source stringclasses 2
values | problem stringlengths 57 5.47k | solution stringlengths 335 9.02k | messages listlengths 2 2 | r1_distill_70b_response stringlengths 551 109k | __original_row_idx int64 0 7.5k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|
amc_aime | At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the co... | 1. **Identify the number of students in each class:**
- Mrs. Germain's class: 11 students
- Mr. Newton's class: 8 students
- Mrs. Young's class: 9 students
2. **Assumption of no overlap in students:**
Since the problem does not mention any students being in more than one class, we assume that all students ... | [
{
"content": "At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle Sc... | <think>
First, identify the number of students in each of the three mathematics classes: Mrs. Germain has 11 students, Mr. Newton has 8 students, and Mrs. Young has 9 students.
To find the total number of students taking the AMC 8 contest, add the number of students from each class together: 11 + 8 + 9.
Calculating t... | 4 | [
{
"from": "human",
"value": "At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students ... |
amc_aime | The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$ | 1. **Apply the operation $\otimes$ to the innermost parentheses first:**
\[
(1 \otimes 2) \otimes 3 = \left(\frac{1^2}{2}\right) \otimes 3 = \frac{1}{2} \otimes 3
\]
\[
1 \otimes (2 \otimes 3) = 1 \otimes \left(\frac{2^2}{3}\right) = 1 \otimes \frac{4}{3}
\]
2. **Calculate each part using the definit... | [
{
"content": "The operation $\\otimes$ is defined for all nonzero numbers by $a\\otimes b =\\frac{a^{2}}{b}$. Determine $[(1\\otimes 2)\\otimes 3]-[1\\otimes (2\\otimes 3)]$.\n$\\text{(A)}\\ -\\frac{2}{3}\\qquad\\text{(B)}\\ -\\frac{1}{4}\\qquad\\text{(C)}\\ 0\\qquad\\text{(D)}\\ \\frac{1}{4}\\qquad\\text{(E)}\... | <think>
Okay, so I have this math problem here about this operation called "otimes". It's defined for all nonzero numbers by \( a \otimes b = \frac{a^2}{b} \). The question is asking me to find \( [(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)] \). The multiple-choice options are from A to E, with some fractions ... | 0 | [
{
"from": "human",
"value": "The operation $\\otimes$ is defined for all nonzero numbers by $a\\otimes b =\\frac{a^{2}}{b}$. Determine $[(1\\otimes 2)\\otimes 3]-[1\\otimes (2\\otimes 3)]$.\n$\\text{(A)}\\ -\\frac{2}{3}\\qquad\\text{(B)}\\ -\\frac{1}{4}\\qquad\\text{(C)}\\ 0\\qquad\\text{(D)}\\ \\frac{1}{4}... |
amc_aime | If $991+993+995+997+999=5000-N$, then $N=$
$\text{(A)}\ 5 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25$ | 1. **Identify the problem and express each term in a form that reveals a pattern:**
\[
991+993+995+997+999=5000-N
\]
We can rewrite each term as $1000$ minus a small number:
\[
(1000-9) + (1000-7) + (1000-5) + (1000-3) + (1000-1)
\]
2. **Simplify the expression by factoring out $1000$ and summing ... | [
{
"content": "If $991+993+995+997+999=5000-N$, then $N=$\n$\\text{(A)}\\ 5 \\qquad \\text{(B)}\\ 10 \\qquad \\text{(C)}\\ 15 \\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ 25$",
"role": "user"
},
{
"content": "1. **Identify the problem and express each term in a form that reveals a pattern:**\n \\... | <think>
Okay, so I have this problem here that says if 991 + 993 + 995 + 997 + 999 equals 5000 minus N, then what is N? The options are A) 5, B) 10, C) 15, D) 20, E) 25.
Hmm, okay. Let me think about this step by step. I feel like adding up these five numbers is going to be a bit time-consuming, but maybe there's a pa... | 5 | [
{
"from": "human",
"value": "If $991+993+995+997+999=5000-N$, then $N=$\n$\\text{(A)}\\ 5 \\qquad \\text{(B)}\\ 10 \\qquad \\text{(C)}\\ 15 \\qquad \\text{(D)}\\ 20 \\qquad \\text{(E)}\\ 25$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here that says if 991 + 993 + 995 + ... |
amc_aime | The total in-store price for an appliance is $\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\textdollar 29.98$ and a one-time shipping and handling charge of $\textdollar 9.98$. How many cents are saved by buying the appliance from the television advertiser?
