sample_index int64 1.02k 250k | difficulty_bucket int64 0 4 | llama8b_solve_rate float64 0 1 | problem stringlengths 15 985 | ground_truth stringlengths 1 186 | chop_fraction float64 0.2 0.8 | partial_solution stringlengths 50 23.5k | ok bool 1
class | think stringlengths 838 2.91k | confidence float64 0 99 | raw_response stringlengths 884 2.95k | error null |
|---|---|---|---|---|---|---|---|---|---|---|---|
242,863 | 4 | 0.9375 | Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is the value of $x - y$? Express your answer as a whole number. | 10 | 0.6 | <think>
To solve the problem, we start by converting the given percentages into equations involving \(x\) and \(y\).
1. The first statement is that two is \(10\%\) of \(x\). This can be written as:
\[
2 = 0.1x
\]
2. The second statement is that two is \(20\%\) of \(y\). This can be written as:
\[
2 = 0... | true | Let me analyze this partial solution step by step.
The problem setup is correct:
- "Two is 10% of x" correctly translates to 2 = 0.1x
- "Two is 20% of y" correctly translates to 2 = 0.2y
The approach is sound: solve each equation separately for x and y, then compute x - y.
For the first equation:
- 2 = 0.1x
- x = 2/... | 0.96 | <think>
Let me analyze this partial solution step by step.
The problem setup is correct:
- "Two is 10% of x" correctly translates to 2 = 0.1x
- "Two is 20% of y" correctly translates to 2 = 0.2y
The approach is sound: solve each equation separately for x and y, then compute x - y.
For the first equation:
- 2 = 0.1x
... | null |
218,181 | 2 | 0.484375 | Find the equation of the directrix of the parabola given by the equation $y = -x^2$. Express your answer as an equation of a line (e.g., $y = c$). | y = \frac{1}{4} | 0.2 | <think>
To find the directrix of the parabola given by the equation \(y = -x^2\), we start by recognizing that the standard form of a parabola that opens upwards or downwards is | true | The partial solution correctly identifies that we need to work with the standard form of a parabola. The equation $y = -x^2$ can be rewritten as $x^2 = -y$, which is indeed a parabola opening downward (due to the negative sign).
The standard form for a vertical parabola is $(x-h)^2 = 4p(y-k)$ where the vertex is at $(... | 0.75 | <think>
The partial solution correctly identifies that we need to work with the standard form of a parabola. The equation $y = -x^2$ can be rewritten as $x^2 = -y$, which is indeed a parabola opening downward (due to the negative sign).
The standard form for a vertical parabola is $(x-h)^2 = 4p(y-k)$ where the vertex ... | null |
47,524 | 3 | 0.671875 | Given that the equation $3x^{t-1}+y-5=0$ is a linear equation in $x$ and $y$, the value of $t$ is ______. | 2 | 0.4 | <think>
To determine the value of \(t\) for which the equation \(3x^{t-1} + y - 5 = 0\) is a linear equation in \(x\) and \(y\), we need to understand the form of a linear equation in two variables. A linear equation in \(x\) and \(y\) can be written as | true | The partial solution correctly identifies that to solve this problem, one must understand what makes an equation "linear in x and y." A linear equation in two variables has the standard form $Ax + By + C = 0$ where $A$, $B$, and $C$ are constants, and crucially, both $x$ and $y$ appear to the first power only.
The par... | 0.94 | <think>
The partial solution correctly identifies that to solve this problem, one must understand what makes an equation "linear in x and y." A linear equation in two variables has the standard form $Ax + By + C = 0$ where $A$, $B$, and $C$ are constants, and crucially, both $x$ and $y$ appear to the first power only.
... | null |
200,032 | 1 | 0.25 | A retail store wants to hire 50 new phone reps to assist with the increased call volume that they will experience over the holiday. Each phone rep will work 8 hours a day and will be paid $14.00 an hour. After 5 days, how much will the company pay all 50 new employees? | 28000 | 0.6 | <think>
To solve this problem, we need to calculate the total hours worked by all phone reps in 5 days and then multiply that by the hourly wage.
1. Each phone rep works 8 hours a day.
2. Each phone rep works for 5 days.
3. The hourly wage for each phone rep is $14.00.
First, calculate the total hours worked by one ... | true | Let me analyze this partial solution step by step.
The problem asks for the total payment to 50 phone reps working 8 hours/day for 5 days at $14/hour.
The partial solution has:
1. Correctly identified the key parameters: 8 hours/day, 5 days, $14/hour
2. Correctly set up the approach: calculate total hours, then multi... | 0.94 | <think>
Let me analyze this partial solution step by step.
The problem asks for the total payment to 50 phone reps working 8 hours/day for 5 days at $14/hour.
The partial solution has:
1. Correctly identified the key parameters: 8 hours/day, 5 days, $14/hour
2. Correctly set up the approach: calculate total hours, th... | null |
195,302 | 3 | 0.75 | If eight apples cost the same as four bananas, and two bananas cost the same as three cucumbers, how many cucumbers can Tyler buy for the price of 16 apples? | 12 | 0.6 | <think>
To solve the problem, we need to establish relationships between the prices of apples, bananas, and cucumbers, and then use these relationships to determine how many cucumbers can be bought for the price of 16 apples.
First, let's define the prices:
- Let \( A \) be the price of one apple.
- Let \( B \) be the... | true | Let me analyze this partial solution step by step.
The problem setup is correct:
- Defined variables A, B, C for prices of apples, bananas, and cucumbers
- This is the standard approach for this type of problem
Step 1 analysis: "8A = 4B"
- This correctly translates "eight apples cost the same as four bananas"
- The s... | 0.94 | <think>
Let me analyze this partial solution step by step.
The problem setup is correct:
- Defined variables A, B, C for prices of apples, bananas, and cucumbers
- This is the standard approach for this type of problem
Step 1 analysis: "8A = 4B"
- This correctly translates "eight apples cost the same as four bananas"... | null |
44,296 | 2 | 0.40625 | The sum of the first $n$ terms of a geometric sequence ${a_n}$ is $S_n$, with $a_1=-\frac{1}{2}$. If $\frac{S_6}{S_3}=\frac{7}{8}$, find $a_2 \cdot a_4$. | a_2 \cdot a_4 = \frac{1}{4} \cdot \frac{1}{16} = \frac{1}{64} | 0.8 | <think>
Given the first term \(a_1 = -\frac{1}{2}\) of a geometric sequence and the ratio of the sum of the first 6 terms to the sum of the first 3 terms as \(\frac{S_6}{S_3} = \frac{7}{8}\), we need to find \(a_2 \cdot a_4\).
First, recall the formula for the sum of the first \(n\) terms of a geometric sequence:
\[
S... | true | Let me analyze this partial solution step by step.
