Multiplication and Division Worksheets: Multiply by Parts (Distributive Property) - 20 problems with solutions in red.

The worksheet you've uploaded is titled "Multiplication and Division Worksheets: Multiply by parts (Distributive property)", and it focuses on using the distributive property of multiplication to solve problems. The distributive property states:

> $ a \times (b + c) = a \times b + a \times c $

This means we can break down one of the numbers into parts (usually tens and units), multiply each part separately, and then add the results.

Let’s go through a few examples step-by-step to show how the distributive property is applied, and verify the answers given in red.

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🔹 Example 1: $ 5 \times 105 $

Break 105 into $ 100 + 5 $

$$
5 \times 105 = 5 \times (100 + 5) = (5 \times 100) + (5 \times 5) = 500 + 25 = 525 \quad ✔
$$

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🔹 Example 2: $ 7 \times 104 $

Break 104 into $ 100 + 4 $

$$
7 \times 104 = 7 \times (100 + 4) = (7 \times 100) + (7 \times 4) = 700 + 28 = 728 \quad ✔
$$

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🔹 Example 3: $ 8 \times 99 $

Break 99 into $ 100 - 1 $

$$
8 \times 99 = 8 \times (100 - 1) = (8 \times 100) - (8 \times 1) = 800 - 8 = 792 \quad ✔
$$

Note: This uses the distributive property with subtraction.

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🔹 Example 4: $ 3 \times 90 $

This is straightforward:
$$
3 \times 90 = 270 \quad ✔
$$
(Or think: $ 3 \times 9 \times 10 = 27 \times 10 = 270 $)

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🔹 Example 5: $ 3 \times 107 $

Break 107 into $ 100 + 7 $

$$
3 \times 107 = 3 \times (100 + 7) = (3 \times 100) + (3 \times 7) = 300 + 21 = 321 \quad ✔
$$

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🔹 Example 6: $ 8 \times 103 $

Break 103 into $ 100 + 3 $

$$
8 \times 103 = 8 \times (100 + 3) = (8 \times 100) + (8 \times 3) = 800 + 24 = 824 \quad ✔
$$

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🔹 Example 7: $ 6 \times 98 $

Break 98 into $ 100 - 2 $

$$
6 \times 98 = 6 \times (100 - 2) = (6 \times 100) - (6 \times 2) = 600 - 12 = 588 \quad ✔
$$

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🔹 Example 8: $ 2 \times 95 $

Break 95 into $ 100 - 5 $

$$
2 \times 95 = 2 \times (100 - 5) = (2 \times 100) - (2 \times 5) = 200 - 10 = 190 \quad ✔
$$

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🔹 Example 9: $ 4 \times 93 $

Break 93 into $ 90 + 3 $ or $ 100 - 7 $. Let's use $ 90 + 3 $:

$$
4 \times 93 = 4 \times (90 + 3) = (4 \times 90) + (4 \times 3) = 360 + 12 = 372 \quad ✔
$$

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🔹 Example 10: $ 4 \times 102 $

Break 102 into $ 100 + 2 $

$$
4 \times 102 = 4 \times (100 + 2) = (4 \times 100) + (4 \times 2) = 400 + 8 = 408 \quad ✔
$$

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🔹 Example 11: $ 8 \times 93 $

Break 93 into $ 90 + 3 $

$$
8 \times 93 = 8 \times (90 + 3) = (8 \times 90) + (8 \times 3) = 720 + 24 = 744 \quad ✔
$$

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🔹 Example 12: $ 3 \times 98 $

Break 98 into $ 100 - 2 $

$$
3 \times 98 = 3 \times (100 - 2) = (3 \times 100) - (3 \times 2) = 300 - 6 = 294 \quad ✔
$$

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🔹 Example 13: $ 7 \times 106 $

Break 106 into $ 100 + 6 $

$$
7 \times 106 = 7 \times (100 + 6) = (7 \times 100) + (7 \times 6) = 700 + 42 = 742 \quad ✔
$$

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🔹 Example 14: $ 6 \times 91 $

Break 91 into $ 90 + 1 $

$$
6 \times 91 = 6 \times (90 + 1) = (6 \times 90) + (6 \times 1) = 540 + 6 = 546 \quad ✔
$$

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🔹 Example 15: $ 8 \times 92 $

Break 92 into $ 90 + 2 $

$$
8 \times 92 = 8 \times (90 + 2) = (8 \times 90) + (8 \times 2) = 720 + 16 = 736 \quad ✔
$$

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🔹 Example 16: $ 9 \times 95 $

Break 95 into $ 100 - 5 $

$$
9 \times 95 = 9 \times (100 - 5) = (9 \times 100) - (9 \times 5) = 900 - 45 = 855 \quad ✔
$$

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🔹 Example 17: $ 8 \times 102 $

Break 102 into $ 100 + 2 $

$$
8 \times 102 = 8 \times (100 + 2) = (8 \times 100) + (8 \times 2) = 800 + 16 = 816 \quad ✔
$$

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🔹 Example 18: $ 5 \times 91 $

Break 91 into $ 90 + 1 $

$$
5 \times 91 = 5 \times (90 + 1) = (5 \times 90) + (5 \times 1) = 450 + 5 = 455 \quad ✔
$$

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🔹 Example 19: $ 6 \times 105 $

Break 105 into $ 100 + 5 $

$$
6 \times 105 = 6 \times (100 + 5) = (6 \times 100) + (6 \times 5) = 600 + 30 = 630 \quad ✔
$$

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🔹 Example 20: $ 8 \times 109 $

Break 109 into $ 100 + 9 $

$$
8 \times 109 = 8 \times (100 + 9) = (8 \times 100) + (8 \times 9) = 800 + 72 = 872 \quad ✔
$$

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✔ All answers are correct!

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✔ Summary: How to Use the Distributive Property

To multiply a number by a two-digit number close to 100 (like 99, 105, etc.), follow these steps:

1. Break the larger number into parts (e.g., 100 + 5, 100 - 2).
2. Multiply each part by the smaller number.
3. Add or subtract the results.

This method simplifies mental math and helps build understanding of multiplication.

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