kreasof-ai/SPL-400K-math-shepherd · Datasets at Hugging Face

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[QUESTION] Mr. Maxwell takes 1 hr to drive to work in the morning but 1 and half hours to drive back home in the evening. If the morning and evening traffic is similar and he drives at an average speed of 30mph in the morning, what is the average speed of his return journey? [STEP] Step 1: Mr. Maxwell takes 1 and half hour to drive back home in the evening so he takes 1+1/2 = <<1+1/2=90>>90 minutes to drive home. [STEP] Step 2: Mr. Maxwell takes 90 minutes to drive home and drives at an average speed of 30mph so he drives a distance of 90/30 = <<90/30=3>>3 miles. [STEP] Step 3: Mr. Maxwell drives 3 miles in the evening at an average speed of 30mph so his average speed for the return journey is 30+3 = <<30+3=33>>33mph. 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[QUESTION] If two distinct numbers are randomly chosen from the set {1, 2, 3, 4, 5}, what is the probability that the smaller number is a divisor of the larger number? Express your answer as a common fraction. [STEP] Step 1: There are a total of $\binom{5}{2}=10$ pairs of numbers that can be chosen. [STEP] Step 2: We can list them all out: (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5). [STEP] Step 3: Out of these pairs, the pairs where the smaller number is a divisor of the larger number are: (1, 2), (1, 3), (1, 4), (1, 5), (2, 4). [STEP] Step 4: So, there are 5 pairs that satisfy the condition. [STEP] Step 5: Therefore, the probability is $\frac{5}{10}=\boxed{\frac{1}{2}}$. 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[QUESTION] Pauline has a garden with vegetables. In it, Pauline has planted 3 kinds of tomatoes - 5 of each kind, 5 kinds of cucumbers - 4 of each kind, and 30 potatoes. In the whole garden, there are 10 rows with 15 spaces in each to plant any vegetable. How many more vegetables could Pauline plant in her garden? [STEP] Step 1: In the garden, Pauline has planted 3 kinds of tomatoes * 5 tomatoes/kind = <<35=15>>15 tomatoes. [STEP] Step 2: Pauline has planted 5 kinds of cucumbers 4 cucumbers/kind = <<54=20>>20 cucumbers. [STEP] Step 3: In total, Pauline has planted 15 tomatoes + 20 cucumbers + 30 potatoes = <<15+20+30=65>>65 vegetables. [STEP] Step 4: There are 10 rows 15 spaces/row = <<10*15=150>>150 spaces to plant any vegetable. [STEP] Step 5: So, Pauline could plant 150 - 65 = <<150-65=85>>85 more vegetables in the garden. 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[QUESTION] If $x = 101$ and $x^3y - 2x^2y + xy = 101000$, what is the value of $y$? [STEP] Step 1: I notice that the equation has only one variable term, $y$, on the left and two terms on the right. [STEP] Step 2: I wonder if I can simplify the equation by expanding the left side or using some algebraic identities. [STEP] Step 3: I recall that $x^3y - 2x^2y + xy = x^2(y - 1) - xy$, so I can rewrite the equation as $x^2(y - 1) - xy = 101000$. [STEP] Step 4: I see that the left side has two factors of $x$, so I can factor them out as $x(x - 1)(y - 1) - xy = 101000$. [STEP] Step 5: I see that the left side is a difference of two squares, so I can factor it further as $(x - 1)(x(y - 1) - xy) = 101000$. [STEP] Step 6: I notice that the left side has a common factor of $x - 1$, so I can divide both sides by it and get $(x - 1)(x(y - 1) - xy) / (x - 1) = 101000 / (x - 