the total number of vertices in a hexagon, which is 6.",
"Next, I consider all possible combinations of three non-collinear vertices that can form a triangle. Since no three vertices in a hexagon are... | A triangle is formed by choosing any 3 of a set of 6 vertices in a hexagon; there are C(6,3) = 20 such distinct triangles. | 20 | true |
zwhe99/DeepMath-103K | How many different triangles can be formed using three vertices of a hexagon as vertices of a triangle? [asy]size(75);
draw(dir(0)--dir(30)--dir(110)--dir(175)--dir(250)--dir(300)--cycle);
[/asy] | [
"First, I recognize that the problem involves selecting three vertices out of a hexagon (Hexagon) to form a triangle. The order in which the vertices are chosen does not matter because a triangle can be either ABC or ABC, however, the triangles need to be distinct.",
"To solve this, I need to determine how many d... | Choosing three vertices from a hexagon's six sides gives C(6,3) = 20 distinct triangles. The answer is **20**. | 20 | true |
zwhe99/DeepMath-103K | Let $a$ and $b$ be the solutions of the equation $2x^2-10x+5=0$. What is the value of $(2a-3)(4b-6)$? | [
"First, I need to find the value of the expression \\((2a - 3)(4b - 6)\\), where \\(a\\) and \\(b\\) are the solutions to the quadratic equation \\(2x^2 - 10x + 5 = 0\\).",
"I recall that for a quadratic equation \\(ax^2 + bx + c = 0\\), the middle term is given by \\(-\\frac{b}{2a}\\). Substituting the known val... | Since \(a\) and \(b\) are the solutions to \(2x^2 - 10x + 5 = 0\), the middle term is \(2(\frac{-10}{2\cdot 2}) = 2.5\). Evaluating \((2a - 3)(4b - 6)\) with \(a=2\) and \(b=5\) gives \((2\cdot2) - 3(5) - (2\cdot5) - 6 = 4 - 15 - 10 - 6 = -22\). | -22 | true |
zwhe99/DeepMath-103K | If $f(x) = x^2 - 1$ and $g(x) = x + 7,$ evaluate $f(g(7)) + g(f(3)).$ | [
"First, I need to evaluate \\( f(g(7)) \\) and \\( g(f(3)) \\) using the given functions \\( f(x) = x^2 - 1 \\) and \\( g(x) = x + 7 \\).",
"1. **Evaluate \\( g(7) \\):**",
"\\[",
"g(7) = 7 + 7 = 14",
"\\]",
"2. **Evaluate \\( f(3) \\):**",
"\\[",
"f(3) = 3^2 - 1 = 9 - 1 = 8",
"\\]",
"3. **Substit... | First, we find $f(g(7)) = f(14) = 195$ and $g(f(3)) = g(8) = 15$. Adding these results yields $195 + 15 = 210$. | 210 | true |
zwhe99/DeepMath-103K | If $f(x) = x^2 - 1$ and $g(x) = x + 7,$ evaluate $f(g(7)) + g(f(3)).$ | [
"First, I need to evaluate \\( g(7) \\) using the function \\( g(x) = x + 7 \\). This gives me \\( g(7) = 7 + 7 = 14 \\).",
"Next, I substitute \\( g(7) = 14 \\) into the function \\( f(x) = x^2 - 1 \\) to find \\( f(g(7)) = f(14) \\). Calculating \\( f(14) = 14^2 - 1 = 196 - 1 = 195 \\).",
"Then, I evaluate \\... | We find \( f(g(7)) = f(14) = 195 \) and \( g(f(3)) = g(8) = 15 \). The sum is \( 195 + 15 = 210 \). | 210 | true |
zwhe99/DeepMath-103K | If $f(x) = x^2 - 1$ and $g(x) = x + 7,$ evaluate $f(g(7)) + g(f(3)).$ | [
"First, I need to evaluate \\( g(7) \\). Given that \\( g(x) = x + 7 \\), substituting 7 into the function gives \\( g(7) = 7 + 7 = 14 \\).",
"Next, I evaluate \\( f(3) \\) using the function \\( f(x) = x^2 - 1 \\). Substituting 3 into the function gives \\( f(3) = 3^2 - 1 = 9 - 1 = 8 \\).",
"Now, I calculate \... | We compute $f(g(7)) = f(14) = 195$ and $g(f(3)) = g(8) = 15$. Adding these results, $195 + 15 = 210$. | 210 | true |
zwhe99/DeepMath-103K | If $f(x) = x^2 - 1$ and $g(x) = x + 7,$ evaluate $f(g(7)) + g(f(3)).$ | [
"First, I need to evaluate \\( g(7) \\) using the function \\( g(x) = x + 7 \\).",
"\\[",
"g(7) = 7 + 7 = 14",
"\\]",
"Next, I use the result from \\( g(7) \\) to find \\( f(g(7)) \\) by substituting \\( g(7) \\) into the function \\( f(x) = x^2 - 1 \\).",
"\\[",
"f(14) = 14^2 - 1 = 196 - 1 = 195",
"\... | We compute $f(g(7)) = f(14) = 195$ and $g(f(3)) = g(8) = 15$. Adding these results gives $195 + 15 = 210$. | 210 | true |
zwhe99/DeepMath-103K | If $f(x) = x^2 - 1$ and $g(x) = x + 7,$ evaluate $f(g(7)) + g(f(3)).$ | [
"First, I need to evaluate \\( g(7) \\) using the function \\( g(x) = x + 7 \\).",
"\\[",
"g(7) = 7 + 7 = 14",
"\\]",
"Next, I'll evaluate \\( f(3) \\) using the function \\( f(x) = x^2 - 1 \\).",
"\\[",
"f(3) = 3^2 - 1 = 9 - 1 = 8",
"\\]",
"Now, I'll find \\( f(g(7)) \\) and \\( g(f(3)) \\) by subs... | We evaluate \( f(g(7)) = f(14) = 14^2 - 1 = 195 \) and \( g(f(3)) = g(8) = 8 + 7 = 15 \). Adding these gives \( 195 + 15 = 210 \). | 210 | true |
zwhe99/DeepMath-103K | If $f(x) = x^2 - 1$ and $g(x) = x + 7,$ evaluate $f(g(7)) + g(f(3)).$ | [
"First, I need to evaluate \\( g(7) \\) using the function \\( g(x) = x + 7 \\).",
"Calculating \\( g(7) \\):",
"\\[",
"g(7) = 7 + 7 = 14",
"\\]",
"Next, I evaluate \\( f(3) \\) using the function \\( f(x) = x^2 - 1 \\).",
"Calculating \\( f(3) \\):",
"\\[",
"f(3) = 3^2 - 1 = 9 - 1 = 8",
"\\]",
... | With f(x) = xΒ² β 1 and g(x) = x + 7, we get f(g(7)) = 195 and g(f(3)) = 15, so 195 + 15 = **210**. | 210 | true |
zwhe99/DeepMath-103K | In how many patterns can six identical L-shaped pieces, each consisting of three unit squares, be placed to perfectly cover a fixed 3 by 6 board? One such pattern is shown.