The setup is correct:
- Formula for sum of geometric sequence: $S_n = a_1 \frac{1-r^n}{1-r}$ β
- Given: $a_1 = -\frac{1}{2}$ and $\frac{S_6}{S_3} = \frac{7}{8}$ β
The algebraic manipulation:
- Setting up $\frac{S_6}{S_3} = \frac{1-r^6}{1-r^3} = \frac{7}{8}$ β
- Facto... | 0.96 | <think>
Let me analyze this partial solution step by step.
The setup is correct:
- Formula for sum of geometric sequence: $S_n = a_1 \frac{1-r^n}{1-r}$ β
- Given: $a_1 = -\frac{1}{2}$ and $\frac{S_6}{S_3} = \frac{7}{8}$ β
The algebraic manipulation:
- Setting up $\frac{S_6}{S_3} = \frac{1-r^6}{1-r^3} = \frac{7}{8}$ β... | null |
21,715 | 0 | 0.046875 | Simplify first, then evaluate: $\frac{x}{{x}^{2}-1}(\frac{x-1}{x}-2)$, where $x=\tan 45^{\circ}+\cos 30^{\circ}$. | -\frac{2\sqrt{3}}{3} | 0.4 | <think>
First, we need to simplify the expression \(\frac{x}{{x}^{2}-1} \left( \frac{x-1}{x} - 2 \right)\). We can start by simplifying the expression inside the parentheses.
1. Simplify \(\frac{x-1}{x} - 2\):
\[
\frac{x-1}{x} - 2 = \frac{x-1}{x} - \frac{2x}{x} = \frac{x-1-2x}{x} = \frac{-x-1}{x} = -\frac{x+1}{x... | true | Let me analyze this partial solution step by step.
The problem asks to simplify $\frac{x}{{x}^{2}-1}(\frac{x-1}{x}-2)$ where $x=\tan 45^{\circ}+\cos 30^{\circ}$.
**Step 1 Analysis:**
The partial solution simplifies $\frac{x-1}{x} - 2$:
- $\frac{x-1}{x} - 2 = \frac{x-1}{x} - \frac{2x}{x} = \frac{x-1-2x}{x} = \frac{-x-... | 0.88 | <think>
Let me analyze this partial solution step by step.
The problem asks to simplify $\frac{x}{{x}^{2}-1}(\frac{x-1}{x}-2)$ where $x=\tan 45^{\circ}+\cos 30^{\circ}$.
**Step 1 Analysis:**
The partial solution simplifies $\frac{x-1}{x} - 2$:
- $\frac{x-1}{x} - 2 = \frac{x-1}{x} - \frac{2x}{x} = \frac{x-1-2x}{x} = \... | null |
238,796 | 3 | 0.625 | Find the coordinates of the vertex of the parabola given by the equation y = -x^2. Express your answer as an ordered pair (x, y). | (0, 0) | 0.2 | <think>
To find the vertex of the parabola given by the equation \( y = -x^2 \), we can use the st | true | The partial solution has barely begun - it only states the intention to find the vertex and mentions "we can use the st..." which appears to be cut off mid-word (likely "standard form" or "standard vertex formula").
For the parabola $y = -x^2$:
- This is already in the form $y = a(x-h)^2 + k$ where $a = -1$, $h = 0$, ... | 0.75 | <think>
The partial solution has barely begun - it only states the intention to find the vertex and mentions "we can use the st..." which appears to be cut off mid-word (likely "standard form" or "standard vertex formula").
For the parabola $y = -x^2$:
- This is already in the form $y = a(x-h)^2 + k$ where $a = -1$, $... | null |
1,067 | 3 | 0.75 | A certain group has 5 boys and 3 girls. Three students are selected to participate in an activity. If $X$ represents the number of girls selected, then $P\left(X\geqslant 2\right)=\_\_\_\_\_\_$. | \frac{2}{7} | 0.4 | <think>
To determine the probability \( P(X \geqslant 2) \), where \( X \) is the number of girls selected from a group of 5 boys and 3 girls, we need to follow these steps:
1. Calculate the total number of ways to select 3 students out of 8.
2. Calculate the number of favorable outcomes where at least 2 girls are sel... | true | Let me analyze this partial solution step by step.
The problem setup is correct:
- 5 boys and 3 girls (total 8 students)
- Selecting 3 students
- X = number of girls selected
- Need to find P(X β₯ 2)
The approach outlined is completely correct:
1. Find total ways to select 3 from 8: $\binom{8}{3}$
2. Find favorable ou... | 0.95 | <think>
Let me analyze this partial solution step by step.
The problem setup is correct:
- 5 boys and 3 girls (total 8 students)
- Selecting 3 students
- X = number of girls selected
- Need to find P(X β₯ 2)
The approach outlined is completely correct:
1. Find total ways to select 3 from 8: $\binom{8}{3}$
2. Find favo... | null |
71,237 | 4 | 0.96875 | A company was contracted to construct three buildings, with the second building being two times as tall as the first building. The third building had to be three times as tall as the combined height of the first and second buildings. The total height of the three buildings together was 7200 feet. How tall was the first... | 600 | 0.6 | <think>
Let's define the height of the first building as \( H_1 \).
Given that the second building is two times as tall as the first building, its height \( H_2 \) can be expressed as:
\[ H_2 = 2 \times H_1 \]
The third building is three times as tall as the combined height of the first and second buildings, so its h... | true | Let me analyze this partial solution step by step.
**Setup and Variables:**
The problem is correctly set up with $H_1$ as the height of the first building.
**Second Building:**
$H_2 = 2H_1$ - This is correct based on "the second building being two times as tall as the first building."
**Third Building:**
The problem... | 0.96 | <think>
Let me analyze this partial solution step by step.
**Setup and Variables:**
The problem is correctly set up with $H_1$ as the height of the first building.
**Second Building:**
$H_2 = 2H_1$ - This is correct based on "the second building being two times as tall as the first building."
**Third Building:**
The... | null |
78,399 | 0 | 0.015625 | A retailer bought three machines with different wholesale prices of $P, $Q, and $R, and sold them after various discounts and profit margins. The first machine was sold after applying a 10% discount on the retail price and making a 20% profit on wholesale price, the second machine was sold after applying a 15% discount... | $453 | 0.8 | <think>
To find the total retail price of all the machines, we need to determine the retail price for each machine individually and then sum them up. Let's denote the retail prices of the first, second, and third machines as \(P_r\), \(Q_r\), and \(R_r\) respectively.
1. **First Machine:**
- Wholesale price (\(P\))... | true | Let me analyze the partial solution's approach and correctness.
The problem setup is clear: For each machine, we know:
- Wholesale price
- Profit margin on wholesale price
- Discount applied to retail price to get selling price
The key relationship is:
Selling Price = Wholesale Price Γ (1 + Profit Margin) = Retail Pr... | 0.95 | <think>
Let me analyze the partial solution's approach and correctness.
The problem setup is clear: For each machine, we know:
- Wholesale price
- Profit margin on wholesale price
- Discount applied to retail price to get selling price
The key relationship is:
Selling Price = Wholesale Price Γ (1 + Profit Margin) = R... | null |
164,959 | 0 | 0.046875 | A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes? | 40 ext{ minutes} | 0.8 | <think>
Let's assume the initial amount of potatoes in each bucket is \(1\) unit. After peeling, the trainee cook collected exactly one bucket of peeled potatoes. Since \(25\%\) of the potatoes went into peels, the remaining \(75\%\) of the potatoes became the peeled potatoes.
