Question Everything changed
Today’s problem — free all day, every step. Every day has one.
The ratio of boys to girls in a chess club was 3 : 4. After 6 boys joined and 2 girls left, the ratio became 3 : 2. How many boys were in the club at first?
Answer
9 boys (and 12 girls at first).
Both amounts change, so no total and no gap stays fixed.
In this one. Scale both end amounts to 6 equal end-parts; on the aligned bars, cancel the shared 6 at-first parts and see that the remaining 6 at-first parts equal the 12 who joined plus the 6 needed to replace those who left.
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.
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