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My solution was slightly different but took the same amount of time:
ReplyDeleteStart with
1 / [1 - (1/x)] = 1 + 1/3
Immediately, you see that
1 + 1/3 = 4/3
Multiply the left side by 4/4 to get:
4 / [4 - (4/x)] = 4/3
So, in comparing numerators, we can see that
4 = 4
Which means that, for the denominators,
4 - (4/x) = 3
There are a number of ways to go from here. I multiplied both sides by x to get rid of x in the denominator:
4x - 4 = 3x
Then just solve for x:
x = 4
QED.