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?

Question

Accepted Answer

Let the number of marbles Anna has be A and Ben has be B.

1. If Anna gives one marble to Ben, then Anna has A - 1 marbles and Ben has B + 1 marbles. They have equal marbles:
   A - 1 = B + 1
   => A - B = 2  ...(i)

2. If Ben gives one marble to Anna, then Anna has A + 1 marbles and Ben has B - 1 marbles. Anna has three times the marbles of Ben:
   A + 1 = 3(B - 1)
   => A + 1 = 3B - 3
   => A - 3B = -4  ...(ii)

From (i): A = B + 2
Substitute in (ii):
   (B + 2) - 3B = -4
   B + 2 - 3B = -4
   -2B + 2 = -4
   -2B = -6
   B = 3

Then A = B + 2 = 3 + 2 = 5

Total marbles = A + B = 5 + 3 = 8

But 8 is not the correct answer according to the options. Let's verify the second condition with these values:

If Ben gives one marble to Anna:
Anna = 5 + 1 = 6
Ben = 3 - 1 = 2
Is Anna's marbles three times Ben's? 6 = 3 * 2, yes.

If Anna gives one marble to Ben:
Anna = 5 - 1 = 4
Ben = 3 + 1 = 4
Equal marbles, yes.

So total is 8, which corresponds to option 2.

Hence, the total number of marbles is 8.

Answer

Answer

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