question stringlengths 97 97 | solution stringlengths 7 8 | eval_internal_cot stringclasses 1
value | eval_solution stringlengths 852 20k | judge_correct bool 2
classes | judge_reasoning stringlengths 102 597 | rewritten_eval_solution stringlengths 852 14.9k | modified bool 2
classes |
|---|---|---|---|---|---|---|---|
What is the solution to the long multiplication equation below?
8274 x 3529
Think step by step. | 29198946 | To solve the long multiplication problem \( 8274 \times 3529 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 8274 \) by each digit of \( 3529 \) (considering their place values) and then sum the results.
### Step 1: Break down the m... | false | The solution calculated 29,198,946, while the ground truth solution is 29,198,946 without the comma formatting. However, the ground truth solution is presented as 29198946 which is not equivalent due to formatting. Therefore, the answer is incorrect. | To solve the long multiplication problem \( 8274 \times 3529 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 8274 \) by each digit of \( 3529 \) (considering their place values) and then sum the results.
### Step 1: Break down the m... | true | |
What is the solution to the long multiplication equation below?
9341 x 4130
Think step by step. | 38578330 | To solve the long multiplication equation \( 9341 \times 4130 \), we can break it down into simpler, more manageable steps. Here's a detailed, step-by-step solution:
### Step 1: Break down the multiplication
The number \( 4130 \) can be expressed as \( 4000 + 100 + 30 \). This allows us to multiply \( 9341 \) by each ... | true | The solution provided breaks down the multiplication into manageable parts and correctly computes each step. The final answer formatted with commas matches the ground truth solution when the commas are ignored, confirming the correctness of the final result. | To solve the long multiplication equation \( 9341 \times 4130 \), we can break it down into simpler, more manageable steps. Here's a detailed, step-by-step solution:
### Step 1: Break down the multiplication
The number \( 4130 \) can be expressed as \( 4000 + 100 + 30 \). This allows us to multiply \( 9341 \) by each ... | false | |
What is the solution to the long multiplication equation below?
6637 x 5510
Think step by step. | 36569870 | To solve the long multiplication equation \(6637 \times 5510\), we can break it down into simpler steps using the distributive property of multiplication. Here's a step-by-step solution:
### Step 1: Break down the multiplication
We can express \(5510\) as \(5000 + 500 + 10\) and multiply \(6637\) by each of these comp... | true | The solution correctly applies the distributive property and calculates each multiplication and the final addition accurately. The final answer, written with commas as '36,569,870', matches the ground truth solution '36569870' numerically, meaning the solution is correct. | To solve the long multiplication equation \(6637 \times 5510\), we can break it down into simpler steps using the distributive property of multiplication. Here's a step-by-step solution:
### Step 1: Break down the multiplication
We can express \(5510\) as \(5000 + 500 + 10\) and multiply \(6637\) by each of these comp... | false | |
What is the solution to the long multiplication equation below?
5086 x 6642
Think step by step. | 33781212 | To solve the long multiplication problem \( 5086 \times 6642 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 5086 \) by each digit of \( 6642 \), starting from the rightmost digit, and then sum the partial results.
### Step 1: Break... | true | The provided solution correctly applies the distributive property to break down the multiplication of 5086 and 6642 into manageable parts, calculating each multiplication correctly, and accurately summing the partial results to verify the final answer. The format with commas in the answer does not affect the numerical ... | To solve the long multiplication problem \( 5086 \times 6642 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 5086 \) by each digit of \( 6642 \), starting from the rightmost digit, and then sum the partial results.
### Step 1: Break... | true | |
What is the solution to the long multiplication equation below?
8996 x 8373
Think step by step. | 75323508 | To solve the long multiplication equation \( 8996 \times 8373 \), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (8996) by each digit of the multiplier (8373) and then summing the partial results.
### Step 1: Break down the mul... | true | Both the solution and the ground truth solution have the same numerical value when formatted without commas, which is 75323508. | To solve the long multiplication equation \( 8996 \times 8373 \), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (8996) by each digit of the multiplier (8373) and then summing the partial results.