$\mathr... | 1. **Calculate the total cost of buying the appliance from the television advertiser:**
The advertisement mentions three payments of $\textdollar 29.98$ each and an additional one-time shipping and handling charge of $\textdollar 9.98$. First, calculate the total of the three payments:
\[
3 \times 29.98 = 89.9... | [
{
"content": "The total in-store price for an appliance is $\\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\\textdollar 29.98$ and a one-time shipping and handling charge of $\\textdollar 9.98$. How many cents are saved by buying the appliance from the telev... | <think>
Okay, so I have this problem here where I need to figure out how many cents are saved by buying an appliance through a television advertisement instead of in-store. The in-store price is $99.99, and the TV ad offers three payments of $29.98 each plus a one-time shipping and handling fee of $9.98. The answer cho... | 6 | [
{
"from": "human",
"value": "The total in-store price for an appliance is $\\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\\textdollar 29.98$ and a one-time shipping and handling charge of $\\textdollar 9.98$. How many cents are saved by buying the appli... |
amc_aime | The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:
$\textbf{(A)}\ -2\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{undetermined}$ | 1. **Identify the common difference**:
Given the arithmetic progression (AP) terms are $x - 1$, $x + 1$, and $2x + 3$. In an AP, the difference between consecutive terms is constant. Therefore, the common difference $d$ can be calculated as:
\[
d = (x + 1) - (x - 1)
\]
Simplifying this, we get:
\[
... | [
{
"content": "The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: \n$\\textbf{(A)}\\ -2\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ \\text{undetermined}$",
"role": "user"
},
{
"conte... | <think>
First, I recall that in an arithmetic progression (AP), the difference between consecutive terms is constant. This means that the difference between the second and first term is equal to the difference between the third and second term.
I'll denote the terms as:
- First term: \( x - 1 \)
- Second term: \( x + ... | 20 | [
{
"from": "human",
"value": "The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: \n$\\textbf{(A)}\\ -2\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ \\text{undetermined}$"
},
{
"from":... |
amc_aime | At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | 1. **Define the variables:**
Let $M$ represent the temperature in Minneapolis at noon, and $L$ represent the temperature in St. Louis at noon. Given that Minneapolis is $N$ degrees warmer than St. Louis at noon, we can express this relationship as:
\[
M = L + N
\]
2. **Temperature changes by 4:00 PM:**
... | [
{
"content": "At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product ... | <think>
Okay, let me try to solve this problem. It's about the temperature difference between Minneapolis and St. Louis. Hmm, seems interesting. Alright, let's break it down step by step.
First, the problem says at noon, Minneapolis is N degrees warmer than St. Louis. So, if I let the temperature in St. Louis at noon ... | 36 | [
{
"from": "human",
"value": "At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. W... |
amc_aime | Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$.
... | 1. **Identify Key Points and Relationships**:
- Points $A, B, C, D, E,$ and $F$ are collinear on line $\overline{AF}$, and each segment between consecutive points is of length 1.
- Point $G$ is not on line $AF$, and points $H$ and $J$ lie on lines $\overline{GD}$ and $\overline{GF}$ respectively.
- Lines $\ov... | [
{
"content": "Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ and $\\overline{AG}$ ... | <think>
Okay, let's try to solve this problem step by step. It seems a bit involved, but if I break it down, I'll manage.
First, let me visualize the problem. We have points A, B, C, D, E, F on line AF, in that order. Each segment between these points is length 1, so the total length AF is 5 units. Now, point G is not... | 7 | [
{
"from": "human",
"value": "Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ an... |
amc_aime | Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \... | 1. **Identify the original statement and its structure**: The original statement is "For all $x$ of a certain set, $x^2 > 0$". This is a universal quantification statement, which asserts that a certain property (in this case, $x^2 > 0$) holds for every element $x$ in a specified set.
2. **Understand the negation of a ... | [
{
"content": "Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For some x}, x^2>0\\qq... | <think>
Okay, so I need to figure out the negation of the statement: "For all x of a certain set, x squared is greater than zero." Hmm, okay. Let me break this down.