1)$. [STEP] Step 7: I see that the left side is still a difference of two squares, so I can factor it further as $(x(y - 1) - xy)(x - 1) / (x - 1) = 101000 / (x - 1)$. [STEP] Step 8: I simplify the left side and get $(xy - x^2)(x - 1) / (x - 1) = 101000 / (x - 1)$. [STEP] Step 9: I cancel out the $(x - 1)$ in the numerator and denominator and get $xy - x^2 = 101000$. [STEP] Step 10: I solve for $xy$ and get $xy = 101000 + x^2$. [STEP] Step 11: I substitute $x = 101$ and get $y = (101000 + 101^2) / (101 + 1) = (101000 + 101) / 102 = 100999$. 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[QUESTION] Given that $x - y = 6$ and $x + y = 12$, what is the value of $x$? [STEP] Step 1: Adding the two equations, we get $2x = 18$, so $x = \boxed{9}$. 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[QUESTION] Find the projection of the vector $\begin{pmatrix} 4 \\ 5 \end{pmatrix}$ onto the vector $\begin{pmatrix} 2 \\ 0 \end{pmatrix}.$ [STEP] Step 1: To find the projection of a vector onto another vector, I need to use the formula $\text{proj}_\mathbf{u} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$, where $\mathbf{u}$ is the vector I'm projecting onto and $\mathbf{v}$ is the vector I'm projecting. [STEP] Step 2: In this case, $\mathbf{u} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$. [STEP] Step 3: So, I need to calculate $\text{proj}_\mathbf{u} \mathbf{v} = \frac{\begin{pmatrix} 4 \\ 5 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 0 \end{pmatrix}}{\begin{pmatrix} 2 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 0 \end{pmatrix}} \begin{pmatrix} 2 \\ 0 \end{pmatrix}$. [STEP] Step 4: To calculate the dot products, I use the rule that $\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \begin{pmatrix} \mathbf{b}_1 \\ \mathbf{b}_2 \end{pmatrix} = \mathbf{a}_1 \mathbf{b}_1 + \mathbf{a}_2 \mathbf{b}_2$. [STEP] Step 5: So, I get $\text{proj}_\mathbf{u} \mathbf{v} = \frac{4 \cdot 2 + 5 \cdot 0}{2 \cdot 2 + 0 \cdot 0} \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \frac{8}{4} \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$. 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[QUESTION] John decides to buy utensils. They come in 30 packs with an equal number of knives, forks, and spoons. How many packs does he need to buy if he wants 50 spoons? [STEP] Step 1: He needs 50/30=<<50/30=1.7>>1.7 packs [STEP] Step 2: Rounding 1.7 up to the next highest whole number gives 2 packs The answer is: 2	[ 50281, 60, 26310, 2449, 62, 2516, 21936, 281, 4489, 2780, 561, 3683, 15, 50276, 3726, 1705, 275, 1884, 30171, 342, 271, 4503, 1180, 273, 36014, 13, 323, 661, 13, 285, 15695, 790, 15, 50276, 2347, 1142, 30171, 1057, 344, 878, 281, 4489, 604, 344, 5605, 2456, 15695, 790, 32, 544, 1267, 13801, 62, 18952, 337, 27, 754, 3198, 2456, 16, 1229, 30, 14193, 1235, 16, 1229, 30, 18, 15, 24, 5064, 18, 15, 24, 30171, 544, 1267, 13801, 62, 18952, 374, 27, 12778, 272, 337, 15, 24, 598, 281, 253, 1735, 4585, 2644, 1180, 4245, 374, 30171, 380, 3662, 310, 27, 374, 50282, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 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[QUESTION] Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. [STEP] Step 1: To find the points where the parabolas intersect, I need to set their $y$-coordinates equal to each other and solve for $x$. [STEP] Step 2: So I have $2x^2-7x+1=8x^2+5x+1$. [STEP] Step 3: I can simplify this by combining like terms: $3x^2-6x=0$. [STEP] Step 4: To solve this quadratic equation, I can use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=3, b=-6, c=0$. [STEP] Step 5: Plugging in these values, I get $x=\frac{6\pm\sqrt{36+0}}{6}$. [STEP] Step 6: Simplifying the square root, I get $x=\frac{6\pm\sqrt{36}}{6}$. [STEP] Step 7: The square root of 36 is 6, so I get $x=\frac{6+6}{6}=\frac{12}{6}=2$. [STEP] Step 8: This means that the parabolas intersect when $x=2$. [STEP] Step 9: So the points where they intersect are $(0,1)$, $(2,2)$, and $(4,4)$. [STEP] Step 10: These are the points in order of increasing $x$-coordinate, separated by semicolons. # Answer 0,1;2,2;4,4	[ 50281, 60, 26310, 2449, 62, 5761, 1061, 357, 16328, 403, 253, 14580, 273, 253, 7424, 370, 90, 30, 19, 89, 63, 19, 14, 24, 89, 12, 18, 5, 285, 370, 90, 30, 25, 89, 63, 19, 12, 22, 89, 12, 18, 1352, 7918, 512, 2792, 835, 597, 23965, 15, 50276, 2765, 253, 2792, 275, 1340, 273, 3629, 370, 89, 2911, 36236, 13, 9070, 407, 3300, 20917, 790, 15, 544, 1267, 13801, 62, 18952, 337, 27, 1916, 1089, 253, 2792, 835, 253, 1061, 357, 16328, 23965, 13, 309, 878, 281, 873, 616, 370, 90, 2911, 29309, 8475, 4503, 281, 1016, 643, 285, 8415, 323, 370, 89, 1352, 544, 1267, 13801, 62, 18952, 374, 27, 1893, 309, 452, 370, 19, 89, 63, 19, 14, 24, 89, 12, 18, 30, 25, 89, 63, 19, 12, 22, 89, 12, 18, 1352, 544, 1267, 13801, 62, 18952, 495, 27, 309, 476, 25636, 436, 407, 16248, 751, 2426, 27, 370, 20, 89, 63, 19, 14, 23, 89, 30, 17, 1352, 544, 1267, 13801, 62, 18952, 577, 27, 1916, 8415, 436, 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[QUESTION] Damien collects glasses for a hobby and decides he wants to put them in display cupboards. His tall cupboard can hold 20 glasses, his wide cupboard can hold twice as many, and his narrow cupboard can hold 15 glasses with the glasses divided equally among the three shelves. As Damien is putting glasses in the narrow cupboard, he accidentally breaks one of the shelves. If all of the cupboards are full, how many glasses are currently being displayed? [STEP] Step 1: The tall cupboard can hold 20 glasses, and the wide and narrow cupboards together can hold 2 x 20 = <<2*20=40>>40 glasses. [STEP] Step 2: If all of the cupboards are full, the tall cupboard is holding 20 glasses, the wide cupboard is holding 40 / 2 = <<40/2=20>>20 glasses, and the narrow cupboard is holding 15 / 2 = <<15/2=7.5>>7.5 glasses. [STEP] Step 3: The narrow cupboard can now only hold 7.5 - 1 = <<7.5-1=6.5>>6.5 glasses. [STEP] Step 4: If all of the cupboards are full, the narrow cupboard was holding 7.5 glasses, and the 6.5 are now being displayed. [STEP] Step 5: If the tall and wide cupboards are both full, the total number of glasses displayed is 20 + 20 = <<20+20=40>>40. [STEP] Step 6: The narrow cupboard holds 6.5 glasses, and the total number of glasses displayed is 40 + 6.5 = <<40+6.5=46.5>>46.5. [STEP] Step 7: There are 46.5 glasses currently being displayed. 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"[QUESTION] Let $x$ and $y$ be positive real numbers such that $x + y = 10.$ Find the minimum value(...TRUNCATED)	[50281,60,26310,2449,62,1281,370,89,5,285,370,90,5,320,2762,1524,3904,824,326,370,89,559,340,426,884(...TRUNCATED)	[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED)	[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1(...TRUNCATED)