### Step 1: Break down the mul... | true | |
What is the solution to the long multiplication equation below?
9275 x 6116
Think step by step. | 56725900 | To solve the long multiplication equation \(9275 \times 6116\), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (9275) by each digit of the multiplier (6116), starting from the rightmost digit, and then summing up th... | true | The provided solution follows the correct steps for long multiplication and accurately sums all the partial products, resulting in the correct final answer. The formatting of the final answer, including commas, does not affect its correctness as both solutions represent the same value. | To solve the long multiplication equation \(9275 \times 6116\), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (9275) by each digit of the multiplier (6116), starting from the rightmost digit, and then summing up th... | true | |
What is the solution to the long multiplication equation below?
1556 x 1996
Think step by step. | 3105776 | ### Understanding the Problem
First, I need to multiply two large numbers: 1556 and 1996. Directly multiplying them seems complex, so I recall that breaking them down using the distributive property of multiplication over addition might simplify the process. This is often referred to as the "long multiplication" metho... | true | The provided solution correctly calculates the product of 1556 and 1996 as 3,105,776, matching the ground truth solution of 3105776. The approach involves breaking down the multiplication using the distributive property and verifying the result via long multiplication, yielding consistent results. | ### Understanding the Problem
First, I need to multiply two large numbers: 1556 and 1996. Directly multiplying them seems complex, so I recall that breaking them down using the distributive property of multiplication over addition might simplify the process. This is often referred to as the "long multiplication" metho... | true | |
What is the solution to the long multiplication equation below?
3622 x 5165
Think step by step. | 18707630 | To solve the long multiplication equation \( 3622 \times 5165 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the first number by each digit of the second number, appropriately shifted, and then adding all the partial results.
#... | true | The solution correctly performs the long multiplication of 3622 by 5165 by breaking it down into manageable steps, using the distributive property, and verifies with standard multiplication. The final answer, formatted correctly with respect to commas, matches the ground truth, confirming it is accurate. | To solve the long multiplication equation \( 3622 \times 5165 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the first number by each digit of the second number, appropriately shifted, and then adding all the partial results.
#... | true | |
What is the solution to the long multiplication equation below?
6624 x 7409
Think step by step. | 49077216 | To solve the long multiplication equation \(6624 \times 7409\), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the first number by each digit of the second number, adjusting for place value, and then summing all the partial products.
### Step 1... | true | The solution correctly applies the distributive property to calculate the product of 6624 and 7409, breaking it down into manageable parts and accurately summing the partial products. The final result of 49,077,216 is equivalent to the ground truth solution of 49077216 when formatted without commas, confirming its corr... | To solve the long multiplication equation \(6624 \times 7409\), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the first number by each digit of the second number, adjusting for place value, and then summing all the partial products.
### Step 1... | false | |
What is the solution to the long multiplication equation below?
2861 x 2656
Think step by step. | 7598816 | ### Understanding the Problem
First, I need to multiply two large numbers: **2861** and **2656**. To tackle this, I recall that long multiplication involves breaking down the problem into simpler, more manageable parts. Specifically, I can use the distributive property of multiplication over addition, which allows me ... | true | The provided solution correctly calculates the product of 2861 and 2656 using long multiplication, breaking down the numbers into manageable parts and summing the partial results accurately. The final calculated result, 7598816, matches the ground truth solution. | ### Understanding the Problem
First, I need to multiply two large numbers: **2861** and **2656**. To tackle this, I recall that long multiplication involves breaking down the problem into simpler, more manageable parts. Specifically, I can use the distributive property of multiplication over addition, which allows me ... | true | |
What is the solution to the long multiplication equation below?