If the initial amount of potatoes in each... | true | Let me analyze this partial solution carefully.
**Problem Setup:**
- Started with 2 buckets of unpeeled potatoes
- Peeled everything in 1 hour
- 25% became peels, so 75% became peeled potatoes
- Question: How long to collect exactly 1 bucket of peeled potatoes?
**Partial Solution Analysis:**
The solution correctly i... | 0.25 | <think>
Let me analyze this partial solution carefully.
**Problem Setup:**
- Started with 2 buckets of unpeeled potatoes
- Peeled everything in 1 hour
- 25% became peels, so 75% became peeled potatoes
- Question: How long to collect exactly 1 bucket of peeled potatoes?
**Partial Solution Analysis:**
The solution cor... | null |
214,059 | 2 | 0.578125 | Given \\(\\tan \\theta = 2\\), simplify the expression \\(\\sin^{2}\\theta + \\sin \\theta \\cos \\theta - 2\\cos^{2}\\theta\\) and provide the numerical value of the result. Express your answer in the form \\(\\boxed{[answer]}\\). | \\frac{4}{5} | 0.4 | <think>
Given \(\tan \theta = 2\), we can use trigonometric identities to simplify the expression \(\sin^2 \theta + \sin \theta \cos \theta - 2 \cos^2 \theta\).
First, recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Since \(\tan \theta = 2\), we can set \(\sin \theta = 2k\) and \(\cos \theta = k\) for s... | true | Let me analyze this partial solution step by step.
The approach taken is valid: using the constraint $\tan \theta = 2$ to express $\sin \theta$ and $\cos \theta$ in terms of a parameter $k$, then using the Pythagorean identity.
**What's correct:**
1. The setup $\sin \theta = 2k$ and $\cos \theta = k$ correctly encode... | 0.94 | <think>
Let me analyze this partial solution step by step.
The approach taken is valid: using the constraint $\tan \theta = 2$ to express $\sin \theta$ and $\cos \theta$ in terms of a parameter $k$, then using the Pythagorean identity.
**What's correct:**
1. The setup $\sin \theta = 2k$ and $\cos \theta = k$ correctl... | null |
156,790 | 0 | 0.015625 | What is the minimum width that an infinite strip of paper must have in order to cut out any triangle with an area of 1? | \sqrt[4]{3} | 0.8 | <think>
To determine the minimum width of the paper required to cut out any triangle with an area of 1, we need to consider the properties of triangles and the dimensions necessary to cut them out.
The area of a triangle is given by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
... | true | The partial solution is attempting to find the minimum width of an infinite strip that can contain any triangle of area 1.
The approach taken here has a critical flaw. The solver is trying to minimize the width by considering a specific type of triangle (right-angled with equal base and height). However, the problem a... | 0.12 | <think>
The partial solution is attempting to find the minimum width of an infinite strip that can contain any triangle of area 1.
The approach taken here has a critical flaw. The solver is trying to minimize the width by considering a specific type of triangle (right-angled with equal base and height). However, the p... | null |
93,228 | 3 | 0.625 | Debelyn had 20 dolls before she gave Andrena some dolls. Christel had 24 dolls before giving Andrena 5 dolls. After all the gifts, Andrena now has 2 more dolls than Christel, and Andrena has 3 more dolls than Debelyn. How many dolls did Debelyn give to Andrena? | 2 | 0.2 | <think>
Let's denote the number of dolls Debelyn gave to Andrena as \( D \) and the number of dolls Christel gave to Andrena as \( C \). According to the problem, \( C = 5 \).
First, we can write the equations based on the information given:
1. Debelyn originally had 20 dolls and gave \( D \) dolls to Andrena. So, Deb... | true | Let me analyze this partial solution carefully.
The problem setup:
- Debelyn: started with 20 dolls, gave D dolls to Andrena
- Christel: started with 24 dolls, gave 5 dolls to Andrena
- After gifts: Andrena has 2 more dolls than Christel
- After gifts: Andrena has 3 more dolls than Debelyn
The partial solution has:
1... | 0.72 | <think>
Let me analyze this partial solution carefully.
The problem setup:
- Debelyn: started with 20 dolls, gave D dolls to Andrena
- Christel: started with 24 dolls, gave 5 dolls to Andrena
- After gifts: Andrena has 2 more dolls than Christel
- After gifts: Andrena has 3 more dolls than Debelyn
The partial solutio... | null |
237,917 | 1 | 0.265625 | Given that $P$ is a moving point on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and the product of the slopes of the lines connecting point $P$ and the two vertices of the major axis of the ellipse is $(-\frac{1}{4})$, find the eccentricity of the ellipse. Express your answer as a numerical va... | \frac{\sqrt{3}}{2} | 0.4 | <think>
Given that $P$ is a point on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can represent $P$ as \((x, y)\). The vertices of the major axis are at \((\pm a, 0)\). The slopes of the lines connecting point \(P\) to these vertices are given by:
1. Slope of the line from \(P\) to \((a, 0)\) is:
\[
... | true | Let me analyze this partial solution step by step.
**What's been done correctly:**
1. The vertices of the major axis are correctly identified as $(\pm a, 0)$
2. The slopes $m_1 = \frac{y}{x-a}$ and $m_2 = \frac{y}{x+a}$ are correctly calculated
3. The product is correctly computed as $\frac{y^2}{x^2-a^2} = -\frac{1}{4... | 0.82 | <think>
Let me analyze this partial solution step by step.
**What's been done correctly:**
1. The vertices of the major axis are correctly identified as $(\pm a, 0)$
2. The slopes $m_1 = \frac{y}{x-a}$ and $m_2 = \frac{y}{x+a}$ are correctly calculated
3. The product is correctly computed as $\frac{y^2}{x^2-a^2} = -\f... | null |
172,306 | 4 | 0.890625 | Three numbers are in the ratio $2: 3: 4$. The sum of the squares of these numbers is 725. What are these numbers? | 10, 15, 20 \text{ or } -10, -15, -20 | 0.4 | <think>
Let's denote the three numbers as \(2x\), \(3x\), and \(4x\) based on the given ratio \(2:3:4\).
According to the problem, the sum of the squares of these numbers is 725. Therefore, we can write the equation as:
\[
(2x)^2 + (3x)^2 + (4x)^2 = 725
\]
Simplify each term:
\[
4x^2 + 9x^2 + 16x^2 = 725
\]
| true | The partial solution sets up the problem correctly by parameterizing the three numbers as $2x$, $3x$, and $4x$ where $x$ is a scaling factor. This is the standard and correct approach for ratio problems.
The equation $(2x)^2 + (3x)^2 + (4x)^2 = 725$ is set up correctly from the problem statement.
