Al and Betsy each have some marbles; in particular, Al has 2 marbles. If the person who has more marbles

Question

(if there is such a person) gives (1/2) of the difference between the number of marbles that each currently has to the person who has less marbles, and then each discards (1/2) of the number of marbles which were donated, Betsy ends up with 12 marbles. How many marbles did Betsy have initially?

Answer 1

Unlike most problems of this type, this one is perhaps easier to solve from the end backwards!

That's because the amounts manipulated decrease going forward through the problem, and therefore you would have to deal with fractional amounts of a variable if you solved it "forwards", but only with integer amounts of a variable if you solve it "backwards".

So, let the variable "x" represent the number of marbles Betsy discards at the end. Then immediately before she discarded, she had (12 + x) marbles.

Next, consider what effect the initial donation had. If the donation was of (1/2) the difference in their initial marbles, then it must have equalized their marble holdings! So, before the final disposal, each had (12 + x) marbles.

Then, since the first donation was of 2x marbles (since the discard was (1/2) of the number donated), Betsy started with (12 + x + 2x) = (12 + 3x) marbles, and Al with (12 + x - 2x) = (12 - x) marbles.

Since Al started with 2 = (12 - x) marbles, x = 10 and Betsy started with (12 + 3*10) = 44 marbles.

But -- in the real world -- why would you want to discard your marbles? Why not instead use all of them to make up an interesting math game? Just sayin'.

-- Cheers, -- Mr. d.