[QUESTION] Mr. Maxwell takes 1 hr to drive to work in the morning but 1 and half hours to drive back home in the evening. If the morning and evening traffic is similar and he drives at an average speed of 30mph in the morning, what is the average speed of his return journey? [STEP] Step 1: Mr. Maxwell takes 1 and half hour to drive back home in the evening so he takes 1+1/2 = <<1+1/2=90>>90 minutes to drive home. [STEP] Step 2: Mr. Maxwell takes 90 minutes to drive home and drives at an average speed of 30mph so he drives a distance of 90/30 = <<90/30=3>>3 miles. [STEP] Step 3: Mr. Maxwell drives 3 miles in the evening at an average speed of 30mph so his average speed for the return journey is 30+3 = <<30+3=33>>33mph. The answer is: 33

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[QUESTION] If two distinct numbers are randomly chosen from the set {1, 2, 3, 4, 5}, what is the probability that the smaller number is a divisor of the larger number? Express your answer as a common fraction. [STEP] Step 1: There are a total of $\binom{5}{2}=10$ pairs of numbers that can be chosen. [STEP] Step 2: We can list them all out: (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5). [STEP] Step 3: Out of these pairs, the pairs where the smaller number is a divisor of the larger number are: (1, 2), (1, 3), (1, 4), (1, 5), (2, 4). [STEP] Step 4: So, there are 5 pairs that satisfy the condition. [STEP] Step 5: Therefore, the probability is $\frac{5}{10}=\boxed{\frac{1}{2}}$. The answer is: \frac{1}{2}

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[QUESTION] Pauline has a garden with vegetables. In it, Pauline has planted 3 kinds of tomatoes - 5 of each kind, 5 kinds of cucumbers - 4 of each kind, and 30 potatoes. In the whole garden, there are 10 rows with 15 spaces in each to plant any vegetable. How many more vegetables could Pauline plant in her garden? [STEP] Step 1: In the garden, Pauline has planted 3 kinds of tomatoes * 5 tomatoes/kind = <<3*5=15>>15 tomatoes. [STEP] Step 2: Pauline has planted 5 kinds of cucumbers * 4 cucumbers/kind = <<5*4=20>>20 cucumbers. [STEP] Step 3: In total, Pauline has planted 15 tomatoes + 20 cucumbers + 30 potatoes = <<15+20+30=65>>65 vegetables. [STEP] Step 4: There are 10 rows * 15 spaces/row = <<10*15=150>>150 spaces to plant any vegetable. [STEP] Step 5: So, Pauline could plant 150 - 65 = <<150-65=85>>85 more vegetables in the garden. The answer is: 85

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[QUESTION] If $x = 101$ and $x^3y - 2x^2y + xy = 101000$, what is the value of $y$? [STEP] Step 1: I notice that the equation has only one variable term, $y$, on the left and two terms on the right. [STEP] Step 2: I wonder if I can simplify the equation by expanding the left side or using some algebraic identities. [STEP] Step 3: I recall that $x^3y - 2x^2y + xy = x^2(y - 1) - xy$, so I can rewrite the equation as $x^2(y - 1) - xy = 101000$. [STEP] Step 4: I see that the left side has two factors of $x$, so I can factor them out as $x(x - 1)(y - 1) - xy = 101000$. [STEP] Step 5: I see that the left side is a difference of two squares, so I can factor it further as $(x - 1)(x(y - 1) - xy) = 101000$. [STEP] Step 6: I notice that the left side has a common factor of $x - 1$, so I can divide both sides by it and get $(x - 1)(x(y - 1) - xy) / (x - 1) = 101000 / (x - 1)$. [STEP] Step 7: I see that the left side is still a difference of two squares, so I can factor it further as $(x(y - 1) - xy)(x - 1) / (x - 1) = 101000 / (x - 1)$. [STEP] Step 8: I simplify the left side and get $(xy - x^2)(x - 1) / (x - 1) = 101000 / (x - 1)$. [STEP] Step 9: I cancel out the $(x - 1)$ in the numerator and denominator and get $xy - x^2 = 101000$. [STEP] Step 10: I solve for $xy$ and get $xy = 101000 + x^2$. [STEP] Step 11: I substitute $x = 101$ and get $y = (101000 + 101^2) / (101 + 1) = (101000 + 101) / 102 = 100999$. The answer is: 100999

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[QUESTION] Given that $x - y = 6$ and $x + y = 12$, what is the value of $x$? [STEP] Step 1: Adding the two equations, we get $2x = 18$, so $x = \boxed{9}$. The answer is: 9