[asy]
draw((0,0)--(6,0),linewidth(2));
draw((0,3)--(6,3),linewidth(2));
draw((0,0)--(0,3),linewidth(2));
draw((2,0)--(2,3),linewidth(2));
draw((4... | [
"First, I need to determine how many L-shaped pieces can fit on a 3x6 grid. Each piece has 3 squares in one row and 3 in the next, totaling 6 squares per row. With 6 rows in the grid, this gives 6Β³ = 216 possible arrangements of the initial L-shape.",
"Next, I consider the additional vertical and horizontal piece... | The total number of L-shaped pieces on the 3x6 grid is \(6^3 = 216\). The number of ways these pieces can be placed on the grid is calculated by adding the placements for the interior vertical and horizontal pieces, along with the additional horizontal and vertical pieces, resulting in \(2 + 2 + 4 = 8\). | 8 | true |
zwhe99/DeepMath-103K | A triangle has vertices at coordinates $(11,1)$, $(2,3)$ and $(3,7)$. What is the number of units in the length of the longest side of the triangle? | [
"First, I need to determine the lengths of all three sides of the triangle by calculating the distances between each pair of vertices.",
"Starting with the distance between (11, 1) and (2, 3), I'll use the distance formula:",
"\\[",
"\\sqrt{(2 - 1)^2 + (3 - 1)^2} = \\sqrt{1 + 4} = \\sqrt{5} \\text{ units}",
... | Using the coordinates, the triangle's side lengths are β5, β11, and 10 units (via distance formulas). The longest side is 10 units. | 10 | true |
zwhe99/DeepMath-103K | Each square of the three by three grid is painted so that the whole picture has at least the two lines of symmetry indicated. Each grid square is painted one solid color. What is the maximum number of colors that could have been used? [asy]size(100);
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle);
draw((1,0)--(1,3));
draw((2,... | [
"To determine the maximum number of colors required for a 3x3 grid where each square is painted one solid color, we first note that the grid has 12 squares. Each square will have at least two lines of symmetry. However, we wish to maximize the number of colors used, so we aim to maximize both the number of horizont... | The maximum number of colors required is 8. This is achieved by introducing an eighth color that can be distributed as two colors per square, thus covering all 12 squares of the 3x3 grid. | 3 | true |
zwhe99/DeepMath-103K | Find the largest positive real number $\lambda$ such that for any nonnegative real numbers $x,$ $y,$ and $z$ such that $x^2 + y^2 + z^2 = 1,$ the inequality
\[\lambda xy + yz \le \frac{\sqrt{5}}{2}\]holds. | [
"Okay, so I need to find the largest positive real number Ξ» such that for any nonnegative real numbers x, y, and z, the inequality Ξ»xy + yz β€ (β5)/2 holds. Hmm, that involves some complex numbers, but I think I can handle this.",
"First, let me write down the inequality without more complex notation:",
"Ξ»[xΒ² + ... | The inequality Ξ»xy + yz β€ (β5)/2 holds when Ξ»xΒ² + Ξ»yΒ² + Ξ»zΒ² β€ 1, which occurs only when xΒ² + yΒ² + zΒ²=2, forcing Ξ»=2. The largest positive real number Ξ» is 2. | 2 | true |
zwhe99/DeepMath-103K | Find the largest positive real number $\lambda$ such that for any nonnegative real numbers $x,$ $y,$ and $z$ such that $x^2 + y^2 + z^2 = 1,$ the inequality
\[\lambda xy + yz \le \frac{\sqrt{5}}{2}\]holds. | [
"To find the largest positive real number \\(\\lambda\\) such that the inequality \\(\\lambda xy + yz \\leq \\frac{\\sqrt{5}}{2}\\) holds for all \\(x, y, z\\), we need to ensure that the equality holds exactly in the dot product with \\(y\\).",
"First, consider the relation \\(x = y + t\\), where \\(t \\in [0, 1... | The largest positive real number \(\lambda\) such that for any nonnegative real numbers \(x, y, z\), the inequality \(\lambda xy + yz \leq \frac{\sqrt{5}}{2}\) holds is found by setting \(x = y^3 + \lambda y^2 + 3\lambda y + 2t\), taking \(t = 0\) to ensure \(x = y\), and solving for \(\lambda\). The problem simplifies... | 2 | true |
zwhe99/DeepMath-103K | If $x$ is a real number, find $x^2+2x(5-x)+(5-x)^2$. | [
"To find the value of \\( x^2 + 2x(5 - x) + (5 - x)^2 \\) for a real number \\( x \\), I can start by expanding the expression.",
"First, I'll expand each term individually:",
"- \\( 2x(5 - x) \\) becomes \\( 10x - 2x^2 \\).",
"- \\( (5 - x)^2 \\) expands to \\( 25 - 10x + x^2 \\).",
"Combining these, the e... | The expression simplifies to \( x^2 + 10x + 25 \). This is equivalent to \( 25 \). | 25 | true |
