Question P6 Everything changed
Today’s problem — free all day, every step. Every day has one.
The ratio of Ali’s money to Billy’s money was 2 : 1. After Ali saved another $60 and Billy spent $150, the ratio became 4 : 1. How much money did Ali have at first?
Answer
Ali had $660 at first. (Billy had $330.)
Both amounts change, so no total and no gap stays fixed.
In this one. Ali's money in the end has two names — 2 units + $60 (from the ratio at first) and 4 units − $600 (because it is 4 copies of Billy's end amount, 1 unit − $150); setting the two names side by side and cancelling the units they share turns the picture into a number.
Both people in a ratio change by different amounts, with words like 'joined' or 'saved more' for one and 'left' or 'spent' for the other.
Children look for a fixed total or a fixed gap out of habit. Neither is fixed here, so only the ratio at the end can give the size of a share.
Worked examples of all 33 patterns and skills, in a booklet you can print.