The multiple of a sum

Lhamo and Norbu each order a vegetable cutlet (₹43) and a rasgulla (₹24). What do they pay?

Each share is 43 + 24. Is the total 2 × 43 + 24? Look at its terms: 2 × 43 and 24. That means 24 more than twice 43. Not what we want! We want twice the whole share: 2 × (43 + 24).

Count it another way: two cutlets and two rasgullas: 2 × 43 + 2 × 24. Same money, so 2 × (43 + 24) = 2 × 43 + 2 × 24.

At the Republic Day parade, boy scouts march in 4 rows of 5, girl guides in 3 rows of 5. Total: 4 × 5 + 3 × 5 = 20 + 15 = 35. Or count the rows first: 4 + 3 = 7 rows of 5: (4 + 3) × 5 = 35. So 4 × 5 + 3 × 5 = (4 + 3) × 5. (But 5 × 4 + 3 is not 5 × (4 + 3): brackets matter.)

Why it always works: 10 × 98 + 3 × 98 is 98 taken 10 times, then 3 more times: 98 taken 13 times = (10 + 3) × 98. It works for differences too: 14 × 10 − 6 × 10 = (14 − 6) × 10.

In one line: the multiple of a sum (or difference) is the sum (or difference) of the multiples. a × (b + c) = a × b + a × c.