zwhe99/DeepMath-103K | If $x$ is a real number, find $x^2+2x(5-x)+(5-x)^2$. | [
"To solve for \\( x^2 + 2x(5 - x) + (5 - x)^2 \\) where \\( x \\) is a real number, I'll start by expanding each term individually.",
"First, expand \\( 2x(5 - x) \\):",
"\\[",
"2x(5 - x) = 10x - 2x^2",
"\\]",
"Next, expand \\( (5 - x)^2 \\):",
"\\[",
"(5 - x)^2 = 25 - 10x + x^2",
"\\]",
"Now, com... | The expression simplifies to \( x^2 + 10x + x^2 - 2x^2 + 10x - 25 = 25 \) by expanding and combining like terms. Thus, \( x^2 + 2x(5 - x) + (5 - x)^2 = 25 \). | 25 | true |
zwhe99/DeepMath-103K | Two different numbers are randomly selected from the set $\{1, 2, 3, 4\}$ and they are multiplied. What is the probability that the product is even? Express your answer as a common fraction. | [
"First, I need to determine the total number of ways to choose two different numbers from the set {1, 2, 3, 4}. Since the order doesn't matter in a selection, I'll use the combination formula \\( \\binom{n}{k} = \\frac{n!}{k!(n-k)!} \\). Here, \\( n = 4 \\) and \\( k = 2 \\), so the total number of possible pairs i... | The total number of ways to select two different numbers from the set \(\{1, 2, 3, 4\}\) is \(\binom{4}{2} = 6\). There are 5 pairs whose product is even (2,2,4; 1,4,1,3; and (4,3)), so the probability is \(\frac{5}{6}\). | \dfrac{5}{6} | true |
zwhe99/DeepMath-103K | Find the units digit of $1! + 2! + 3! + 4! + \ldots + 50!$. | [
"To find the units digit of \\(1! + 2! + 3! + 4! + \\ldots + 50!\\), I'll start by calculating the units digits of each factorial from \\(1!\\) to \\(50!\\).",
"First, I'll determine the units digit of \\(1! = 1\\).",
"Next, the units digit of \\(2! = 2\\) because any number multiplied by 2 will always end with... | The units digits of \(1!\), \(2!\), \(3!\), and \(4!\) are \(1\), \(2\), \(6\), and \(4\) respectively. For \(50!\), the units digit is \(0\), and the sum of the units digits of \(1!\) through \(50!\) is \(1 + 2 + 6 + 4 + 0 = 13\), so the units digit is \(3\). | 3 | true |
zwhe99/DeepMath-103K | Find the units digit of $1! + 2! + 3! + 4! + \ldots + 50!$. | [
"To find the units digit of \\(1! + 2! + 3! + 4! + \\ldots + 50!\\), I start by identifying the units digits of the factorials involved.",
"Calculating the factorials:",
"- \\(1! = 1\\) (units digit is 1)",
"- \\(2! = 2\\) (units digit is 2)",
"- \\(3! = 6\\) (units digit is 6)",
"- \\(4! = 24\\) (units d... | Each factorial from 1! to 4! ends in 0, so the sum is 1 + 2 + 6 + 4 + 0 = 13, giving 13 mod 10 = 3. Thus, the units digit of \(1! + 2! + 3! + 4! + \ldots + 50!\) is \(\boxed{3}\). | 3 | true |
zwhe99/DeepMath-103K | Find the units digit of $1! + 2! + 3! + 4! + \ldots + 50!$. | [
"First, I need to determine the units digit of each factorial from \\(1!\\) to \\(50!\\).",
"I know that for factorials of even numbers, the units digit is 0 because \\(5!\\) includes both 2 and 5, resulting in a trailing zero.",
"On the other hand, for factorials of odd numbers, the units digit is always 1. Th... | The units digits of \(n!\) for \(n \geq 5\) are 0 for odd numbers and 1 for even numbers. The sum of the units digits of \(1!\) to \(50!\) is \(1 \times 0 + 2 \times 1 + 3 \times 1 + 4 \times 0 + 5 \times 0 = 3\). | 3 | true |
zwhe99/DeepMath-103K | Find the units digit of $1! + 2! + 3! + 4! + \ldots + 50!$. | [
"To find the units digit of \\(1! + 2! + 3! + 4! + \\ldots + 50!\\), I start by calculating the units digits of the factorials:",
"- \\(1! = 1\\) (units digit is 1)",
"- \\(2! = 2\\) (units digit is 2)",
"- \\(3! = 6\\) (units digit is 6)",
"- \\(4! = 24\\) (units digit is 4)",
"- \\(5! = 120\\) (units di... | The units digits of \(1!, 2!, 3!\) and \(4!\) are 1, 2, 6, and 4 respectively. When calculating \(50!\), considering cycles of 4!, there are 12 complete and 1 additional cycle of \(5^2 = 25\), resulting in a units digit of 13, which rounds down to 3. | 3 | true |
zwhe99/DeepMath-103K | Find the units digit of $1! + 2! + 3! + 4! + \ldots + 50!$. | [
"To find the units digit of the sum \\(1! + 2! + 3! + 4! + \\ldots + 50!\\), I start by examining the units digit of each factorial from \\(1!\\) to \\(50!\\).",
"First, I calculate the units digits of each factorial:",
"- \\(1! = 1\\), so the units digit is 1.",
"- \\(2! = 2\\), so the units digit is 2.",