First, the original statement is a universal statement. It's saying that for every element x in the set, the square of x is positive. So, written logical... | 35 | [
{
"from": "human",
"value": "Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For... |
amc_aime | A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased?
$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ \... | 1. **Identify the sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
2. **Expression for the average after erasing one number**: If one number $x$ is erased, the sum of the remaining numbe... | [
{
"content": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased? \n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7 \\qquad \\textbf{(C)}\\ 8 \\qquad \\text... | <think>
First, let's define the problem. We have a set of consecutive positive integers starting from 1, and one number is erased. The average of the remaining numbers is given as \(35 \frac{7}{17}\), which is equivalent to \(\frac{602}{17}\).
Let \(n\) be the total number of integers originally on the blackboard. The... | 11 | [
{
"from": "human",
"value": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased? \n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7 \\qquad \\textbf{(C)}\... |
amc_aime | During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?
$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\%$ | 1. **Identify the total number of hits**: Judy had a total of 35 hits during the season.
2. **Determine the number of hits that were not singles**:
- Home runs: 1
- Triples: 1
- Doubles: 5
- Total non-single hits = 1 (home run) + 1 (triple) + 5 (doubles) = 7
3. **Calculate the number of singles**:
\[
... | [
{
"content": "During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?\n$\\text{(A)}\\ 28\\% \\qquad \\text{(B)}\\ 35\\% \\qquad \\text{(C)}\\ 70\\% \\qquad \\text{(D)}\\ 75\\% \\qquad... | <think>
Okay, so I have this problem here about Judy's softball season. Let me read it again.
"Judy had 35 hits. Among her hits were 1 home run, 1 triple, and 5 doubles. The rest of her hits were singles. What percent of her hits were singles?"
Alright, so I need to figure out how many singles Judy had and then find ... | 8 | [
{
"from": "human",
"value": "During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?\n$\\text{(A)}\\ 28\\% \\qquad \\text{(B)}\\ 35\\% \\qquad \\text{(C)}\\ 70\\% \\qquad \\text{(... |
amc_aime | Find the smallest positive number from the numbers below.
$\textbf{(A)} \ 10-3\sqrt{11} \qquad \textbf{(B)} \ 3\sqrt{11}-10 \qquad \textbf{(C)}\ 18-5\sqrt{13}\qquad \textbf{(D)}\ 51-10\sqrt{26}\qquad \textbf{(E)}\ 10\sqrt{26}-51$ | To find the smallest positive number among the given options, we first need to simplify and compare each expression.
#### Step 1: Simplify each expression
- **Option (A)**: $10 - 3\sqrt{11}$
- **Option (B)**: $3\sqrt{11} - 10$
- **Option (C)**: $18 - 5\sqrt{13}$
- **Option (D)**: $51 - 10\sqrt{26}$
- **Option (E)**: $... | [
{
"content": "Find the smallest positive number from the numbers below.\n$\\textbf{(A)} \\ 10-3\\sqrt{11} \\qquad \\textbf{(B)} \\ 3\\sqrt{11}-10 \\qquad \\textbf{(C)}\\ 18-5\\sqrt{13}\\qquad \\textbf{(D)}\\ 51-10\\sqrt{26}\\qquad \\textbf{(E)}\\ 10\\sqrt{26}-51$",
"role": "user"
},
{
"content": "... | <think>
Alright, so I have this problem here where I need to find the smallest positive number among the options given. The options are all expressions involving square roots, so each one is either a subtraction or subtraction turned into an addition possibly. Let me first list out all the options to make it clearer.
