TAUR-dev/v3_12d_eval

Dataset card for v3_12d_eval

This dataset was made with Curator.

Dataset details

A sample from the dataset:

{
    "question": "What is the solution to the long multiplication equation below?\n\n468474295138 x 549622370782\n\nThink step by step.",
    "solution": "257483952744173935857916",
    "eval_prompt": "What is the solution to the long multiplication equation below?\n\n468474295138 x 549622370782\n\nThink step by step.",
    "eval_internal_cot": "",
    "eval_solution": "### Understanding the Problem\n\nFirst, I need to understand what the problem is asking. We have two very large numbers:\n\n- **Multiplicand (M):** 468,474,295,138\n- **Multiplier (N):** 549,622,370,782\n\nThe goal is to find the product of these two numbers, i.e., M \u00d7 N.\n\n### Breaking Down the Problem\n\nMultiplying two 12-digit numbers directly seems daunting. However, I recall that there are methods to simplify such large multiplications. One common technique is the **long multiplication** method, which breaks down the problem into smaller, more manageable parts.\n\n### Long Multiplication Basics\n\nLong multiplication involves multiplying each digit of the second number (multiplier) by the first number (multiplicand) and then adding all the partial results, taking into account the positional values (i.e., powers of 10).\n\nBut multiplying a 12-digit number by another 12-digit number directly would result in 12 partial products, each with up to 24 digits. That's complex, but perhaps manageable if I take it step by step.\n\n### Step 1: Break Down the Multiplier\n\nTo simplify, I can break the multiplier (N) into its constituent digits and multiply M by each digit, shifted according to its place value.\n\nBut before diving into that, perhaps using the **distributive property of multiplication over addition** (also known as the FOIL method) would be more efficient. This means expressing N as a sum of numbers that are easier to multiply.\n\n### Step 2: Express N in Expanded Form\n\nLet's express N in terms of powers of 10:\n\nN = 549,622,370,782  \n= 500,000,000,000 (5 \u00d7 10^11)  \n+ 40,000,000,000 (4 \u00d7 10^10)  \n+ 9,000,000,000 (9 \u00d7 10^9)  \n+ 600,000,000 (6 \u00d7 10^8)  \n+ 20,000,000 (2 \u00d7 10^7)  \n+ 2,000,000 (2 \u00d7 10^6)  \n+ 300,000 (3 \u00d7 10^5)  \n+ 70,000 (7 \u00d7 10^4)  \n+ 0 (0 \u00d7 10^3)  \n+ 700 (7 \u00d7 10^2)  \n+ 80 (8 \u00d7 10^1)  \n+ 2 (2 \u00d7 10^0)\n\n### Step 3: Multiply M by Each Component of N\n\nNow, I can multiply M by each term in the expanded form of N and then sum all the partial results.\n\n#### Term 1: 5 \u00d7 10^11\nM \u00d7 500,000,000,000 = 468,474,295,138 \u00d7 5 \u00d7 10^11  \nFirst, multiply M by 5:  \n468,474,295,138 \u00d7 5 = 2,342,371,475,690  \nNow, multiply by 10^11 (add 11 zeros):  \n2,342,371,475,690 \u00d7 10^11 = 234,237,147,569,000,000,000,000\n\n#### Term 2: 4 \u00d7 10^10\nM \u00d7 40,000,000,000 = 468,474,295,138 \u00d7 4 \u00d7 10^10  \nFirst, multiply M by 4:  \n468,474,295,138 \u00d7 4 = 1,873,897,180,552  \nNow, multiply by 10^10:  \n1,873,897,180,552 \u00d7 10^10 = 18,738,971,805,520,000,000,000\n\n#### Term 3: 9 \u00d7 10^9\nM \u00d7 9,000,000,000 = 468,474,295,138 \u00d7 9 \u00d7 10^9  \nFirst, multiply M by 9:  \n468,474,295,138 \u00d7 9 = 4,216,268,656,242  \nNow, multiply by 10^9:  \n4,216,268,656,242 \u00d7 10^9 = 4,216,268,656,242,000,000,000\n\n#### Term 4: 6 \u00d7 10^8\nM \u00d7 600,000,000 = 468,474,295,138 \u00d7 6 \u00d7 10^8  \nFirst, multiply M by 6:  \n468,474,295,138 \u00d7 6 = 2,810,845,770,828  \nNow, multiply by 10^8:  \n2,810,845,770,828 \u00d7 10^8 = 281,084,577,082,800,000,000\n\n#### Term 5: 2 \u00d7 10^7\nM \u00d7 20,000,000 = 468,474,295,138 \u00d7 2 \u00d7 10^7  \nFirst, multiply M by 2:  \n468,474,295,138 \u00d7 2 = 936,948,590,276  \nNow, multiply by 10^7:  \n936,948,590,276 \u00d7 10^7 = 9,369,485,902,760,000,000\n\n#### Term 6: 2 \u00d7 10^6\nM \u00d7 2,000,000 = 468,474,295,138 \u00d7 2 \u00d7 10^6  \nFirst, multiply M by 2:  \nSame as above, 936,948,590,276  \nNow, multiply by 10^6:  \n936,948,590,276 \u00d7 10^6 = 936,948,590,276,000,000\n\n#### Term 7: 3 \u00d7 10^5\nM \u00d7 300,000 = 468,474,295,138 \u00d7 3 \u00d7 10^5  \nFirst, multiply M by 3:  \n468,474,295,138 \u00d7 3 = 1,405,422,885,414  \nNow, multiply by 10^5:  \n1,405,422,885,414 \u00d7 10^5 = 140,542,288,541,400,000\n\n#### Term 8: 7 \u00d7 10^4\nM \u00d7 70,000 = 468,474,295,138 \u00d7 7 \u00d7 10^4  \nFirst, multiply M by 7:  \n468,474,295,138 \u00d7 7 = 3,279,320,065,966  \nNow, multiply by 10^4:  \n3,279,320,065,966 \u00d7 10^4 = 32,793,200,659,660,000\n\n#### Term 9: 0 \u00d7 10^3\nM \u00d7 0 = 0  \nThis term contributes nothing to the final product.\n\n#### Term 10: 7 \u00d7 10^2\nM \u00d7 700 = 468,474,295,138 \u00d7 7 \u00d7 10^2  \nFirst, multiply M by 7:  \nSame as above, 3,279,320,065,966  \nNow, multiply by 10^2:  \n3,279,320,065,966 \u00d7 10^2 = 327,932,006,596,600\n\n#### Term 11: 8 \u00d7 10^1\nM \u00d7 80 = 468,474,295,138 \u00d7 8 \u00d7 10^1  \nFirst, multiply M by 8:  \n468,474,295,138 \u00d7 8 = 3,747,794,361,104  \nNow, multiply by 10^1:  \n3,747,794,361,104 \u00d7 10^1 = 37,477,943,611,040\n\n#### Term 12: 2 \u00d7 10^0\nM \u00d7 2 = 936,948,590,276  \nThis is the last term, so no multiplication by 10 needed.\n\n### Step 4: Sum All Partial Results\n\nNow, I need to add all the partial results obtained from each term. This is a massive addition, so I'll proceed step by step.\n\nLet's list all the partial results:\n\n1. 234,237,147,569,000,000,000,000\n2. + 18,738,971,805,520,000,000,000\n   = 252,976,119,374,520,000,000,000\n3. + 4,216,268,656,242,000,000,000\n   = 257,192,388,030,762,000,000,000\n4. + 281,084,577,082,800,000,000\n   = 257,473,472,607,844,800,000,000\n5. + 9,369,485,902,760,000,000\n   = 257,482,842,093,747,560,000,000\n6. + 936,948,590,276,000,000\n   = 257,483,779,042,337,836,000,000\n7. + 140,542,288,541,400,000\n   = 257,483,919,584,626,377,400,000\n8. + 32,793,200,659,660,000\n   = 257,483,952,377,827,037,060,000\n9. + 0\n   = No change\n10. + 327,932,006,596,600\n    = 257,483,952,705,759,043,656,600\n11. + 37,477,943,611,040\n    = 257,483,952,743,236,987,267,640\n12. + 936,948,590,276\n    = 257,483,952,744,173,935,857,916\n\n### Verification\n\nTo ensure accuracy, I can cross-verify using the **difference of squares** method, but given the complexity, I'll stick with the distributive method for now.\n\n### Final Answer\n\nAfter performing all the partial multiplications and additions, the product of the two numbers is:\n\n**257,483,952,744,173,935,857,916**",
    "raw_eval_prompt": "What is the solution to the long multiplication equation below?\n\n468474295138 x 549622370782\n\nThink step by step.",
    "judge_correct": true,
    "judge_reasoning": "The solution performed the multiplication correctly using the distributive property, and the final answer matches the ground truth solution exactly. Thus, it is accurate."
}

Loading the dataset

You can load this dataset using the following code:

from datasets import load_dataset

dataset = load_dataset("TAUR-dev/v3_12d_eval")