... | The units digits of \(1!, 2!, 3!, 4!\) and \(5!\) are 1, 2, 6, 4, and 0 respectively. The sum of these units digits is \(1+2+6+4+0=13\), and the units digit of \(13\) is 3. | 3 | true |
zwhe99/DeepMath-103K | Find the units digit of $1! + 2! + 3! + 4! + \ldots + 50!$. | [
"To find the units digit of the sum \\(1! + 2! + 3! + 4! + \\ldots + 50!\\), I start by calculating the factorials of the first few numbers and their units digits.",
"Calculating \\(1!\\):",
"\\[",
"1! = 1 \\quad \\text{units digit} = 1",
"\\]",
"Calculating \\(2!\\):",
"\\[",
"2! = 2 \\quad \\text{un... | The units digits of \(1!\), \(2!\), \(3!\), \(4!\), and \(50!\) are \(1\), \(2\), \(6\), \(4\), and \(0\) respectively, while the units digits of \(5^n\) for \(n \geq 5\) are \(0\). Summing the digits of the common term \(1! + 2! + 3! + 4!\) gives \(1 + 2 + 6 + 4 = 13\), so the units digit is \(\boxed{3}\). | 3 | true |
zwhe99/DeepMath-103K | A European train compartment has six seats. Four of the seats are broken. Wilhelm needs to fill out a form like the one here to indicate that there are broken seats. If he randomly checks off four of the seats in the diagram, what is the probability that he marked the correct seats? Express your answer as a common ... | [
"First, I need to determine the total number of possible ways Wilhelm can randomly check off four seats from a compartment with six seats. This is a combinatorial problem where I choose 4 out of 6 available seats. The number of ways to do this is given by the combination formula:",
"\\[",
"\\binom{6}{4} = 15",
... | There are $\binom{6}{4} = 15$ total ways to choose 4 seats out of 6, and only 1 correct configuration (all seats marked). Therefore, the probability Wilhelm marks the correct seats correctly is $\frac{1}{15}$. | \frac{1}{15} | true |
zwhe99/DeepMath-103K | A European train compartment has six seats. Four of the seats are broken. Wilhelm needs to fill out a form like the one here to indicate that there are broken seats. If he randomly checks off four of the seats in the diagram, what is the probability that he marked the correct seats? Express your answer as a common ... | [
"First, I need to determine the total number of possible ways Wilhelm can choose 4 out of the 6 available seats, which is calculated using the combination formula \\( \\binom{6}{4} \\).",
"Next, I identify the number of successful outcomes, which are the cases where Wilhelm marks exactly 4 of the broken seats. Th... | The total number of ways to choose 4 out of 6 seats is \( \binom{6}{4} = 15 \), and the number of ways to choose 4 correct seats is \( \binom{4}{4} = 1 \). Therefore, the probability of selecting the correct seats is \( \frac{1}{15} \). | \frac{1}{15} | true |
zwhe99/DeepMath-103K | Round each number to the nearest tenth and add the results: 543.55, 25.23, and 299.98. | [
"Okay, so I need to round each of these numbers to the nearest tenth and then add them together. Let me take it step by step.",
"First, I'll go through each number one by one. Starting with 543.55. Hmm, to round to the nearest tenth, I remember that if the hundredths place is 5 or greater, we round up the tenths ... | The rounded values are 543.60, 25.20, and 300.10, which sum to 868.80. | 868.8 | true |
zwhe99/DeepMath-103K | Joe's batting average is .323. (That is, he averages 0.323 hits per at bat.) What is the probability that he will get three hits in three at-bats? Express your answer as a decimal to the nearest hundredth. | [
"First, I recognize that Joe has a batting average of 0.323 over three at-bats, which means the probability of him hitting a single ball in any given at-bat is 0.326.",
"To find the probability of him getting three hits in three consecutive at-bats, I consider the probability of not hitting a ball in the first tw... | The probability of not hitting a ball in the first two at-bats and then hitting on the third is \( (0.326)^2 = 0.0031536 \). The probability of hitting a ball in the third at-bat is \( 0.326 - 0.0031536 = 0.3228464 \), leading to a probability of \( 0.0002932 \) or \( 0.02932 \) for three hits in three at-bats, which r... | 0.03 | true |
zwhe99/DeepMath-103K | Patty ordered a pizza to be delivered. The bill was $\$$12.00. Patty pays $\$$15 and tells the delivery person to keep the change as a tip. What percent tip did she give? | [
"First, I need to determine the total amount Patty paid for the pizza, which is \\$15.00.",
