The second partial product in a two-digit multiplication is generally larger because it represents multiplication by the tens digit, which is inherently a greater value than the ones digit used in the first partial product.
Here's a breakdown:
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Understanding Partial Products: When multiplying a number by a two-digit number, you break down the two-digit number into its tens and ones place values. You then multiply the original number by each of these place values separately. These individual multiplication results are called partial products.
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Place Value Matters: Consider multiplying 123 by 45.
- The first partial product comes from multiplying 123 by 5 (the ones digit).
- The second partial product comes from multiplying 123 by 40 (the tens digit).
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The Tens Digit Advantage: Since the tens digit is always a multiple of 10 (e.g., 10, 20, 30, 40, etc.), multiplying by it is the same as multiplying by the ones digit and then by 10. Multiplying by 10 effectively shifts the number one place value higher, thereby increasing its overall value.
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Example: In the 123 x 45 example:
- 123 x 5 = 615 (First partial product)
- 123 x 40 = 4920 (Second partial product)
Clearly, 4920 > 615 because 40 is significantly larger than 5.
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Why "Generally" Larger? The statement holds true unless the tens digit is zero. If you multiply by a number like 05, the second partial product will be zero and therefore smaller than the first.
In short, the second partial product, being a result of multiplication by a value in the tens place (or higher), will almost always be larger than the first partial product, which is a result of multiplication by a value in the ones place.