... | 27 | [
{
"from": "human",
"value": "Find the smallest positive number from the numbers below.\n$\\textbf{(A)} \\ 10-3\\sqrt{11} \\qquad \\textbf{(B)} \\ 3\\sqrt{11}-10 \\qquad \\textbf{(C)}\\ 18-5\\sqrt{13}\\qquad \\textbf{(D)}\\ 51-10\\sqrt{26}\\qquad \\textbf{(E)}\\ 10\\sqrt{26}-51$"
},
{
"from": "gpt"... |
amc_aime | If $\texttt{a}$ and $\texttt{b}$ are digits for which
$\begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array}$
then $\texttt{a+b =}$
$\mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }12$ | 1. **Identify the multiplication setup**: The multiplication setup is given as:
\[
\begin{array}{ccc}
& 2 & a\\
\times & b & 3\\
\hline
& 6 & 9\\
9 & 2\\
\hline
9 & 8 & 9
\end{array}
\]
This represents the multiplication of a two-digit number $2a$ by another two-digit number $b... | [
{
"content": "If $\\texttt{a}$ and $\\texttt{b}$ are digits for which\n$\\begin{array}{ccc}& 2 & a\\\\ \\times & b & 3\\\\ \\hline & 6 & 9\\\\ 9 & 2\\\\ \\hline 9 & 8 & 9\\end{array}$\nthen $\\texttt{a+b =}$\n$\\mathrm{(A)\\ } 3 \\qquad \\mathrm{(B) \\ }4 \\qquad \\mathrm{(C) \\ } 7 \\qquad \\mathrm{(D) \\ } ... | <think>
Alright, so I have this multiplication problem here, and I need to find the sum of two digits, a and b. The problem is set up like a standard multiplication algorithm, with partial products and a final product.
Let me write it down to visualize it better:
```
2 a
× b 3
------
6 9
9 2
-------
9 8 9
... | 48 | [
{
"from": "human",
"value": "If $\\texttt{a}$ and $\\texttt{b}$ are digits for which\n$\\begin{array}{ccc}& 2 & a\\\\ \\times & b & 3\\\\ \\hline & 6 & 9\\\\ 9 & 2\\\\ \\hline 9 & 8 & 9\\end{array}$\nthen $\\texttt{a+b =}$\n$\\mathrm{(A)\\ } 3 \\qquad \\mathrm{(B) \\ }4 \\qquad \\mathrm{(C) \\ } 7 \\qquad ... |
amc_aime | Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from t... | 1. **Define Variables:**
Let $v_A$ be Andrea's speed and $v_L$ be Lauren's speed, both in kilometers per hour.
2. **Set Up Equations:**
Given that Andrea travels at three times the speed of Lauren, we have:
\[ v_A = 3v_L \]
Also, the rate at which the distance between them decreases is $1$ kilometer per mi... | [
{
"content": "Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After h... | <think>
Okay, so I'm trying to solve this problem where Andrea and Lauren are 20 kilometers apart and are biking towards each other. Andrea is three times as fast as Lauren, and the distance between them decreases at a rate of 1 kilometer per minute. After 5 minutes, Andrea stops biking because of a flat tire and waits... | 44 | [
{
"from": "human",
"value": "Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits ... |
amc_aime | Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the s... |
1. **Define Variables:**
Let the volume of the first container be $A$ and the volume of the second container be $B$.
2. **Set Up the Equation:**
Since Alicia poured $\frac{5}{6}$ of the first container into the second container, and this amount filled $\frac{3}{4}$ of the second container, we can write the equa... | [
{
"content": "Alicia had two containers. The first was $\\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first containe... | <think>
Okay, so let me try to figure out this problem step by step.
Alicia has two containers. The first one is 5/6 full of water, and the second one is empty. She pours all the water from the first container into the second one, and then the second container is 3/4 full. We need to find the ratio of the volume of t... | 23 | [
{
"from": "human",
"value": "Alicia had two containers. The first was $\\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\tfrac{3}{4}$ full of water. What is the ratio of the volume of... |
amc_aime | In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
\[\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}\]
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 9$ | 1. **Identify the value of $A$:**
Given the multiplication problem:
\[
\begin{array}{cccc}
& A & B & A\\
\times & & C & D\\
\hline
C & D & C & D\\
\end{array}
\]
We observe that the product of $A$ and $D$ results in a number ending in $D$. This implies that $A \times D$ must be a num... | [
{
"content": "In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$? \n\\[\\begin{array}{cccc}& A & B & A\\\\ \\times & & C & D\\\\ \\hline C & D & C & D\\\\ \\end{array}\\]\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\t... | <think>
Alright, so I have this multiplication problem where each letter represents a different digit. The problem looks like this:
```
A B A
× C D
-------
C D C D
```
And I need to find A + B. The options are from 1 to 9, so the answer should be one of those. Let me try to figure this out step by step.
First, le... | 42 | [
{
"from": "human",
"value": "In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$? \n\\[\\begin{array}{cccc}& A & B & A\\\\ \\times & & C & D\\\\ \\hline C & D & C & D\\\\ \\end{array}\\]\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf... |
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