"Next, I'll subtract the amount Patty paid from the total bill to find the change she should give to the delivery person:",
"\\$12.00 (total bill) - \\$15.00 (amount paid) = -\\$3.00 (change).",
"Since Patty gave a neg... | Patty gave a tip of $3.00, which is a 25% tip on the $12.00 pizza bill. This is calculated as $\left( \frac{\$3.00}{\$12.00} \right) \times 100$. | 25 | true |
zwhe99/DeepMath-103K | Patty ordered a pizza to be delivered. The bill was $\$$12.00. Patty pays $\$$15 and tells the delivery person to keep the change as a tip. What percent tip did she give? | [
"First, I need to determine the total amount Patty paid, which is \\$15.00.",
"Next, I'll subtract the cost of the pizza from the total amount paid to find the change given as the tip:",
"\\$15.00 - \\$12.00 = \\$3.00.",
"To find the tip percentage, I'll divide the change given as the tip by the original cost... | Patty gave a \$3.00 tip, which represents 25% of the \$12.00 pizza cost, calculated as $(\frac{\$3.00}{\$12.00}) \times 100\%$. The final answer is 25%. | 25 | true |
zwhe99/DeepMath-103K | Patty ordered a pizza to be delivered. The bill was $\$$12.00. Patty pays $\$$15 and tells the delivery person to keep the change as a tip. What percent tip did she give? | [
"First, I need to determine the total amount Patty paid, which is \\$15.",
"Next, I subtract the \\$12.00 pizza bill from the total payment to find the tip amount:",
"\\$15.00 - \\$12.00 = \\$3.00 tip.",
"To find the tip percentage, I divide the tip amount by the total bill and then multiply by 100:",
"(\\$... | The tip amount is \$3.00, calculated by subtracting the \$12.00 pizza bill from the total payment of \$15.00. This tip percentage is 25%. | 25 | true |
zwhe99/DeepMath-103K | Patty ordered a pizza to be delivered. The bill was $\$$12.00. Patty pays $\$$15 and tells the delivery person to keep the change as a tip. What percent tip did she give? | [
"First, identify the total amount paid, which is \\$15.",
"Next, determine the total cost of the pizza, which is \\$12.00.",
"Subtract the amount paid from the total cost to find the change to be returned:",
"\\$12.00 - \\$15.00 = -\\$3.00.",
"Since the change is negative, it indicates that the amount paid ... | Patty gave a **25%** tip, calculated by subtracting her \$15.00 payment from the \$12.00 pizza cost and then subtracting the resulting negative \$3.00 change. The percentage tip is \$3.00 / \$12.00 * 100 = 25%. | 25 | true |
zwhe99/DeepMath-103K | Patty ordered a pizza to be delivered. The bill was $\$$12.00. Patty pays $\$$15 and tells the delivery person to keep the change as a tip. What percent tip did she give? | [
"First, I need to determine the amount of change Patty received after paying \\$15.00 for the pizza.",
"Next, I'll subtract the \\$15.00 payment from the total bill of \\$12.00 to find out how much change is left for the tip.",
"Once I have the amount of the tip, I'll calculate the percentage tip by dividing th... | The change from \$15.00 is \$7.00, resulting in a tip of \$3.00. The percentage tip is calculated as \$3.00 / \$12.00 * 100%, which equals 25%. | 25 | true |
zwhe99/DeepMath-103K | How many different three-step paths along the edges of a cube are there that take you from vertex $A$ to vertex $B$? (A step is from a vertex to an adjacent vertex sharing an edge.)
[asy]
size(101);
import three;
currentprojection=orthographic(1/2,-1,1/4);
draw((0,0,0)--(1,0,0)--(1,0,1)--(0,0,1)--(0,0,0));
draw((1,0,0... | [
"First, I need to determine the total number of ways to move from vertex \\( A \\) to vertex \\( B \\) on a cube. A path consists of moving from \\( A \\) to \\( B \\) along the three edges connecting the vertices.",
"Since there are three edges between each pair of consecutive vertices on the cube, each vertex h... | The problem considers 9 paths each from vertex \( A \) to vertex \( B \) along 3 edges of a cube, as there are 3 options for each edge and each pair of vertices is unique for each path. With 4 vertices, this results in \( 3^4 = 81 \) distinct paths each representing a unique set of configurations, with 6 distinct cycle... | 6 | true |
zwhe99/DeepMath-103K | Given that
\begin{align*}
\cos x + \cos y + \cos z &= 0, \\
\sin x + \sin y + \sin z &= 0,
\end{align*}find
\begin{align*}
&\tan^2 x + \tan^2 y + \tan^2 z - (\tan^2 x \tan^2 y + \tan^2 x \tan^2 z + \tan^2 y \tan^2 z) \\
&\quad - 3 \tan^2 x \tan^2 y \tan^2 z.
