Xuan had 40 buttons. ²₅ were grey and the rest were gold and black. The ratio of the number of gold buttons to the number of black buttons was 1 : 3. Xuan decided to buy some more grey buttons to increase the number of grey buttons to half of the total number of buttons.

1. How many less gold buttons than grey buttons did he have at first?
2. How many more grey buttons did he have to buy?

Grey buttons	Gold buttons	Black buttons	Total
2x4	3x4
	1x3	3x3
8 u	3 u	9 u	40

	Grey buttons	Gold buttons	Black buttons
Before	8 u	3 u	9 u
Change	+ ?
After	12 u	12 u
Comparing grey, gold and black buttons in the end	1	1

(a)
The total number of gold buttons and black buttons is repeated. Make the total number of gold buttons and black buttons the same.  LCM of 3 and 4 is 12.

Total number of buttons at first
= 8 u + 3 u + 9 u
= 20 u

20 u = 40

1 u = 40 ÷ 20 = 2

Number of less gold buttons than grey buttons at first
= 8 u - 3 u
= 5 u
= 5 x 2
= 10

(b)
Number of more grey buttons that Xuan had to buy
= 12 u - 8 u
= 4 u
= 4 x 2
= 8

Answer(s): (a) 10; (b) 8