Leon and Henian had a total of 468 sweets. Leon gave 1/4 of his sweets to Henian. Henian then gave 1/3 of his sweets to Leon. In the end, each of them had the same number of sweets. How many sweets did Leon have at first?
Let's start by letting L be the number of sweets Leon had at first, and H be the number of sweets Henian had at first. We know that:
L + H = 468
We also know that Leon gave 1/4 of his sweets to Henian, so he gave away 1/4L. This means that he had 3/4L left. Henian now has H + 1/4L sweets.
Next, Henian gave 1/3 of his sweets to Leon. This means he gave away (1/3)(H + 1/4L), and he now has 2/3(H + 1/4L) left. Leon now has 3/4L + (1/3)(H + 1/4L) sweets.
We also know that in the end, they both had the same number of sweets, so:
3/4L + (1/3)(H + 1/4L) = 2/3(H + 1/4L)
Multiplying both sides by 12 (the least common multiple of 4 and 3) to get rid of the fractions, we get:
9L + 4H = 561