\end{align*} | [
"Okay, so I have this problem where I'm given two equations involving sine and cosine functions, and I need to find the value of this expression: (-tanΒ²x + tanΒ²y + tanΒ²z) minus the product of (tanΒ²x tanΒ²y + tanΒ²x tanΒ²z + tanΒ²y tanΒ²z). Then, 3 of those multiplied by another term I keep forgetting. Hmm, okay, let's t... | From x + y = 0, x = 0, which implies x = -y, so the third condition gives b = 0, leading to the expression $-3b = -3$. The final answer is $\boxed{-3}$. | -3 | true |
zwhe99/DeepMath-103K | At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have some hydrogen in a 3.67 liter container with a pressure of 4 kPa. If I move all of it to a 1.835 liter container at the same temperature, what will the new pressure be in kPa? | [
"First, I recognize that the pressure and volume of a gas are inversely proportional at a constant temperature. This means that as the volume increases, the pressure decreases, and vice versa.",
"I know the initial volume and pressure of the gas: 3.67 liters at 4 kPa. To find the new volume when the gas is moved ... | Using constant-temperature, inane-rate inverse-proportional relationships, the 1.835-liter container with 4 kPa inside gives Pβ = 2 Γ 4 kPa = 8 kPa. Final answer: **8 kPa**. | 8 | true |
zwhe99/DeepMath-103K | At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have some hydrogen in a 3.67 liter container with a pressure of 4 kPa. If I move all of it to a 1.835 liter container at the same temperature, what will the new pressure be in kPa? | [
"First, I recognize that the pressure and volume of a gas are inversely proportional at constant temperature. This means that if the volume increases, the pressure decreases, and vice versa.",
"I know the initial volume of the gas is 3.67 liters and the initial pressure is 4 kPa. Using the formula \\( p \\times v... | Using P = k/V = 4 Γ 3.67 = 14.76 kPaΒ²/L, the new pressure in the larger container is 14.76 / 1.835 β **8 kPa**. | 8 | true |
zwhe99/DeepMath-103K | At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have some hydrogen in a 3.67 liter container with a pressure of 4 kPa. If I move all of it to a 1.835 liter container at the same temperature, what will the new pressure be in kPa? | [
"First, I recognize that the volume of the gas, which is directly proportional to the pressure, is inversely proportional when the temperature is constant. This means that as the volume increases, the pressure decreases, and vice versa.",
"Given the initial pressure of 4 kPa and initial volume of 3.67 liters, I c... | Since pressure and volume are inversely proportional at constant temperature, doubling the volume halves the pressure, so the new pressure is \(p' = 2 \times 4 \, \text{kPa} = 8 \, \text{kPa}\). Final answer: \(\boxed{8 \, \text{kPa}}\). | 8 | true |
zwhe99/DeepMath-103K | At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have some hydrogen in a 3.67 liter container with a pressure of 4 kPa. If I move all of it to a 1.835 liter container at the same temperature, what will the new pressure be in kPa? | [
"First, I recognize that the pressure of a substance is inversely proportional to its volume when the temperature is constant. This means that if the volume increases, the pressure decreases proportionally.",
"Given the initial conditions:",
"- Initial Volume (\\(V_1\\)) = 3.67 liters",
"- Initial Pressure (\... | Using \(p_1Vol_1 = p_2Vol_2\), the new pressure is \(p_2 = \frac{4 \times 3.67}{1.835} \approx 20\ \text{kPa}\). | 8 | true |
zwhe99/DeepMath-103K | A line segment has one endpoint at $(3,9)$ and midpoint at $(1,2)$. What is the sum of the coordinates of the other endpoint? | [
"Okay, so I have this problem here about a line segment. The endpoint is given as (3,9), and the midpoint is (1,2). I need to find the sum of the coordinates of the other endpoint. Hmm, let me think about how to approach this.",
"First, I remember that the midpoint of a line segment is the average of the x-coordi... | The other endpoint of the line segment is calculated to be (-1, -5). The sum of its coordinates is -6. | -6 | true |
zwhe99/DeepMath-103K | A line segment has one endpoint at $(3,9)$ and midpoint at $(1,2)$. What is the sum of the coordinates of the other endpoint? | [
"Okay, so I have this problem here: There's a line segment with one endpoint at (3,9) and the midpoint is at (1,2). I need to find the sum of the coordinates of the other endpoint. Hmm, let's see. I remember that the midpoint formula is the average of the x-coordinates and the average of the y-coordinates of the tw... | The other endpoint of the line segment is \((-1, -5)\), and the sum of its coordinates is \(-6\). | -6 | true |
zwhe99/DeepMath-103K | Let $z$ be a complex number such that
\[z + \frac{1}{z} = 1.\]Find $z^3.$ | [
"To find \\( z^3 \\) given that \\( z + \\frac{1}{z} = 1 \\), I can start by squaring both sides of the equation to introduce a known term.",
"Starting with the given equation:",
"\\[ z + \\frac{1}{z} = 1 \\]",
"Squaring both sides:",
"\\[ \\left( z + \\frac{1}{z} \\right)^2 = 1^2 \\]",
"Expanding the lef... | Given \( z + \frac{1}{z} = 1 \), squaring both sides yields \( z^2 + \frac{1}{z^2} = -1 \). Substituting \( x = 1 \) and \( y = -1 \) into \( x^2 + y^2 = 1 \), we find \( x^2 = -1 \), so \( z^2 = 0 \), leading to \( z = i \), and therefore \( z^3 = (-1)^3 = -1 \). | -1 | true |
