Distributive property

The idea

The multiple of a sum (difference) is the same as the sum (difference) of the multiples: multiplying a number by a bracket equals multiplying the number by each term in the bracket.

Lhamo and Norbu

Lhamo and Norbu each order a vegetable cutlet (₹43) and a rasgulla (₹24). Together they pay 2 × (43 + 24).

They could also count 2 cutlets and 2 rasgullas: 2 × 43 + 2 × 24. Will the total be the same?

• Yes, the same
• No, it will differ
• Only for some numbers

Count it two ways

Rows of dots are split into two groups. Count group by group, or count all the rows at once. Do the two counts ever disagree?

Same total, two ways

4 × 5 + 3 × 5 counts group by group. (4 + 3) × 5 counts all the rows at once. Both are 35.

The multiple of a sum (difference) is the same as the sum (difference) of the multiples. Multiplying a number by a bracket is the same as multiplying it by each term in the bracket. This is the distributive property.

Brackets still matter: 5 × 4 + 3 = 23, but 5 × (4 + 3) = 35.

• Group by group | All at once
• 4 × 5 + 3 × 5 | (4 + 3) × 5 = 35
• 98 × 10 + 98 × 3 | 98 × (10 + 3) = 1274
• 2 × 43 + 2 × 24 | 2 × (43 + 24) = 134
• 14 × 10 − 6 × 10 | (14 − 6) × 10 = 80

Notes

The multiple of a sum (difference) is the same as the sum (difference) of the multiples: multiplying a number by a bracket is the same as multiplying it by each term in the bracket. This is the distributive property.

Check yourself

Given 53 × 18 = 954, find 63 × 18.

Answer: 1134

63 × 18 = (53 + 10) × 18 = 53 × 18 + 10 × 18 = 954 + 180 = 1134.

Which is the same as 97 × 25?

Use (100 − 2) × 15 to find 98 × 15.

Answer: 1470

98 × 15 = 100 × 15 − 2 × 15 = 1500 − 30 = 1470.

Find 47 × 6 + 53 × 6 in one step.

Answer: 600

47 × 6 + 53 × 6 = (47 + 53) × 6 = 100 × 6 = 600.

• 100 × 25 − 3. That takes away only 3. You need to take away 3 groups of 25.
• 100 × 25 − 3 × 25 — correct. Yes! 97 = 100 − 3, so 97 × 25 = 2500 − 75 = 2425.
• 97 × 20 + 5. That adds only 5. 97 × 25 needs 5 more groups of 97.