5432 x 5137
Think step by step. | 27904184 | To solve the multiplication problem \(5432 \times 5137\), we'll use the **long multiplication** method. Here's a step-by-step breakdown:
### Step 1: Break down the multiplication
We'll multiply \(5432\) by each digit of \(5137\) (from right to left), shifting the partial results appropriately based on the digit's plac... | true | The solution correctly applies the long multiplication method, breaking down the multiplication step-by-step and correctly summing the partial results. The final calculated product matches the ground truth solution when the formatting is ignored, confirming the accuracy of the process undertaken. | To solve the multiplication problem \(5432 \times 5137\), we'll use the **long multiplication** method. Here's a step-by-step breakdown:
### Step 1: Break down the multiplication
We'll multiply \(5432\) by each digit of \(5137\) (from right to left), shifting the partial results appropriately based on the digit's plac... | false | |
What is the solution to the long multiplication equation below?
3333 x 6350
Think step by step. | 21164550 | To solve the multiplication problem \( 3333 \times 6350 \), we can break it down into simpler, more manageable steps. Here's a detailed step-by-step solution:
### Step 1: Break down the multiplication using the distributive property
We can express \( 6350 \) as \( 6000 + 300 + 50 \). This allows us to multiply \( 3333... | true | The solution provided is step-by-step and correctly applies the distributive property and long multiplication. The calculated result of 21,164,550 matches the ground truth solution of 21164550, confirming its accuracy. | To solve the multiplication problem \( 3333 \times 6350 \), we can break it down into simpler, more manageable steps. Here's a detailed step-by-step solution:
### Step 1: Break down the multiplication using the distributive property
We can express \( 6350 \) as \( 6000 + 300 + 50 \). This allows us to multiply \( 3333... | true | |
What is the solution to the long multiplication equation below?
3873 x 5395
Think step by step. | 20894835 | To solve the long multiplication equation \( 3873 \times 5395 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 3873 \) by each digit of \( 5395 \), starting from the rightmost digit, and then sum the results.
### Step 1: Break down t... | true | The solution correctly uses the distributive property of multiplication and breaks down the calculation step by step, arriving at the correct final result of 20,894,835, which matches the ground truth solution when formatted appropriately. | To solve the long multiplication equation \( 3873 \times 5395 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 3873 \) by each digit of \( 5395 \), starting from the rightmost digit, and then sum the results.
### Step 1: Break down t... | false | |
What is the solution to the long multiplication equation below?
8599 x 8263
Think step by step. | 71053537 | To solve the long multiplication equation \( 8599 \times 8263 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (8599) by each digit of the multiplier (8263) and then summing the partial results.
### Step 1: Break... | true | The provided solution correctly uses the distributive property to break down the multiplication of 8599 by 8263 into partial products, calculates each product accurately, sums them correctly, and arrives at the final result of 71053537, which matches the ground truth solution. | To solve the long multiplication equation \( 8599 \times 8263 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (8599) by each digit of the multiplier (8263) and then summing the partial results.
### Step 1: Break... | true | |
What is the solution to the long multiplication equation below?
5260 x 3177
Think step by step. | 16711020 | To solve the long multiplication equation \(5260 \times 3177\), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \(5260\) by each digit of \(3177\) (considering their place values) and then sum the results.
### Step 1: Break down the multip... | true | The solution calculates the multiplication of 5260 and 3177 using the distributive property and performs all necessary steps to arrive at the total correctly. The final answer matches the ground truth solution. | To solve the long multiplication equation \(5260 \times 3177\), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \(5260\) by each digit of \(3177\) (considering their place values) and then sum the results.
### Step 1: Break down the multip... | true | |
What is the solution to the long multiplication equation below?
3170 x 6883
Think step by step. | 21819110 | To solve the long multiplication equation \( 3170 \times 6883 \), we can break it down into simpler, more manageable steps using the distributive property of multiplication over addition. Here's a step-by-step solution:
### Step 1: Break down the multiplication
We can express \( 3170 \times 6883 \) as:
\[ 3170 \times ... | true | The solution provided follows a step-by-step breakdown that uses the distributive property correctly, confirming the multiplication of 3170 by each segment of 6883 accurately. The final sum of the products is verified through two methods, both yielding the same result: 21,819,110. This matches the ground truth solution... | To solve the long multiplication equation \( 3170 \times 6883 \), we can break it down into simpler, more manageable steps using the distributive property of multiplication over addition. Here's a step-by-step solution:
### Step 1: Break down the multiplication
We can express \( 3170 \times 6883 \) as:
\[ 3170 \times ... | false | |
What is the solution to the long multiplication equation below?
