We Distribute Yet Things Multiply: Thinking Puzzles

The distributive law splits a hard product into easy ones. The same idea counts fares, schedules and seats step by step.

Key ideas

a × (b + c) = a × b + a × c, so 98 × 15 = 100 × 15 − 2 × 15.
Products of numbers just above and below a round number use (a + b)(a − b) = a² − b².
A common factor can be taken out of a long sum before adding.
For a train, the passengers on each stretch are those on board after people get off and on at each stop.
For a work schedule, count the days each worker works first, then multiply by what they do each day.
An area model shows (r + k)(c + m) as four rectangles added together.

Remember

Distributive law	a(b + c) = ab + ac; a(b − c) = ab − ac
Example	98 × 15 = 1,500 − 30 = 1,470
Difference of squares	(a + b)(a − b) = a² − b²: 53 × 47 = 2,500 − 9 = 2,491
Square of a sum	(a + b)² = a² + 2ab + b²: 52² = 2,704
Shortcut	(x + y)² − (x − y)² = 4xy
Common factor	m + 2m + … + nm = m × n(n + 1) ÷ 2
Two brackets	(r + k)(c + m) = rc + rm + kc + km

Watch out

Forgetting the middle term: (a + b)² is not a² + b².
Charging the fare of a stretch to people who got off before it.
Multiplying only the first number inside the bracket.

Worked example

A train runs from P to S, stopping at Q and R. 60 passengers board at P. At Q, 30 get off and 45 get on. At R, 30 get off and 10 get on. Everyone left gets off at S. A ticket costs ₹25 from P to Q, ₹20 from Q to R and ₹20 from R to S, for each stretch travelled. How many passengers travel (a) from Q to R (b) from R to S?

Working: 60 − … at each station: subtract those who get off, add those who get on

Answer: (a) 75; (b) 55