The expansion is cor... | 0.95 | <think>
The partial solution sets up the problem correctly by parameterizing the three numbers as $2x$, $3x$, and $4x$ where $x$ is a scaling factor. This is the standard and correct approach for ratio problems.
The equation $(2x)^2 + (3x)^2 + (4x)^2 = 725$ is set up correctly from the problem statement.
The expansio... | null |
247,216 | 1 | 0.25 | Find the sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$. Express your answer as a single integer. | 0 | 0.4 | <think>
To solve for the sum of the numerical coefficients in the complete expansion of \((x^2 - 2xy + y^2)^7\), we can leverage a simple trick. The sum of the numerical coefficients in the expansion of any polynomial \(P(x, y)\) can be found by evaluating \(P(1, 1)\).
Given the polynomial | true | The partial solution correctly identifies the key technique: to find the sum of all numerical coefficients in a polynomial expansion, substitute all variables equal to 1. This is a standard and reliable method because when $x = y = 1$, each term in the expansion contributes exactly its numerical coefficient to the sum.... | 0.98 | <think>
The partial solution correctly identifies the key technique: to find the sum of all numerical coefficients in a polynomial expansion, substitute all variables equal to 1. This is a standard and reliable method because when $x = y = 1$, each term in the expansion contributes exactly its numerical coefficient to ... | null |
222,936 | 3 | 0.796875 | Given the operation on $\mathbb{R}$ defined as $$\begin{vmatrix} a & c \\ b & d\end{vmatrix} =ad-bc$$, find the range of $x$ satisfying the inequality $$\begin{vmatrix} x & 3 \\ -x & x\end{vmatrix} < \begin{vmatrix} 2 & 0 \\ 1 & 2\end{vmatrix}$$. Express your answer as an interval. | (-4,1) | 0.2 | <think>
To solve the problem, we need to evaluate both sides of the inequality using the given operation.
First, let's evaluate the determinant on the left-hand side of the inequality:
$$
\begin{vmatrix} x & 3 \\ -x & x \end{vmatrix} = (x \cdot x) - (3 \cdot (-x)) = x^2 + 3x
$$
Next, let's evaluate the determinant | true | The partial solution correctly identifies the need to evaluate both determinants and begins this process. Let me verify the first calculation:
For $\begin{vmatrix} x & 3 \\ -x & x \end{vmatrix}$, using the formula $ad - bc$ where $a=x, b=-x, c=3, d=x$:
- $ad = x \cdot x = x^2$
- $bc = (-x) \cdot 3 = -3x$
- So $ad - bc... | 0.88 | <think>
The partial solution correctly identifies the need to evaluate both determinants and begins this process. Let me verify the first calculation:
For $\begin{vmatrix} x & 3 \\ -x & x \end{vmatrix}$, using the formula $ad - bc$ where $a=x, b=-x, c=3, d=x$:
- $ad = x \cdot x = x^2$
- $bc = (-x) \cdot 3 = -3x$
- So ... | null |
226,685 | 3 | 0.625 | Given a complex number $z=\frac{1+2i}{2-i}+1$, calculate the conjugate of $z$, denoted as $\overline{z}$. Express your answer as a complex number in the form $a + bi$, where $i$ is the imaginary unit. | 1 - i | 0.8 | <think>
To solve for the conjugate of the complex number \( z = \frac{1+2i}{2-i} + 1 \), we will follow these steps:
1. Simplify the fraction \(\frac{1+2i}{2-i}\).
2. Add 1 to the simplified fraction.
3. Find the conjugate of the resulting complex number.
### Step 1: Simplify \(\frac{1+2i}{2-i}\)
To simplify the frac... | true | Let me verify the partial solution step by step.
**Step 1 verification:**
The approach is correct - multiplying by the conjugate of the denominator.
Denominator calculation:
$(2-i)(2+i) = 4 - i^2 = 4 - (-1) = 5$ β
Numerator calculation:
$(1+2i)(2+i) = 1(2) + 1(i) + 2i(2) + 2i(i)$
$= 2 + i + 4i + 2i^2$
$= 2 + 5i + 2(... | 0.98 | <think>
Let me verify the partial solution step by step.
**Step 1 verification:**
The approach is correct - multiplying by the conjugate of the denominator.
Denominator calculation:
$(2-i)(2+i) = 4 - i^2 = 4 - (-1) = 5$ β
Numerator calculation:
$(1+2i)(2+i) = 1(2) + 1(i) + 2i(2) + 2i(i)$
$= 2 + i + 4i + 2i^2$
$= 2 +... | null |
124,458 | 1 | 0.390625 | one paper sheet can be used to create 8 paper boats or 6 paper planes . if total exact 80 paper toys are to be made , minimum how many sheets need to be used to make paper boats ? | 10 | 0.6 | <think>
To determine the minimum number of paper sheets needed to make exactly 80 paper toys, we need to consider the ratio of sheets required to make boats versus planes. We know that:
- 1 sheet can be used to create 8 boats.
- 1 sheet can be used to create 6 planes.
Given that we need to create a total of 80 paper ... | true | Let me analyze this partial solution carefully.
The problem asks for the MINIMUM number of sheets needed to make paper BOATS, given that exactly 80 paper toys total must be made.
The partial solution correctly sets up the equation:
- Let x = sheets for boats, y = sheets for planes
- 8x + 6y = 80 (correct)
- Simplifie... | 0.88 | <think>
Let me analyze this partial solution carefully.
The problem asks for the MINIMUM number of sheets needed to make paper BOATS, given that exactly 80 paper toys total must be made.
The partial solution correctly sets up the equation:
- Let x = sheets for boats, y = sheets for planes
- 8x + 6y = 80 (correct)
- S... | null |
144,533 | 2 | 0.453125 | Ivan has a bird feeder in his yard that holds two cups of birdseed. Every week, he has to refill the emptied feeder. Each cup of birdseed can feed fourteen birds, but Ivan is constantly chasing away a hungry squirrel that steals some birdseed from the feeder every week. Ivan's bird feeder feeds 21 birds weekly. What is... | 1:4 | 0.8 | <think>
Let's denote the amount of birdseed stolen by the squirrel as \( S \). We know that the total birdseed in the feeder is 2 cups, and each cup feeds 14 birds, so the total number of birds that the feeder would feed without any theft is \( 2 \times 14 = 28 \) birds.
Given that the feeder feeds 21 birds weekly, th... | true | Let me analyze this partial solution step by step.
**Problem Setup Understanding:**
- Feeder holds 2 cups of birdseed
- Each cup feeds 14 birds
- Without theft: 2 Γ 14 = 28 birds could be fed
- With theft: only 21 birds are actually fed
- Need to find: ratio of stolen birdseed to total birdseed
**Partial Solution Ana... | 0.94 | <think>
Let me analyze this partial solution step by step.