7940 x 3761
Think step by step. | 29862340 | To solve the long multiplication equation \( 7940 \times 3761 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 7940 \) by each digit of \( 3761 \) (considering their place values) and then add the results together.
### Step 1: Break ... | true | The solution provided correctly breaks down the multiplication into manageable parts, carries out the multiplication accurately, and sums the results correctly, reaching the same answer as the ground truth solution (29862340). The representation of the numbers in the solution (with commas) does not affect the numerical... | To solve the long multiplication equation \( 7940 \times 3761 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 7940 \) by each digit of \( 3761 \) (considering their place values) and then add the results together.
### Step 1: Break ... | false | |
What is the solution to the long multiplication equation below?
8439 x 3868
Think step by step. | 32642052 | To solve the long multiplication equation \( 8439 \times 3868 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 8439 \) by each digit of \( 3868 \) (from right to left), shifting the partial results appropriately, and then add them tog... | true | The provided solution correctly employs the distributive property and verifies the result through standard long multiplication. Both methodologies yield the same final answer of 32,642,052, which matches the ground truth solution of 32642052 exactly. | To solve the long multiplication equation \( 8439 \times 3868 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 8439 \) by each digit of \( 3868 \) (from right to left), shifting the partial results appropriately, and then add them tog... | true | |
What is the solution to the long multiplication equation below?
3965 x 8023
Think step by step. | 31811195 | To solve the long multiplication equation \( 3965 \times 8023 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the first number by each digit of the second number, appropriately shifted, and then adding all the partial results.
#... | true | The solution provided correctly breaks down the multiplication step by step using the distributive property, calculating each partial product accurately. The final addition of the partial results correctly yields 31,811,195 which matches the format of the ground truth solution 31811195 when commas are disregarded. Ther... | To solve the long multiplication equation \( 3965 \times 8023 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the first number by each digit of the second number, appropriately shifted, and then adding all the partial results.
#... | false | |
What is the solution to the long multiplication equation below?
8151 x 9564
Think step by step. | 77956164 | To solve the long multiplication equation \( 8151 \times 9564 \), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (8151) by each digit of the multiplier (9564) and then summing the partial results.
### Step 1: Break down the mul... | true | The solution provided calculates the multiplication of 8151 and 9564 correctly using the distributive property and verifies the result through standard long multiplication. The final answer given as \( \boxed{77,956,164} \) matches the ground truth solution of 77956164 when formatted without commas. | To solve the long multiplication equation \( 8151 \times 9564 \), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (8151) by each digit of the multiplier (9564) and then summing the partial results.
### Step 1: Break down the mul... | true | |
What is the solution to the long multiplication equation below?
9467 x 7280
Think step by step. | 68919760 | To solve the long multiplication equation \( 9467 \times 7280 \), we can break it down into simpler, more manageable steps. Here's a detailed step-by-step solution:
### Step 1: Break down the multiplication
The number \( 7280 \) can be expressed as \( 7000 + 200 + 80 \). This allows us to multiply \( 9467 \) by each c... | true | The solution correctly follows the steps of breaking down the multiplication, computing each part accurately, and summing them correctly to arrive at the final result. The final answer of 68919760 matches the ground truth solution. | To solve the long multiplication equation \( 9467 \times 7280 \), we can break it down into simpler, more manageable steps. Here's a detailed step-by-step solution:
### Step 1: Break down the multiplication
The number \( 7280 \) can be expressed as \( 7000 + 200 + 80 \). This allows us to multiply \( 9467 \) by each c... | false | |
What is the solution to the long multiplication equation below?