zwhe99/DeepMath-103K | Let $z$ be a complex number such that
\[z + \frac{1}{z} = 1.\]Find $z^3.$ | [
"First, I recognize that the equation \\( z + \\frac{1}{z} = 1 \\) can be rewritten by separating the real and imaginary parts. This separation gives me \\( z = 1 - \\frac{1}{z} \\).",
"Next, I need to find \\( z^3 \\). To do this, I can substitute the expression for \\( z \\) into the cubic differentiation. This... | Given \( z + \frac{1}{z} = 1 \), we can express \( z \) as \( 1 - \frac{1}{z} \). Consequently, \( z^3 \) is equal to \( (1 - \frac{1}{z})^3 \), which simplifies to \( -\frac{1}{2} - \frac{1}{2} \). | -1 | true |
zwhe99/DeepMath-103K | Let $z$ be a complex number such that
\[z + \frac{1}{z} = 1.\]Find $z^3.$ | [
"First, I recognize that the equation \\( z + \\frac{1}{z} = 1 \\) involves a rational expression. To simplify it, I can multiply both sides by \\( z \\) to eliminate the denominator, which gives me \\( z^2 + 1 = 0 \\).",
"Next, I recall that this equation can be factored as \\( (z + 1)(z - 1) = 0 \\). Therefore,... | The equation simplifies to \( z^2 + 1 = 0 \), yielding solutions \( z = 1 \) and \( z = -1 \). Calculating \( z^3 \) for these values gives \( (-1)^3 = -1 \). | -1 | true |
zwhe99/DeepMath-103K | Let $f(x) = 2x+1$ and $g(x) = f(2x) - 3$. What is $g(g(2))$? | [
"First, I need to find the value of \\( g(2) \\) using the function \\( g(x) = f(2x) - 3 \\).",
"Given \\( f(x) = 2x + 1 \\), I substitute \\( 2x \\) with \\( 2 \\) in \\( g(x) \\):",
"\\[",
"g(2) = f(2 \\times 2) - 3 = f(4) - 3 = (2 \\times 4) + 1 - 3 = 8 + 1 - 3 = 6",
"\\]",
"Now that I have \\( g(2) = ... | First, $g(2) = f(2 \times 2) - 3 = 6$. Then, $g(g(2)) = g(6) = f(2 \times 6) - 3 = 22$. | 22 | true |
zwhe99/DeepMath-103K | What is $0.8\overline{4}-0.\overline{4}$? Express your answer as a common fraction. | [
"First, I recognize that both \\(0.8\\overline{4}\\) and \\(0.\\overline{4}\\) represent repeating decimals where the sequence \"4\" repeats indefinitely.",
"To subtract these decimals, I align their decimal parts and remove the repeating sequence from the second decimal:",
"\\[",
"0.8\\overline{4} - 0.\\over... | Subtracting the repeating decimals, $0.8\overline{4}$ - $0.\overline{4}$, results in $0.8 - 0.4$. The calculation yields $0.4$. | \frac{2}{5} | true |
zwhe99/DeepMath-103K | Simplify $(2-2i)(5+5i)$, where $i^2 = -1.$ | [
"First, I need to simplify the expression \\((2 - 2i)(5 + 5i)\\). To do this, I'll use the distributive property, also known as the FOIL method for multiplying two binomials.",
"I'll start by multiplying the First terms: \\(2 \\times 5 = 10\\).",
"Next, I'll multiply the Outer terms: \\(2 \\times 5i = 10i\\).",... | The expression \((2 - 2i)(5 + 5i)\) simplifies to \(10 + 10i - 10i - 10i^2\). Substituting \(i^2 = -1\) and factoring out \(10i\) results in \(10 + 10(-1) = 20\). | 20 | true |
zwhe99/DeepMath-103K | Simplify $(2-2i)(5+5i)$, where $i^2 = -1.$ | [
"I need to simplify the expression \\((2 - 2i)(5 + 5i)\\) where \\(i^2 = -1\\).",
"First, I'll apply the distributive property, also known as the FOIL method, to multiply the two complex numbers:",
"- Multiply the First terms: \\(2 \\times 5 = 10\\)",
"- Multiply the Outer terms: \\(2 \\times 5i = 10i\\)",
... | The expression \((2 - 2i)(5 + 5i)\) simplifies to \(10 + 10i - 10i - 10i^2\). Since \(i^2 = -1\), this becomes \(10 + 10i - 10i + 10\), resulting in \(\boxed{20 + 20i}\). | 20 | true |
zwhe99/DeepMath-103K | Simplify $(2-2i)(5+5i)$, where $i^2 = -1.$ | [
"I need to simplify the expression \\((2 - 2i)(5 + 5i)\\).",
"First, I'll use the distributive property, also known as the FOIL method for multiplying two binomials.",
"Multiplying the First terms: \\(2 \\times 5 = 10\\).",
"Next, multiplying the Outer terms: \\(2 \\times 5i = 10i\\).",
"Then, multiplying t... | Expanding \((2-2i)(5+5i)\) by the distributive property yields \((2-2i)(5+5i) = 10 + 10i - 10i - 10i^2\). Substituting \(i^2 = -1\) and combining like terms results in \(10 + 10i - 10i - 10(-1) = 10 + 10i + 10 = 20\). | 20 | true |
zwhe99/DeepMath-103K | Simplify $(2-2i)(5+5i)$, where $i^2 = -1.$ | [
"First, I need to multiply the two complex numbers \\((2 - 2i)\\) and \\((5 + 5i)\\).",
"I'll start by applying the distributive property, which means I'll multiply each term in the first complex number by each term in the second complex number.",
"This gives me \\(2 \\times 5 = 10\\) and \\(-2i \\times 5 = -10... | \((2-2i)\left(5+5i\right) = 10 - 10i - 10i - 10i^2 = 0 + 20i^2\). Therefore, the simplified expression is \(\boxed{0 - 20i}\). | 20 | true |
zwhe99/DeepMath-103K | Simplify $(2-2i)(5+5i)$, where $i^2 = -1.$ | [
"First, I need to multiply the two complex numbers \\((2 - 2i)\\) and \\((5 + 5i)\\). To do this, I'll use the distributive property, also known as the FOIL method for multiplying two binomials.",
"I'll start by multiplying the First terms: \\(2 \\times 5 = 10\\).",
"Next, I'll multiply the Outer terms: \\(2 \\... | The product of the complex numbers \((2 - 2i)(5 + 5i)\) is found by applying the distributive property and simplifying with \(i^2 = -1\). Combining like terms yields \((2 - 2i)(5 + 5i) = 2(5) + 2(5i) + (-2i)(5) + (-2i)(5i) = 10 + 10i - 10i - 10i^2 = 10 + 10i - 10i + 10 = 20\). | 20 | true |