**Problem Setup Understanding:**
- Feeder holds 2 cups of birdseed
- Each cup feeds 14 birds
- Without theft: 2 Γ 14 = 28 birds could be fed
- With theft: only 21 birds are actually fed
- Need to find: ratio of stolen birdseed to total birdseed
**Partial Solu... | null |
14,214 | 1 | 0.28125 | Simplify: $$\frac {a-b}{a+2b}Γ· \frac {a^{2}-b^{2}}{a^{2}+4ab+4b^{2}}-2.$$ | - \frac {a}{a+b} | 0.4 | <think>
To simplify the expression \(\frac {a-b}{a+2b} Γ· \frac {a^2-b^2}{a^2+4ab+4b^2} - 2\), we should first simplify the division part of the expression. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal.
1. Rewrite the division as multiplication by the reciprocal:
\[
\frac {a-b}{a+2b... | true | Let me analyze this partial solution step by step.
The problem is to simplify: $$\frac {a-b}{a+2b}Γ· \frac {a^{2}-b^{2}}{a^{2}+4ab+4b^{2}}-2$$
**What the partial solution has done correctly:**
1. **Correctly identified the division-to-multiplication conversion**: The partial solution correctly states that dividing by... | 0.88 | <think>
Let me analyze this partial solution step by step.
The problem is to simplify: $$\frac {a-b}{a+2b}Γ· \frac {a^{2}-b^{2}}{a^{2}+4ab+4b^{2}}-2$$
**What the partial solution has done correctly:**
1. **Correctly identified the division-to-multiplication conversion**: The partial solution correctly states that div... | null |
227,736 | 3 | 0.78125 | Find the integer closest to the real number $\sqrt{17}-1$. Express your answer as a single integer. | 3 | 0.4 | <think>
To solve for the integer closest to \(\sqrt{17} - 1\), we first need to estimate \(\sqrt{17}\).
We know that:
\[ \sqrt{16} = 4 \]
\[ \sqrt{25} = 5 \]
Since \(17\) is between \(16\) and \(25\), it follows that:
\[ 4 < \sqrt{17} < 5 \]
To get a more precise estimate, we can consider that:
\[ 4.5^2 = 20.25 \]
\... | true | Let me analyze this partial solution step by step.
The solver is trying to estimate $\sqrt{17}$ to determine the integer closest to $\sqrt{17} - 1$.
**Correct steps so far:**
1. Established that $\sqrt{16} = 4$ and $\sqrt{25} = 5$, so $4 < \sqrt{17} < 5$ β
2. Computed $4.5^2 = 20.25$ β
3. Computed $4.4^2 = 19.36$ β
... | 0.72 | <think>
Let me analyze this partial solution step by step.
The solver is trying to estimate $\sqrt{17}$ to determine the integer closest to $\sqrt{17} - 1$.
**Correct steps so far:**
1. Established that $\sqrt{16} = 4$ and $\sqrt{25} = 5$, so $4 < \sqrt{17} < 5$ β
2. Computed $4.5^2 = 20.25$ β
3. Computed $4.4^2 = 19... | null |
165,585 | 1 | 0.21875 | Let $\mathrm{a}$, $\mathrm{b}$, and $\mathrm{c}$ be three integers. Find all pairs of integers $x, y$ (possibly with a negative sign) such that $a x + b y = c$. | (x_0 + kb, y_0 - ka) | 0.2 | <think>
To solve the equation \(ax + by = c\) for all pairs of integers \(x, y\) (including negative integers), we need to consider the properties of linear Diophantine equations. A linear Diophantine equation has integer solutions if and only if the greatest common divisor (gcd) of the coefficients of \(x\) and \(y\) ... | true | The partial solution correctly identifies that this is a linear Diophantine equation and states the fundamental existence criterion: $ax + by = c$ has integer solutions if and only if $\gcd(a,b) \mid c$. This is the correct first step.
However, the problem asks to "find all pairs of integers $x, y$" satisfying the equ... | 0.28 | <think>
The partial solution correctly identifies that this is a linear Diophantine equation and states the fundamental existence criterion: $ax + by = c$ has integer solutions if and only if $\gcd(a,b) \mid c$. This is the correct first step.
However, the problem asks to "find all pairs of integers $x, y$" satisfying... | null |
178,032 | 1 | 0.265625 | Given \(\overrightarrow{AB} = 6 \vec{i} + \vec{j}\), \(\overrightarrow{BC} = x \vec{i} + y \vec{j}\), \(\overrightarrow{CD} = -2 \vec{i} - 3 \vec{j}\), and \(\overrightarrow{BC} \parallel \overrightarrow{DA}\). Find the relationship between \(x\) and \(y\). | x + 2y = 0 | 0.4 | <think>
To determine the relationship between \(x\) and \(y\), we need to analyze the given vectors and the condition that \(\overrightarrow{BC}\) is parallel to \(\overrightarrow{DA}\).
First, let's denote the position vectors of points \(A\), \(B\), \(C\), and \(D\) as \(\overrightarrow{A}\), \(\overrightarrow{B}\),... | true | The partial solution has correctly set up the problem by expressing each vector in terms of position vectors. The approach is sound:
1. They've correctly written $\overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} = 6\vec{i} + \vec{j}$
2. They've correctly written $\overrightarrow{BC} = \overrightarrow{C} -... | 0.85 | <think>
The partial solution has correctly set up the problem by expressing each vector in terms of position vectors. The approach is sound:
1. They've correctly written $\overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} = 6\vec{i} + \vec{j}$
2. They've correctly written $\overrightarrow{BC} = \overrightar... | null |
208,427 | 0 | 0.03125 | Given $0οΌΞ²οΌΞ±οΌ\frac{Ο}{2}$, and $cos({Ξ±-Ξ²})=\frac{{12}}{{13}}$, $cos2Ξ²=\frac{3}{5}$, calculate the value of $\sin \left(\alpha +eta
ight)$. Express your answer as a fraction in lowest terms. | \frac{63}{65} | 0.8 | <think>
Given:
- \( 0 < \beta < \alpha < \frac{\pi}{2} \)
- \( \cos(\alpha - \beta) = \frac{12}{13} \)
- \( \cos 2\beta = \frac{3}{5} \)
We need to find the value of \( \sin(\alpha + \beta) \).
First, let's find \(\sin(\alpha - \beta)\):
Since \(\cos^2(\alpha - \beta) + \sin^2(\alpha - \beta) = 1\),
\[
\sin^2(\alpha ... | true | Let me analyze this partial solution step by step.
**What's been done correctly:**
1. Found $\sin(\alpha - \beta) = \frac{5}{13}$ from $\cos(\alpha - \beta) = \frac{12}{13}$ β
2. Used $\cos 2\beta = \frac{3}{5}$ to find $\cos\beta = \frac{2\sqrt{5}}{5}$ β
3. Found $\sin\beta = \frac{\sqrt{5}}{5}$ β
4. Found $\sin 2\be... | 0.96 | <think>
Let me analyze this partial solution step by step.