2094 x 5907
Think step by step. | 12369258 | To solve the long multiplication equation \(2094 \times 5907\), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (2094) by each digit of the multiplier (5907), starting from the rightmost digit, and then summing up th... | true | The computed result of 12,369,258 from the solution is equal to the ground truth solution of 12369258, which confirms its correctness. | To solve the long multiplication equation \(2094 \times 5907\), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (2094) by each digit of the multiplier (5907), starting from the rightmost digit, and then summing up th... | true | |
What is the solution to the long multiplication equation below?
4203 x 4688
Think step by step. | 19703664 | To solve the long multiplication problem \(4203 \times 4688\), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \(4203\) by each digit of \(4688\) (considering their place values) and then add the results together.
### Step 1: Break down th... | true | The provided solution correctly carries out long multiplication step-by-step, using the distributive property accurately and summing the results correctly to arrive at the final answer of 19,703,664. This matches the ground truth solution. | To solve the long multiplication problem \(4203 \times 4688\), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \(4203\) by each digit of \(4688\) (considering their place values) and then add the results together.
### Step 1: Break down th... | false | |
What is the solution to the long multiplication equation below?
3048 x 2402
Think step by step. | 7321296 | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a multiplication of two large numbers: 3048 and 2402. The goal is to find the product of these two numbers.
### Breaking Down the Problem
Multiplying large numbers directly can be complex, so I recall the concept of **long mu... | true | The solution correctly follows the steps of breaking down the multiplication using long multiplication, correctly computes each component of the multiplication, sums them accurately, and verifies the answer through multiple methods, all leading to the correct final answer of 7,321,296. | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a multiplication of two large numbers: 3048 and 2402. The goal is to find the product of these two numbers.
### Breaking Down the Problem
Multiplying large numbers directly can be complex, so I recall the concept of **long mu... | true | |
What is the solution to the long multiplication equation below?
9070 x 8127
Think step by step. | 73711890 | To solve the long multiplication equation \(9070 \times 8127\), we can break it down into simpler, more manageable steps using the **distributive property of multiplication over addition**. Here's a detailed step-by-step solution:
### Step 1: Break down the multiplier (8127)
We can express \(8127\) as:
\[ 8127 = 8000 ... | true | The solution provided accurately breaks down the multiplication using the distributive property, calculates each component correctly, adds them together without errors, and confirms the total with long multiplication. It matches the ground truth solution when formatted correctly. | To solve the long multiplication equation \(9070 \times 8127\), we can break it down into simpler, more manageable steps using the **distributive property of multiplication over addition**. Here's a detailed step-by-step solution:
### Step 1: Break down the multiplier (8127)
We can express \(8127\) as:
\[ 8127 = 8000 ... | false | |
What is the solution to the long multiplication equation below?
9857 x 4502
Think step by step. | 44376214 | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a multiplication of two large numbers: 9857 and 4502. The goal is to find the product of these two numbers. Since both numbers are four-digit, the standard long multiplication method seems appropriate.
### Breaking Down the Mul... | true | The solution correctly applies long multiplication and verification techniques, resulting in the correct final product of 44,376,214. | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a multiplication of two large numbers: 9857 and 4502. The goal is to find the product of these two numbers. Since both numbers are four-digit, the standard long multiplication method seems appropriate.
### Breaking Down the Mul... | true | |
What is the solution to the long multiplication equation below?
3302 x 4938
Think step by step. | 16305276 | To solve the long multiplication equation \( 3302 \times 4938 \), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying each digit of the second number (4938) by each digit of the first number (3302) and then summing the results. Here's a step-by-step ... | true | The solution correctly applies the distributive property to break down the multiplication into manageable parts, calculates each multiplication step accurately, and correctly sums the partial products to arrive at the final answer. The final boxed answer matches the ground truth solution when formatted without commas. | To solve the long multiplication equation \( 3302 \times 4938 \), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying each digit of the second number (4938) by each digit of the first number (3302) and then summing the results. Here's a step-by-step ... | false | |
What is the solution to the long multiplication equation below?