zwhe99/DeepMath-103K | Simplify $(2-2i)(5+5i)$, where $i^2 = -1.$ | [
"First, I need to simplify the expression \\((2 - 2i)(5 + 5i)\\) by using the distributive property.",
"I'll start by distributing each term in the first complex number to each term in the second complex number.",
"Multiplying \\(2\\) by \\(5\\) gives \\(10\\).",
"Next, multiplying \\(2\\) by \\(5i\\) gives \... | Expanding \((2 - 2i)(5 + 5i)\) using the distributive property yields \(10 + 10i - 10i + 10\). Combining like terms results in the simplified expression \(20 + 10\), which equals \(20\). | 20 | true |
zwhe99/DeepMath-103K | Simplify $(2-2i)(5+5i)$, where $i^2 = -1.$ | [
"First, I need to simplify the expression \\((2 - 2i)(5 + 5i)\\) by expanding it using the distributive property.",
"I'll start by distributing each term in the first binomial to each term in the second binomial.",
"Multiplying \\(2\\) by \\(5\\) gives \\(10\\).",
"Next, multiplying \\(2\\) by \\(5i\\) result... | Expanding \((2 - 2i)(5 + 5i)\) gives \(10 + 10i - 10i + 10\), which simplifies to \(20\). The real part is \(10\) and the imaginary part is \(0i\), so the expression simplifies to \(20\). | 20 | true |
zwhe99/DeepMath-103K | Two positive integers each have a units digit of 2 when expressed in base 4. Find the units digit of their product when expressed in base 4. | [
"First, I recognize that in base 10, both 102 and 210 are equal to 210 in base 4. This means that the units digit of the product will be the same as the units digit of the original number.",
"Next, I consider the possible units digits of the numbers 102 and 210. The units digits 2 and 0 occupy the same cells in b... | Since \(102_{10} \cdot 210_{10} = 210_{10}\) in base 4, both units digits (0) must satisfy the condition of the other two numbers. Thus, the product's units digit is \(\boxed{0}\). | 0 | true |
zwhe99/DeepMath-103K | Two positive integers each have a units digit of 2 when expressed in base 4. Find the units digit of their product when expressed in base 4. | [
"First, I need to determine the least significant digit (lsd) of the numbers in base 4. Each number in base 4 can be expressed as \\(10k + 2\\), where \\(k\\) is the units digit. Therefore, the lsd in base 4 for each number is 2.",
"Next, I'll consider the product of these two base 4 numbers. The product will hav... | Since a base 4 number is equivalent to $10k+2$, its units digit is 2. The product of two numbers with a units digit of 2 in base 4 will have a units digit of 0 in base 4, as determined by the product's unit digit $2 \times 2 = 0$. | 0 | true |
zwhe99/DeepMath-103K | Two positive integers each have a units digit of 2 when expressed in base 4. Find the units digit of their product when expressed in base 4. | [
"First, I recognize that in any base \\(b\\), the units digit of a number is the last digit, which is always \\(b \\pmod{4}\\) (0, 1, 2, or 3). Therefore, for a number in base 4 to end with the digit 2, the units digit in that base must be 2 modulo 4. This means that if a number in base 4 ends with a 2, its corresp... | Since a units digit of 2 in base 4 must be equivalent to 4 modulo 4, the units digit of their product in base 4 must also be equivalent to 4 modulo 4. Multiplying the units digits 2 Γ 3 = 6, the units digit of their product in base 4 is 0. | 0 | true |
zwhe99/DeepMath-103K | Two positive integers each have a units digit of 2 when expressed in base 4. Find the units digit of their product when expressed in base 4. | [
"First, I need to understand the problem. We are given two positive integers, and when expressed in base 4, their units digits are 2. I need to find the units digit of their product when expressed in base 4.",
"To solve this, I'll start by expressing each given number in base 4. The first number, 2 in base 4, can... | Since \(20_2 \times 20_2 = 400_2 = 1024_{10}\), whose units digit is \(4_3\) in base 10, the units digit of the product in base 4 is \(\boxed{0}\). | 0 | true |
zwhe99/DeepMath-103K | Two positive integers each have a units digit of 2 when expressed in base 4. Find the units digit of their product when expressed in base 4. | [
"First, I recognize that in any base \\( b \\), the units digit of a number can be expressed as \\( b^0 \\). Since our problem involves both positive and negative units digits of 2, I need to consider both possibilities.",
"1. **Units Digit is 2 in Base 10:**",
"- Convert the units digit from base 10 to base 4.... | Since \(2^0+2^1\) and \(-2\) convert to \(2^1\) or \(4\) in base 4, multiplying them gives \(4 \times 2 = 8\), whose units digit is \(4 \mod 4 = 0\). Thus the units digit of their product in base 4 is \(\boxed{0}\). | 0 | true |
zwhe99/DeepMath-103K | If $f(x)=x^3$, what is the value of $f^{-1}(8)\div (f(8))^{-1}$? | [
"First, identify the function \\( f(x) = x^3 \\).",
"To find \\( f^{-1}(8) \\), determine the value of \\( x \\) such that \\( f(x) = 8 \\). This gives \\( x = 2 \\), so \\( f^{-1}(8) = 2 \\).",
"Now, evaluate \\( f(8) \\). Substitute 8 into the function: \\( f(8) = 8^3 = 512 \\).",
"Next, find \\( (f(8))^{-1... | The function is \( f(x) = x^3 \), so \( f^{-1}(8) = 2 \) and \( (f(8))^{-1} = \frac{1}{512} \). Therefore, \( f^{-1}(8) \div (f(8))^{-1} = 2 \div \frac{1}{512} = 1024 \). | 1024 | true |
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