**What's been done correctly:**
1. Found $\sin(\alpha - \beta) = \frac{5}{13}$ from $\cos(\alpha - \beta) = \frac{12}{13}$ β
2. Used $\cos 2\beta = \frac{3}{5}$ to find $\cos\beta = \frac{2\sqrt{5}}{5}$ β
3. Found $\sin\beta = \frac{\sqrt{5}}{5}$ β
4. Found $\... | null |
234,050 | 4 | 0.875 | Solve the linear equation $8x = 2x - 6$ for $x$. Express your answer as a single value. | -1 | 0.4 | <think>
To solve the equation \(8x = 2x - 6\) for \(x\), we need to isolate \(x\). Let's follow the steps:
1. Start with the given equation:
\[
8x = 2x - 6
\]
2. Subtract \(2x\) from both sides to combi | true | The partial solution is solving a simple linear equation $8x = 2x - 6$. Let me analyze what's been done and what remains.
The approach is correct: the solver has identified that they need to isolate $x$ and has begun the standard method of collecting like terms.
Step 1 is just restating the problem - correct.
Step 2... | 0.95 | <think>
The partial solution is solving a simple linear equation $8x = 2x - 6$. Let me analyze what's been done and what remains.
The approach is correct: the solver has identified that they need to isolate $x$ and has begun the standard method of collecting like terms.
Step 1 is just restating the problem - correct... | null |
107,070 | 4 | 0.8125 | If p # q denotes the least common multiple of p and q, then w equals the least common multiple of the least common multiple of 12 and 16 and the least common multiple of 18 and 24. What is the value of w? | 144 | 0.4 | <think>
To solve this problem, we will follow the given notation step-by-step. We are given that \( w \) is defined as the least common multiple (LCM) of the LCM of 12 and 16, and the LCM of 18 and 24.
1. **Calculate the LCM of 12 and 16:**
- The prime factorizations are:
\[
12 = 2^2 \times 3
\]
... | true | The partial solution is computing the LCM of 12 and 16 using prime factorization, which is the correct approach.
Let me verify:
- 12 = 2Β² Γ 3 β
- 16 = 2β΄ β
- LCM(12, 16) = 2β΄ Γ 3 = 16 Γ 3 = 48 β
The calculation appears to be cut off at "16 \t" but the method is correct and the work shown so far is accurate.
The rema... | 0.88 | <think>
The partial solution is computing the LCM of 12 and 16 using prime factorization, which is the correct approach.
Let me verify:
- 12 = 2Β² Γ 3 β
- 16 = 2β΄ β
- LCM(12, 16) = 2β΄ Γ 3 = 16 Γ 3 = 48 β
The calculation appears to be cut off at "16 \t" but the method is correct and the work shown so far is accurate.
... | null |
201,608 | 0 | 0 | When the number $2^{1000}$ is divided by $13$, the remainder in the division is | $3$ | 0.4 | <think>
To find the remainder when \(2^{1000}\) is divided by 13, we can use Fermat's Little Theorem. Fermat's Little Theorem states that if \(p\) is a prime number and \(a\) is any integer not divisible by \(p\), then:
\[ a^{p-1} \equiv 1 \pmod{p} \]
Here, \(p = 13\) and \(a = 2\). According to Fermat's Little Theor... | true | The partial solution correctly identifies Fermat's Little Theorem as the appropriate tool. Let me verify the setup:
1. **Correct application of FLT**: Since 13 is prime and gcd(2,13) = 1, we have $2^{12} \equiv 1 \pmod{13}$. This is correct.
2. **Strategy outlined**: The solver recognizes they need to express 1000 in... | 0.94 | <think>
The partial solution correctly identifies Fermat's Little Theorem as the appropriate tool. Let me verify the setup:
1. **Correct application of FLT**: Since 13 is prime and gcd(2,13) = 1, we have $2^{12} \equiv 1 \pmod{13}$. This is correct.
2. **Strategy outlined**: The solver recognizes they need to express... | null |
99,775 | 1 | 0.359375 | James does chores around the class. There are 3 bedrooms, 1 living room, and 2 bathrooms to clean. The bedrooms each take 20 minutes to clean. The living room takes as long as the 3 bedrooms combined. The bathroom takes twice as long as the living room. He also cleans the outside which takes twice as long as clean... | 6 | 0.2 | <think>
First, let's calculate the time it takes to clean each room and the outside:
- Each bedroom takes 20 minutes to clean, and there are 3 bedrooms.
\[
\text{Time for bedrooms} = 3 \times 20 \text{ minutes} = 60 \text{ minutes}
\]
- The living room takes as long as the 3 bedrooms combined, which is 60 minut... | true | Let me analyze this partial solution step by step.
The problem asks for how long James works in hours after splitting chores with his 2 siblings.
The partial solution has correctly:
1. Calculated bedroom cleaning time: 3 Γ 20 = 60 minutes β
2. Identified that living room = 3 bedrooms combined = 60 minutes β
What rem... | 0.65 | <think>
Let me analyze this partial solution step by step.
The problem asks for how long James works in hours after splitting chores with his 2 siblings.
The partial solution has correctly:
1. Calculated bedroom cleaning time: 3 Γ 20 = 60 minutes β
2. Identified that living room = 3 bedrooms combined = 60 minutes β
... | null |
236,719 | 2 | 0.53125 | Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n. If a_6/a_5 = 2/3, find the ratio S_{11}/S_{9}. Express your answer as a fraction. | \frac{22}{27} | 0.4 | <think>
To solve the problem, let's begin by understanding the given information and the properties of the arithmetic sequence. We are given that \( \frac{a_6}{a_5} = \frac{2}{3} \).
For an arithmetic sequence, the \(n\)-th term \(a_n\) can be expressed as:
\[ a_n = a_1 + (n-1)d \]
where \(a_1\) is the first term and ... | true | Let me analyze this partial solution carefully.
The problem asks for $S_{11}/S_9$ given that $a_6/a_5 = 2/3$ in an arithmetic sequence.
**What the partial solution has done:**
1. Correctly set up the general form $a_n = a_1 + (n-1)d$
2. Correctly expressed the condition: $\frac{a_1 + 5d}{a_1 + 4d} = \frac{2}{3}$
3. C... | 0.88 | <think>
Let me analyze this partial solution carefully.
The problem asks for $S_{11}/S_9$ given that $a_6/a_5 = 2/3$ in an arithmetic sequence.
**What the partial solution has done:**
1. Correctly set up the general form $a_n = a_1 + (n-1)d$
2. Correctly expressed the condition: $\frac{a_1 + 5d}{a_1 + 4d} = \frac{2}{... | null |
206,664 | 4 | 0.84375 | Find the interval(s) where the function f(x) = x^3 - x^2 - x is decreasing. Express your answer as interval(s) in the format (a, b) or (-β, c) βͺ (d, β). | (-\frac{1}{3}, 1) | 0.6 | <think>
To determine the intervals where the function \( f(x) = x^3 - x^2 - x \) is decreasing, we need to analyze its first derivative.
