A library has two shelves, X and Y. If 12 books are shifted from shelf Y to shelf X, the numbers of books on the two shelves are interchanged. If instead 6 books are shifted from shelf X to shelf Y, shelf Y then holds three times as many books as shelf X. How many books are on X and Y respectively?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 18 and 30
Moving 12 from shelf Y to shelf X swaps the counts only if shelf Y has 12 more than shelf X. The second move then gives one equation.
Let shelf X have \(a\); then shelf Y has \(a+12\)
After 6 are moved from shelf X to shelf Y: shelf X has \(a-6\), shelf Y has \(a+18\)
\(a+18=3(a-6)\), so \(a+18=3 a-18\)
\(a=18\), and shelf Y \(=18+12=30\)
Check: moving 6 gives 12 and 36, and 36 is three times 12
Shelf X has 18 and shelf Y has 30 books.
Why the other options are incorrect
Option (1) 23 and 35 — Incorrect. Moving 6 from shelf X to shelf Y gives 17 and 41, and 41 is not three times 17.
Option (2) 30 and 18 — Incorrect. Moving 12 from shelf Y to shelf X gives 42 and 6, not interchanged.
Option (3) 28 and 40 — Incorrect. Moving 6 from shelf X to shelf Y gives 22 and 46, and 46 is not three times 22.