Decomposition: Breaking Problems Down

Decomposition breaks a big problem into smaller parts that are easier to solve, then puts the answers together.

Key ideas

Split a big task into sub-problems, solve each one, then combine them, like planning a picnic: place, food and transport.
In a “Who am I?” puzzle, each clue removes the numbers that do not fit until only one is left.
When things are passed between friends, the total they have together stays the same: a quick check on your working.
For the fewest coins, use as many of the biggest coin as possible, then the next biggest, and so on.
Comparison clues link into a chain: if A is taller than B and B is taller than C, then A is taller than C.
Elimination crosses out what a clue rules out; often the last item left must be the answer.

Remember

Fewest coins for ₹37 (₹10, ₹5, ₹2, ₹1)	3 × ₹10 + 1 × ₹5 + 1 × ₹2 = 5 coins
Multiples of a up to N	whole part of N ÷ a
Multiples of a but not of b	multiples of a − multiples of both (both = multiples of the LCM)
Example	multiples of 4 but not 6 up to 60: 15 − 5 = 10
Reading a table	find the row and the column together

Watch out

Reading the clues out of order, or skipping one, and ending with more than one answer.
Changing only one person’s amount in a transfer: the giver loses what the receiver gains.
Counting multiples of both numbers twice, or forgetting to take them away.

Worked example

(a) Priya, Zoya, Isha each have a different one of: guitar, drum, flute. Priya does not have the guitar. Isha has the guitar. Zoya does not have the drum. Who has the flute?
(b) How many numbers from 1 to 70 are multiples of 5 but NOT multiples of 2?

Working: (a) only one way fits every clue: Priya – drum, Zoya – flute, Isha – guitar; (b) 14 multiples of 5, of which 7 are also multiples of 2 (multiples of 10)

Answer: (a) Zoya; (b) 7