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Storybooks packed into boxes of 5 and 7
A librarian packs storybooks into boxes of 5, with 3 books left over. If she packs them into boxes of 7 instead, she is short of 5 books for the last box. How many storybooks does she have?
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23 storybooks (in 4 boxes).
The pattern behind it: Gap and difference
Share the same things two ways. One way leaves too many, the other too few.
In this one. The number of boxes is the same both ways, so the swing from 3 spare to 5 short is a gap of 3 + 5 = 8 books — and that gap is built 2 books at a time, one box at a time (7 − 5 = 2), so there are 8 ÷ 2 = 4 boxes.
How to spot it
Objects are grouped two different ways with a leftover in one arrangement and a shortage in the other, using words like 'left over' and 'short of'.
Common mistake
Children combine the leftover and the shortfall in the wrong direction. The gap must then be divided by the difference in group size, not by either group size.
Try one like it
Challenging Pencils — boxes of 6 leave some over, boxes of 8 fall short A teacher packs pencils into boxes of 6 and has 4 pencils left over. If she uses boxes of 8, she is short of 6 pencils. How many p…The 33 problem patterns
All 33 patterns, one to a page — the 19 behind Primary 5 and 6 word problems, and the skills the younger years are built on. The bar model, how to spot it in the question, and the mistake children make. A booklet you can print. Free.
A PDF, and a note when new solutions go up.