Jeremy has a carton containing some bunches of purple grapes and bunches of green grapes. If he adds in 37 bunches of purple grapes, 0.7 of the bunches of grapes in the carton will be bunches of green grapes. If he adds in 53 bunches of green grapes, ³₄ of the bunches of grapes in the carton will be green bunches of grapes. How many bunches of grapes are there in the carton?

	Bunches of green grapes	Bunches of purple grapes	Bunches of green grapes	Bunches of purple grapes
Before	7 u	3 u - 37	3 p - 53	1 p
Change		+ 37	+ 53
After	7 u	3 u	3 p	1 p

0.7 = ⁷₁₀ = ⁷₁₀

If he adds 37 bunches of purple grapes,
Number of bunches of purple grapes in the end
= 10 u - 7 u
= 3 u

If he adds 53 bunches of green grapes,
Number of bunches of purple grapes in the end
= 4 p - 3 p
= 1 p

The number of each type in both scenarios remain unchanged at first.

7 u = 3 p - 53 --- (1)

3 u - 37 = 1 p
3 u = 1 p + 37 --- (2)

Make u the same.
(1)_x3
21 u = 9 p - 159 --- (3)

(2)_x7
21 u = 7 p + 259 --- (4)

(3) = (4)
9 p - 159 = 7 p + 259
9 p - 7 p = 159 + 259
2 p = 418

1 p = 418 ÷ 2 = 209

Number of bunches of grapes at first
= 1 p + 3 p - 53
= 4 p - 53
= 4 x 209 - 53
= 836 - 53
= 783

Answer(s): 783