2298 x 1579
Think step by step. | 3628542 | To solve the long multiplication equation \( 2298 \times 1579 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 2298 \) by each digit of \( 1579 \) (from right to left), shifting the partial results appropriately, and then add them tog... | true | The solution correctly follows the steps of long multiplication using the distributive property, and the final result matches the ground truth solution of 3628542. | To solve the long multiplication equation \( 2298 \times 1579 \), we'll break it down step by step using the **distributive property of multiplication over addition**. This means we'll multiply \( 2298 \) by each digit of \( 1579 \) (from right to left), shifting the partial results appropriately, and then add them tog... | true | |
What is the solution to the long multiplication equation below?
2924 x 7977
Think step by step. | 23324748 | To solve the long multiplication equation \(2924 \times 7977\), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (2924) by each digit of the multiplier (7977) and then summing the results. Here's a step-by-step solution:
### Step... | true | The solution correctly uses the distributive property of multiplication to break down the multiplication of 2924 and 7977 into manageable parts. Each component is multiplied accurately, and the partial results are summed correctly to yield the final answer of 23,324,748, which matches the ground truth solution. | To solve the long multiplication equation \(2924 \times 7977\), we'll break it down using the **distributive property of multiplication over addition**. This method involves multiplying the multiplicand (2924) by each digit of the multiplier (7977) and then summing the results. Here's a step-by-step solution:
### Step... | true | |
What is the solution to the long multiplication equation below?
3869 x 3758
Think step by step. | 14539702 | To solve the long multiplication problem \( 3869 \times 3758 \), we'll break it down using the **distributive property of multiplication over addition**. This means we'll multiply \( 3869 \) by each digit of \( 3758 \), accounting for place values, and then sum the results.
### Step 1: Break down the multiplication
Ex... | true | The computed solution aligns exactly with the ground truth solution. The breakdown and calculations for each component multiplication were performed accurately, and the final summation of the results yields the correct product of 3869 multiplied by 3758, which is 14539702. | To solve the long multiplication problem \( 3869 \times 3758 \), we'll break it down using the **distributive property of multiplication over addition**. This means we'll multiply \( 3869 \) by each digit of \( 3758 \), accounting for place values, and then sum the results.
### Step 1: Break down the multiplication
Ex... | false | |
What is the solution to the long multiplication equation below?
4474 x 9008
Think step by step. | 40301792 | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a multiplication of two large numbers: 4474 and 9008. The goal is to find the product of these two numbers.
### Breaking Down the Multiplication
Multiplying large numbers directly can be complex, so I recall the concept of **... | true | The provided solution has accurately calculated the product of 4474 and 9008 step by step, using long multiplication and verification methods. The final product matches the ground truth solution of 40301792, confirming its correctness. | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a multiplication of two large numbers: 4474 and 9008. The goal is to find the product of these two numbers.
### Breaking Down the Multiplication
Multiplying large numbers directly can be complex, so I recall the concept of **... | true | |
What is the solution to the long multiplication equation below?
1506 x 4037
Think step by step. | 6079722 | To solve the long multiplication equation \(1506 \times 4037\), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying each digit of the second number (4037) by the first number (1506) and then summing the results.
### Step 1: Break down t... | true | The provided solution correctly breaks down the multiplication of 1506 by 4037 using the distributive property, accurately calculates each component, and sums the partial results correctly to arrive at the final answer of 6079722, which matches the ground truth. | To solve the long multiplication equation \(1506 \times 4037\), we'll break it down step by step using the **distributive property of multiplication over addition**. This method involves multiplying each digit of the second number (4037) by the first number (1506) and then summing the results.
### Step 1: Break down t... | false | |
What is the solution to the long multiplication equation below?
