Cathy had some purple and yellow marbles. If she gives away 42 purple marbles, the number of purple marbles will be 20% of the total number of marbles left. If she gives away 12 yellow marbles, the number of purple marbles will be 25% of the total number of marbles left.

1. How many marbles does Cathy have altogether?
2. What fraction of Cathy's marbles is yellow?

	Scenario 1		Scenario 2
	Purple	Yellow	Purple	Yellow
Before	1 u + 42	4 u	1 p	3 p + 12
Change	- 42	No change	No change	- 12
After	1 u	4 u	1 p	3 p

(a)
20% = ²⁰₁₀₀ = ¹₅

25% = ²⁵₁₀₀ = ¹₄

1 u + 42 = 1 p --- (1)

4 u = 3 p + 12
4 u - 12 = 3 p --- (2)

Make p the same.

(1)_x3
3 u + 126 = 3 p --- (3) 

(2) = (3)
4 u - 12 = 3 u + 126 
4 u - 3 u = 126 + 12

1 u = 138

Number of marbles that Cathy had at first
= 1 u + 42 + 4 u
= 5 u + 42
= 5 x 138 + 42
= 690 + 42
= 732

(b)
Number of yellow marbles
= 4 u
= 4 x 138
= 552

Fraction of Cathy's marbles that is yellow 
= ⁵⁵²₇₃₂
= ⁴⁶₆₁

Answer(s): (a) 732; (b) ⁴⁶₆₁