In April, Gwen and David had buttons in the ratio 13 : 8. In May, each of them gave away the same number of buttons. Gwen then had twice as many buttons as David.

1. Find the ratio of the number of buttons David had in April to the number of buttons he had in May.
2. Find the ratio of the total number of buttons both had in April to the total number of buttons both had in May.

	Gwen	David	Difference
Before	13x1 = 13 u	8x1 = 8 u	5x1 = 5 u
Change	- 3 u	- 3 u
After	2x5 = 10 u	1x5 = 5 u	1x5 = 5 u

(a)
Since Gwen and David gave away the same number of buttons, the difference in the number of buttons between Gwen and David at first and in the end remains the same.

Make the difference in the number of buttons at first and in the end the same. LCM of 5 and 1 is 5.

David's buttons in April : David's buttons in May
8 : 5

(b)
Total number of buttons in April
= 13 u + 8 u
= 21 u

Total number of buttons in May
= 10 u + 5 u
= 15 u

Total buttons in April : Total buttons in May
21 : 15 _(÷3)
7 : 5 

Answer(s): (a) 8 : 5; (b) 7 : 5