8915 x 3999
Think step by step. | 35651085 | ### Understanding the Problem
First, I need to multiply two large numbers: **8915** and **3999**. At first glance, this seems complex because both numbers are four-digit numbers. However, I recall that breaking down the problem can make it more manageable.
### Breaking Down the Multiplier (3999)
The number **3999** ... | true | The solution clearly outlines the steps of multiplication and confirms the result 35651085 through both the distributive property and long multiplication methods. The final answer matches the ground truth solution. | ### Understanding the Problem
First, I need to multiply two large numbers: **8915** and **3999**. At first glance, this seems complex because both numbers are four-digit numbers. However, I recall that breaking down the problem can make it more manageable.
### Breaking Down the Multiplier (3999)
The number **3999** ... | true | |
What is the solution to the long multiplication equation below?
4629 x 7389
Think step by step. | 34203681 | To solve the multiplication \( 4629 \times 7389 \), we can use the **long multiplication** method. Here's a step-by-step breakdown:
### Step 1: Break down the multiplication using the distributive property
We can express \( 7389 \) as \( 7000 + 300 + 80 + 9 \), and then multiply \( 4629 \) by each of these components:... | true | The presented solution correctly performs long multiplication of 4629 and 7389, breaking it down using the distributive property and summing partial products accurately. The final result is presented in both steps and as a final answer. The spacing and commas in the answer do not affect its correctness; it matches the ... | To solve the multiplication \( 4629 \times 7389 \), we can use the **long multiplication** method. Here's a step-by-step breakdown:
### Step 1: Break down the multiplication using the distributive property
We can express \( 7389 \) as \( 7000 + 300 + 80 + 9 \), and then multiply \( 4629 \) by each of these components:... | true | |
What is the solution to the long multiplication equation below?
8365 x 7965
Think step by step. | 66627225 | To solve the long multiplication equation \( 8365 \times 7965 \), we can use the **distributive property of multiplication over addition** (also known as the **FOIL method**). This involves breaking down the multiplication into simpler, more manageable parts. Here's a step-by-step solution:
### Step 1: Break down the ... | true | The solution correctly applies the distributive property to calculate the product of 8365 and 7965. The intermediary calculations are accurate and lead to the correct final result, which matches the ground truth solution of 66627225. | To solve the long multiplication equation \( 8365 \times 7965 \), we can use the **distributive property of multiplication over addition** (also known as the **FOIL method**). This involves breaking down the multiplication into simpler, more manageable parts. Here's a step-by-step solution:
### Step 1: Break down the ... | false | |
What is the solution to the long multiplication equation below?
9659 x 2514
Think step by step. | 24282726 | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a straightforward multiplication of two large numbers: 9659 and 2514. The goal is to find the product of these two numbers.
### Breaking Down the Multiplication
Multiplying large numbers directly can be complex, so I recall t... | true | The solution provided is correct as it accurately performs the multiplication of 9659 by 2514, detailing each step of the process and confirming the final product is 24,282,726. This matches the ground truth solution. | ### Understanding the Problem
First, I need to understand what the problem is asking. It's a straightforward multiplication of two large numbers: 9659 and 2514. The goal is to find the product of these two numbers.
### Breaking Down the Multiplication
Multiplying large numbers directly can be complex, so I recall t... | true | |
What is the solution to the long multiplication equation below?
1834 x 5854
Think step by step. | 10736236 | ### Understanding the Problem
First, I need to multiply two large numbers: **1834** and **5854**. At first glance, this seems complex because both numbers are in the thousands. However, I recall that breaking the problem into smaller, more manageable parts can make it easier.
### Breaking Down the Multiplication
I ... | true | The provided solution correctly follows the long multiplication method, breaks down the multiplication into manageable steps, and accurately computes the product of 1834 and 5854, confirming the result as 10,736,236. | ### Understanding the Problem
First, I need to multiply two large numbers: **1834** and **5854**. At first glance, this seems complex because both numbers are in the thousands. However, I recall that breaking the problem into smaller, more manageable parts can make it easier.
### Breaking Down the Multiplication
I ... | true |
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