Numsprite數靈精Play free
中文

Maths olympiad question · P5–P6★★★★ Expert

Sum and multiple: olympiad question 7

An expert question on sum and multiple for P5–P6, with a step-by-step solution, diagrams and the answer. Sum-and-multiple and difference-and-multiple questions appear in the Hua Xia Cup's P3 to P5 first rounds, qualifying rounds and finals.

The question

Classes A, B and C each have some books in their reading corners.

Each of the three loans below starts from the books the classes have now: if class A lends class B 17 books, B will have 3 times as many as A; if B lends C 15 books, C will have twice as many as B; if C lends A 15 books, A will have twice as many as C.

How many books does class A have now?

Show the hints (3)
  1. Hint 1: Call what class A has left after lending 17 one part. How many parts does B have after receiving them — and how many before?
  2. Hint 2: Keep going: write B's and C's books as "so many parts ± so many books", then look at A and C after C lends 15.
  3. Hint 3: After lending 15, C has 6 parts less 94 books; 2 times that equals A's 1 part plus 32. Compare: 11 parts = 220 books.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Each loan starts afresh from the books now. Say A has 1 part left after lending 17: A now = 1 part + 17.
  2. 2.With 17 more, B is 3 × A, 3 parts; so B now = 3 parts − 17.
  3. 3.B lending 15 leaves 3 parts − 32; C with 15 more is 2 times that, 6 parts − 64. So C now = 6 parts − 79.
  4. 4.C lending 15 leaves 6 parts − 94; A with 15 more has 1 part + 32, which is 2 times that: 1 part + 32 = 12 parts − 188.
  5. 5.Compare: 12 − 1 = 11 parts = 32 + 188 = 220 books, so 1 part = 220 ÷ 11 = 20.
  6. 6.Class A has 20 + 17 = 37 books.
  7. 7.Check: B has 3 × 20 − 17 = 43, C has 41. A lends 17: 20 and 60 = 3 × 20 ✓; B lends 15: 28 and 56 ✓; C lends 15: 52 = 2 × 26 ✓
  8. ⚠️Common mistake: treating the loans as happening one after another. Each starts from the books now. Another: mixing up "after receiving" with "now" and adding where you should take away.

Answer: 37 books

Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download