1. **Compute the first derivative of \( f(x) \):**
\[
f(x) = x^3 - x^2 - x
\]
Using the power rule, we find the first derivative \( f'(x) \):
\[
f'(x) = \frac{d}... | true | The partial solution has correctly:
1. Identified that finding where f is decreasing requires analyzing f'(x)
2. Computed f'(x) = 3xΒ² - 2x - 1 (correct differentiation)
3. Set f'(x) = 0 to find critical points
4. Applied the quadratic formula correctly with a=3, b=-2, c=-1
5. Calculated the discriminant: 4 + 12 = 16
6.... | 0.94 | <think>
The partial solution has correctly:
1. Identified that finding where f is decreasing requires analyzing f'(x)
2. Computed f'(x) = 3xΒ² - 2x - 1 (correct differentiation)
3. Set f'(x) = 0 to find critical points
4. Applied the quadratic formula correctly with a=3, b=-2, c=-1
5. Calculated the discriminant: 4 + 12... | null |
47,272 | 0 | 0.015625 | The range of the inclination angle of the line $x\cos \theta +\sqrt{3}y+2=0$ is __________. | \left[0,\dfrac{\pi}{6} \right]\cup\left[\dfrac{5\pi}{6},\pi\right) | 0.4 | <think>
The inclination angle of a line is given by the angle $\alpha$ that the line makes with the positive direction of the x-axis, and it is related to the slope of the line by $\tan(\alpha) = m$, where $m$ is the slope of the line. The slope $m$ of the line $Ax + By + C = 0$ is given by $m = -\frac{A}{B}$.
For th... | true | The partial solution correctly identifies the key relationship: for a line $Ax + By + C = 0$, the slope is $m = -\frac{A}{B}$, and the inclination angle $\alpha$ satisfies $\tan(\alpha) = m$.
For the line $x\cos\theta + \sqrt{3}y + 2 = 0$, we have:
- $A = \cos\theta$
- $B = \sqrt{3}$
- Therefore $m = -\frac{\cos\theta... | 0.42 | <think>
The partial solution correctly identifies the key relationship: for a line $Ax + By + C = 0$, the slope is $m = -\frac{A}{B}$, and the inclination angle $\alpha$ satisfies $\tan(\alpha) = m$.
For the line $x\cos\theta + \sqrt{3}y + 2 = 0$, we have:
- $A = \cos\theta$
- $B = \sqrt{3}$
- Therefore $m = -\frac{\c... | null |
170,504 | 1 | 0.296875 | Let \( z_{1} \) and \( z_{2} \) be a pair of conjugate complex numbers. If \(\left|z_{1} - z_{2}\right| = \sqrt{6}\) and \(\frac{z_{1}}{z_{2}^{2}}\) is a real number, then find \(\left|z_{1}\right|\). | \sqrt{2} | 0.8 | <think>
Given that \( z_1 \) and \( z_2 \) are conjugate complex numbers, we can express them as \( z_1 = a + bi \) and \( z_2 = a - bi \) where \( a \) and \( b \) are real numbers.
1. **Compute the magnitude condition:**
\[
\left|z_1 - z_2\right| = \left| (a + bi) - (a - bi) \right| = \left| 2bi \right| = 2|b... | true | Let me analyze this partial solution step by step.
**Step 1 Analysis (Magnitude condition):**
The work is correct. Given $z_1 = a + bi$ and $z_2 = a - bi$ (conjugates), we have:
- $z_1 - z_2 = 2bi$
- $|z_1 - z_2| = 2|b| = \sqrt{6}$
- Therefore $|b| = \frac{\sqrt{6}}{2}$
This is flawless.
**Step 2 Analysis (Computing... | 0.32 | <think>
Let me analyze this partial solution step by step.
**Step 1 Analysis (Magnitude condition):**
The work is correct. Given $z_1 = a + bi$ and $z_2 = a - bi$ (conjugates), we have:
- $z_1 - z_2 = 2bi$
- $|z_1 - z_2| = 2|b| = \sqrt{6}$
- Therefore $|b| = \frac{\sqrt{6}}{2}$
This is flawless.
**Step 2 Analysis (C... | null |
90,331 | 4 | 1 | A number is doubled and 15 is added. If the resultant is trebled, it becomes 75. What is that number? | 5 | 0.6 | <think>
Let's denote the unknown number by \( x \).
According to the problem:
1. The number is doubled and 15 is added: \( 2x + 15 \).
2. The resultant is trebled (multiplied by 3): \( 3(2x + 15) \).
We are told that this trebled result equals 75:
\[ 3(2x + 15) = 75 \]
First, let's solve this | true | The partial solution has correctly translated the word problem into an algebraic equation. Let me verify each step:
1. "The number is doubled and 15 is added" β $2x + 15$ β (correct)
2. "If the resultant is trebled" β $3(2x + 15)$ β (correct)
3. "it becomes 75" β $3(2x + 15) = 75$ β (correct)
The equation setup is co... | 0.95 | <think>
The partial solution has correctly translated the word problem into an algebraic equation. Let me verify each step:
1. "The number is doubled and 15 is added" β $2x + 15$ β (correct)
2. "If the resultant is trebled" β $3(2x + 15)$ β (correct)
3. "it becomes 75" β $3(2x + 15) = 75$ β (correct)
The equation set... | null |
4,915 | 1 | 0.390625 | Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+y) = f(x) + f(y) + 2xy$ for all $x, y \in \mathbb{R}$, and given that $f(1) = 2$, calculate $f(-3)$. | 6 | 0.2 | <think>
Given the functional equation \( f(x+y) = f(x) + f(y) + 2xy \), we first need to find a form for the function \( f(x) \). Let's start by setting \( x = 0 \) and \( y = 0 \) into the equation:
\[ f(0+0) = f(0) + f(0) + 2 \cdot 0 \cdot 0 \]
\[ f(0) = 2f(0) \]
\[ f(0) = 0 \]
Next, we will f | true | The partial solution has correctly identified the functional equation $f(x+y) = f(x) + f(y) + 2xy$ and taken the first natural step: finding $f(0)$ by substituting $x = y = 0$.
The calculation is correct:
- $f(0) = f(0) + f(0) + 0$
- This gives $f(0) = 2f(0)$, so $f(0) = 0$
This is a good start. The standard approach... | 0.72 | <think>
The partial solution has correctly identified the functional equation $f(x+y) = f(x) + f(y) + 2xy$ and taken the first natural step: finding $f(0)$ by substituting $x = y = 0$.
The calculation is correct:
- $f(0) = f(0) + f(0) + 0$
- This gives $f(0) = 2f(0)$, so $f(0) = 0$
This is a good start. The standard ... | null |
160,963 | 2 | 0.578125 | Today is December 19, 2010. What is the integer part of the sum $\frac{2010}{1000}+\frac{1219}{100}+\frac{27}{10}$? | 16 | 0.4 | <think>
To find the integer part of the sum \(\frac{2010}{1000} + \frac{1219}{100} + \frac{27}{10}\), we need to evaluate each fraction separately and then sum them up.