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[QUESTION] Find the projection of the vector $\begin{pmatrix} 4 \\ 5 \end{pmatrix}$ onto the vector $\begin{pmatrix} 2 \\ 0 \end{pmatrix}.$ [STEP] Step 1: To find the projection of a vector onto another vector, I need to use the formula $\text{proj}_\mathbf{u} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$, where $\mathbf{u}$ is the vector I'm projecting onto and $\mathbf{v}$ is the vector I'm projecting. [STEP] Step 2: In this case, $\mathbf{u} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$. [STEP] Step 3: So, I need to calculate $\text{proj}_\mathbf{u} \mathbf{v} = \frac{\begin{pmatrix} 4 \\ 5 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 0 \end{pmatrix}}{\begin{pmatrix} 2 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 0 \end{pmatrix}} \begin{pmatrix} 2 \\ 0 \end{pmatrix}$. [STEP] Step 4: To calculate the dot products, I use the rule that $\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \begin{pmatrix} \mathbf{b}_1 \\ \mathbf{b}_2 \end{pmatrix} = \mathbf{a}_1 \mathbf{b}_1 + \mathbf{a}_2 \mathbf{b}_2$. [STEP] Step 5: So, I get $\text{proj}_\mathbf{u} \mathbf{v} = \frac{4 \cdot 2 + 5 \cdot 0}{2 \cdot 2 + 0 \cdot 0} \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \frac{8}{4} \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$. The answer is: $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$

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[QUESTION] John decides to buy utensils. They come in 30 packs with an equal number of knives, forks, and spoons. How many packs does he need to buy if he wants 50 spoons? [STEP] Step 1: He needs 50/30=<<50/30=1.7>>1.7 packs [STEP] Step 2: Rounding 1.7 up to the next highest whole number gives 2 packs The answer is: 2

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[QUESTION] Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. [STEP] Step 1: To find the points where the parabolas intersect, I need to set their $y$-coordinates equal to each other and solve for $x$. [STEP] Step 2: So I have $2x^2-7x+1=8x^2+5x+1$. [STEP] Step 3: I can simplify this by combining like terms: $3x^2-6x=0$. [STEP] Step 4: To solve this quadratic equation, I can use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=3, b=-6, c=0$. [STEP] Step 5: Plugging in these values, I get $x=\frac{6\pm\sqrt{36+0}}{6}$. [STEP] Step 6: Simplifying the square root, I get $x=\frac{6\pm\sqrt{36}}{6}$. [STEP] Step 7: The square root of 36 is 6, so I get $x=\frac{6+6}{6}=\frac{12}{6}=2$. [STEP] Step 8: This means that the parabolas intersect when $x=2$. [STEP] Step 9: So the points where they intersect are $(0,1)$, $(2,2)$, and $(4,4)$. [STEP] Step 10: These are the points in order of increasing $x$-coordinate, separated by semicolons. # Answer 0,1;2,2;4,4

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[QUESTION] Damien collects glasses for a hobby and decides he wants to put them in display cupboards. His tall cupboard can hold 20 glasses, his wide cupboard can hold twice as many, and his narrow cupboard can hold 15 glasses with the glasses divided equally among the three shelves. As Damien is putting glasses in the narrow cupboard, he accidentally breaks one of the shelves. If all of the cupboards are full, how many glasses are currently being displayed? [STEP] Step 1: The tall cupboard can hold 20 glasses, and the wide and narrow cupboards together can hold 2 x 20 = <<2*20=40>>40 glasses. [STEP] Step 2: If all of the cupboards are full, the tall cupboard is holding 20 glasses, the wide cupboard is holding 40 / 2 = <<40/2=20>>20 glasses, and the narrow cupboard is holding 15 / 2 = <<15/2=7.5>>7.5 glasses. [STEP] Step 3: The narrow cupboard can now only hold 7.5 - 1 = <<7.5-1=6.5>>6.5 glasses. [STEP] Step 4: If all of the cupboards are full, the narrow cupboard was holding 7.5 glasses, and the 6.5 are now being displayed. [STEP] Step 5: If the tall and wide cupboards are both full, the total number of glasses displayed is 20 + 20 = <<20+20=40>>40. [STEP] Step 6: The narrow cupboard holds 6.5 glasses, and the total number of glasses displayed is 40 + 6.5 = <<40+6.5=46.5>>46.5. [STEP] Step 7: There are 46.5 glasses currently being displayed. The answer is: 46.5

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"[QUESTION] Let $x$ and $y$ be positive real numbers such that $x + y = 10.$ Find the minimum value(...TRUNCATED)

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