Jeremy has a carton containing some bunches of red grapes and bunches of purple grapes. If he adds in 29 bunches of red grapes, 0.7 of the bunches of grapes in the carton will be bunches of purple grapes. If he adds in 46 bunches of purple grapes, ⁶₇ 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 purple grapes	Bunches of red grapes	Bunches of purple grapes	Bunches of red grapes
Before	7 u	3 u - 29	6 p - 46	1 p
Change		+ 29	+ 46
After	7 u	3 u	6 p	1 p

0.7 = ⁷₁₀ = ⁷₁₀

If he adds 29 bunches of red grapes,
Number of bunches of red grapes in the end
= 10 u - 7 u
= 3 u

If he adds 46 bunches of purple grapes,
Number of bunches of red grapes in the end
= 7 p - 6 p
= 1 p

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

7 u = 6 p - 46 --- (1)

3 u - 29 = 1 p
3 u = 1 p + 29 --- (2)

Make u the same.
(1)_x3
21 u = 18 p - 138 --- (3)

(2)_x7
21 u = 7 p + 203 --- (4)

(3) = (4)
18 p - 138 = 7 p + 203
18 p - 7 p = 138 + 203
11 p = 341

1 p = 341 ÷ 11 = 31

Number of bunches of grapes at first
= 1 p + 6 p - 46
= 7 p - 46
= 7 x 31 - 46
= 217 - 46
= 171

Answer(s): 171