Anna and Ben each have a collection of marbles. If Anna gives one marble to Ben, they will both have an equal number of marbles. Conversely, if Ben gives one marble to Anna, she will end up with three times the number of marbles that Ben has. What is the total number of marbles they both have?

Correct — A. Let Anna hold A marbles and Ben hold B.

Anna gives one away, so both counts move: A − 1 = B + 1, which gives A = B + 2.

Ben gives one instead: A + 1 = 3(B − 1)
(B + 2) + 1 = 3B − 3
B + 3 = 3B − 3
2B = 6, so B = 3 and A = 5

Check: Anna gives one → 4 and 4, equal. Ben gives one → 6 and 2, and 6 = 3 × 2.

Total = 5 + 3 = 8 → option (a).

Why the others are wrong

• (b)
  11 — 11 is odd. The first condition forces A = B + 2, so the total is 2B + 2 — always even. Splitting 11 with a gap of 2 gives A = 6.5.
• (c)
  9 — 9 fails the same way: A + B = 9 with A − B = 2 gives A = 5.5, and marbles come whole.
• (d)
  10 — 10 does pass the first test, as A = 6 and B = 4. It fails the second — Ben gives one and Anna has 7 against Ben's 3, but 3 × 3 = 9, not 7.

Two sentences, two unknowns, and the whole difficulty is in the translation.

A transfer changes both counts at once. "Anna gives one to Ben" is A − 1 on one side and B + 1 on the other, in the same equation — not A − 1 = B.

The second sentence describes the state after its own transfer, so the multiple applies to the new counts: A + 1 = 3(B − 1), not A = 3B.

Substituting the first result into the second leaves one linear equation, and it solves in a line.

Key facts

• The first condition, A − 1 = B + 1, means Anna has exactly 2 more marbles than Ben.
• The second condition, after Ben's transfer, is A + 1 = 3(B − 1).
• The only pair satisfying both is Anna 5 and Ben 3.
• Any total consistent with the first condition alone must be even, since it equals 2B + 2.

Common traps

• Writing A − 1 = B, forgetting that the marble arrives on Ben's side.
• Writing A = 3(B − 1) for the second condition, forgetting the marble Anna receives.
• Solving correctly to A = 5 and marking that instead of the total.

No directly related past PYQ was found.