📚 Working backwards through the library

A closer look at one question · SMC 2024, Question 31 (bank id C6)

A library had some storybooks. Betty added 17 and removed 36. Then she received 3 TIMES as many new books as were left on the shelves. She arranged all the books equally between 2 sections, and each section ended up with 458 books. How many books were in the library at first?

1 This question is easier read from the END backwards

🏁 End: 2 sections × 458 = 916 books

↑ undo "received 3× as many again"

Shelves held 229 (before the 3× arrived)

↑ undo "+17 then −36"

🚩 Start: unknown number of books

2 "3 times as many AGAIN" means 4 equal shares, not 3

If the shelves held 1 share and Betty received 3 times as many again, the total became 1 + 3 = 4 shares.

916 books
1 share1 share1 share1 share

One share = 916 ÷ 4 = 229 — that is what the shelves held just before the new books arrived.

3 Undo the +17 and −36

Adding 17 then removing 36 is a NET LOSS of 36 − 17 = 19 books. To undo a loss, add it back: 229 + 19 = 248.

248 books

were in the library at first.

The trap: reading "3 times as many new books as what were left" as making the total 3 TIMES bigger. It ADDS 3 shares to the 1 already there, giving 4 shares in all.