## Multiply by Two-Digit Numbers

When multiplying a one-digit number by a multiple of 10, 100, or 1,000, the Associative Property makes it easier to complete the multiplication mentally.

Examples:

3 × 80 = 3 × (8 × 10)
= (3 × 8) × 10
= 24 × 10
= 240
9 × 200 = 9 × (2 × 100)
= (9 × 2) × 100
= 18 × 100
= 1,800
5 × 6,000 = 5 × (6 × 1,000)
= (5 × 6) × 1,000
= 30 × 1,000
= 30,000

Such multiplications can be extended to include multiplying a multiple of 10 by a multiple of 10 or 100. In this process both the Associative and Commutative properties are used.

Examples:

30 × 80 = 30 × (8 × 10)
= (3 × 10) × (8 × 10)
= 3 × (10 × 8) × 10
= 3 × (8 × 10) × 10
= (3 × 8) × (10 × 10)
= 24 × 100
= 2,400
90 × 200 = 90 × (2 × 100)
= (9 × 10) × (2 × 100)
= 9 × (10 × 2) × 100
= 9 × (2 × 10) × 100
= (9 × 2) × (10 × 100)
= 18 × 1,000
= 18,000

These products can be found by first multiplying the leading digits and then writing the total number of zeros in the factors after that product.

Examples:

3 × 800

Think: 3 × 8 = 24. There are 2 zeros.
Write: 2,400
90 × 200

Think: 9 × 2 = 18. There are 3 zeros.
Write: 18,000

Estimating Products
When estimating products involving a one-digit factor and a multidigit factor, round the multidigit factor to the nearest multiple of 10, 100, or 1,000 with only one nonzero digit. Then multiply.

Examples:

Estimate
Round:
Multiply:
7 × 8,253.
7 × 8,000
(7 × 8) × 1,000
56,000
Estimate
Round:
Multiply:
5 × \$38.10.
5 × \$40
(5 × \$4) × 10
\$200

Teaching Model 7.2: Estimate Products