Today’s problem — free all day, every step. Every day has one.
Mei had some savings. She spent of it on a novel and of the remainder on a backpack. After that, her grandmother gave her $160. The ratio of the total amount she had at the end to the amount she had at first was 5 : 4. How much did Mei have at first?
Answer
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The second fraction is of one piece, not of the whole.
In this one. Cut the starting amount into 20 units — 4 × 5 — so that the quarter, the fifths of the remainder, and the 5 : 4 ratio all land on whole boxes; the $160 gift then shows up as a clean gap of 16 units.
A second fraction arrives after the first and is attached to a piece rather than to the whole — 'of the remainder', 'of what was left', or 'of the ones that were red'.
Children take the second fraction of the original total instead of the piece it was named against, which makes the amount come out too big.
Worked examples of all 33 patterns and skills, in a booklet you can print.