First, let's simplify \(\frac{2010}{1000}\):
\[
\frac{2010}{1000} = \frac{2010 \div 10}{1000 \div 10} = \frac{201}{100} = 2.01 | true | The partial solution has correctly computed the first term: $\frac{2010}{1000} = 2.01$. Let me verify this is correct: $2010 \div 1000 = 2.010$, yes that's right.
The problem requires finding the integer part of $\frac{2010}{1000}+\frac{1219}{100}+\frac{27}{10}$.
The approach shown is straightforward and correct: eva... | 0.88 | <think>
The partial solution has correctly computed the first term: $\frac{2010}{1000} = 2.01$. Let me verify this is correct: $2010 \div 1000 = 2.010$, yes that's right.
The problem requires finding the integer part of $\frac{2010}{1000}+\frac{1219}{100}+\frac{27}{10}$.
The approach shown is straightforward and corr... | null |
47,349 | 3 | 0.65625 | Given a rectangle ABCD with an area of 12 and a diagonal length of 5, find the width of the rectangle. | 3 | 0.8 | <think>
To solve for the width of the rectangle, we need to use the given information: the area of the rectangle (12) and the length of the diagonal (5). Let's denote the length of the rectangle as \( l \) and the width as \( w \).
From the area, we know:
\[ l \times w = 12 \]
From the Pythagorean theorem applied to... | true | Let me analyze this partial solution step by step.
The setup is correct:
- Area: $lw = 12$
- Diagonal: $l^2 + w^2 = 25$
The algebraic manipulation is flawless:
1. Substitution $l = \frac{12}{w}$ is correct
2. Substituting into the Pythagorean equation: $\frac{144}{w^2} + w^2 = 25$ β
3. Multiplying by $w^2$: $144 + w^... | 0.98 | <think>
Let me analyze this partial solution step by step.
The setup is correct:
- Area: $lw = 12$
- Diagonal: $l^2 + w^2 = 25$
The algebraic manipulation is flawless:
1. Substitution $l = \frac{12}{w}$ is correct
2. Substituting into the Pythagorean equation: $\frac{144}{w^2} + w^2 = 25$ β
3. Multiplying by $w^2$: $... | null |
146,542 | 0 | 0.171875 | Dave initially had 24 files and 13 apps on his phone. He then deleted 5 files and 3 apps, but later added 7 more files and 4 apps. After that, he deleted another 10 files and 4 apps, and finally added 5 files and 7 apps. At the end, Dave had 17 apps and 21 files left on his phone. How many total files did he delete thr... | 15 | 0.8 | <think>
To determine the total number of files Dave deleted throughout the process, we will track the changes in the number of files step by step.
Initially, Dave had:
- 24 files
- 13 apps
First, he deleted 5 files and added 7 files:
- Files remaining: \(24 - 5 + 7 = 26\)
Next, he deleted 10 files:
- Files remaining... | true | Let me analyze this partial solution carefully.
The problem asks for the total number of files deleted throughout the process. Let me first verify what the correct answer should be:
Initial files: 24
- Delete 5 files: 24 - 5 = 19
- Add 7 files: 19 + 7 = 26
- Delete 10 files: 26 - 10 = 16
- Add 5 files: 16 + 5 = 21 β ... | 0.94 | <think>
Let me analyze this partial solution carefully.
The problem asks for the total number of files deleted throughout the process. Let me first verify what the correct answer should be:
Initial files: 24
- Delete 5 files: 24 - 5 = 19
- Add 7 files: 19 + 7 = 26
- Delete 10 files: 26 - 10 = 16
- Add 5 files: 16 + 5... | null |
22,201 | 0 | 0.078125 | Given the proposition $p$: $\exists x \in \mathbb{R}$, $x > \sin x$, then the negation of $p$ is ______. | \neg p: \forall x \in \mathbb{R}, x \leqslant \sin x | 0.4 | <think>
To find the negation of the proposition \( p \), we first need to understand the logical structure of the statement. The proposition \( p \) is given as:
\[ p: \exists x \in \mathbb{R}, x > \sin x. \]
The negation of a statement involving an existential quantifier \(\exists\) is a stateme | true | The partial solution correctly identifies the key task: finding the negation of an existential statement. The proposition is:
$$p: \exists x \in \mathbb{R}, x > \sin x$$
The student has started to explain the logical structure and is about to state the rule for negating existential quantifiers. The standard rule is:
$... | 0.88 | <think>
The partial solution correctly identifies the key task: finding the negation of an existential statement. The proposition is:
$$p: \exists x \in \mathbb{R}, x > \sin x$$
The student has started to explain the logical structure and is about to state the rule for negating existential quantifiers. The standard ru... | null |
29,678 | 4 | 0.90625 | Calculate: $-2-(-3)=$____. | 1 | 0.8 | <think>
To solve the expression \(-2 - (-3)\), we need to understand the operation of subtracting a negative number. Subtracting a negative number is equivalent to adding the positive counterpart of that number.
So, \(-2 - (-3)\) can be rewritten as:
\[
-2 + 3
\]
Next, we need to perform the addition \(-2 + 3\). We ... | true | The partial solution correctly identifies the key transformation: subtracting a negative is equivalent to adding the positive. The rewriting of $-2 - (-3)$ as $-2 + 3$ is algebraically correct and represents the crucial conceptual step.
The solution then sets up the correct arithmetic problem: $-2 + 3$. The number lin... | 0.99 | <think>
The partial solution correctly identifies the key transformation: subtracting a negative is equivalent to adding the positive. The rewriting of $-2 - (-3)$ as $-2 + 3$ is algebraically correct and represents the crucial conceptual step.
The solution then sets up the correct arithmetic problem: $-2 + 3$. The nu... | null |
143,447 | 2 | 0.578125 | When Jane visited Iran, she visited 6 more rehabilitation centers than a certain multiple of the number of rehabilitation centers Han visited. Han visited 2 less than twice as many rehabilitation centers as Jude did. Jude visited half fewer rehabilitation centers as Lisa did. Lisa visited 6 rehabilitation centers. They... | 2 | 0.4 | <think>
Let's break down the problem step by step.
1. **Lisa's Rehabilitation Centers:** Lisa visited 6 rehabilitation centers.
\[
L = 6
\]
2. **Jude's Rehabilitation Centers:** Jude visited half fewer rehabilitation centers than Lisa. This means Jude visited half as many as Lisa did.
\[
J = L - \frac{... | true | Let me analyze the partial solution provided.
**What's been done:**
1. Lisa's centers: L = 6 β (correctly identified from problem)
2. Jude's centers: The phrase "half fewer" is interpreted as "half as many"
- Calculation: J = L/2 = 6/2 = 3 β (arithmetic correct given the interpretation)
3. Han's centers: Started bu... | 0.88 | <think>
Let me analyze the partial solution provided.
**What's been done:**
1. Lisa's centers: L = 6 β (correctly identified from problem)
2. Jude's centers: The phrase "half fewer" is interpreted as "half as many"
- Calculation: J = L/2 = 6/2 = 3 β (arithmetic correct given the interpretation)
3. Han's centers: St... | null |
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