Paul had ³₄ as many stamps as Ivan. Ivan had ⁴₇ as many stamps as Elyse. After Paul and Elyse gave 10 stamps to Ivan, Ivan had thrice as many stamps as Paul. Elyse then had the same number of stamps as Ivan.

1. What was the ratio of the number of stamps Elyse had to the number of stamps Paul had in the beginning? Answer in simplest form.
2. How many stamps did Ivan have in the end?

Paul	Ivan	Elyse
3	4
	4	7
3	4	7

(a)
The number of stamps that Ivan is repeated.  

Ratio of the number of stamps that Elyse had to the number of stamps that Paul had in the beginning = 7 : 3

	Paul	Ivan	Elyse	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 10	+ 20	- 10
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

The total number of stamps at first and in the end remains unchanged. Make the total number of stamps the same. LCM of 14 and 7 is 14.

Number of stamps that Ivan received from Paul and Elyse
= 2 x 10
= 20

Number of stamps that Ivan received from Paul and Elyse
= 2 x 1 u
= 2 u

2 u = 10

1 u = 10 ÷ 2 = 5

Number of stamps that Ashley had in the end
= 6 u
= 6 x 5
= 30