Inverse ratio problems are challenging and require careful teaching.

Using Number Properties

Problem: “Jane, Julie, and Atanga plan to paint a house in 24 days. Jane gets sick, so Julie and Atanga have to paint the house without her. How long will it take the pair of them to do the painting?”

Discuss how long 1 person would take to do the job alone.
(Answer: If 3 people take 24 days, 1 person takes 3 times as long = 3 x 24 = 72 days.)

Discuss how long 2 people would take.
(Answer: 72 ÷ 2 = 36.)

Examples: Worksheet (Material Master 8–30).

Understanding Number Properties:

It takes z painters y days to paint a house. How long will it take v painters? (Answer: zy/v days.)

Hard Problem: “3 friends paint 231 bicycles in 7 days. How long will it take 8 friends to paint 616 bicycles?”
(Answer: 3 _ 7 = 21. So 21 “friend days” of work produces 231 bicycles. So 1 “friend  day” of work produces 231 ÷ 21 = 11 painted bicycles. So 8 friends will paint 88  bicycles per day. So 616 bikes will take 616 ÷ 88 = 7 days.)

Understanding Number Properties (hard):

It takes z painters y days to paint n  houses. How long will it take v painters to paint r houses? (Answer: zyr/vn days)