Linda had 280 pencils. ²₇ were purple and the rest were grey and gold. The ratio of the number of grey pencils to the number of gold pencils was 1 : 3. Linda decided to buy some more purple pencils to increase the number of purple pencils to half of the total number of pencils.

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

Purple pencils	Grey pencils	Gold pencils	Total
2x4	5x4
	1x5	3x5
8 u	5 u	15 u	280

	Purple pencils	Grey pencils	Gold pencils
Before	8 u	5 u	15 u
Change	+ ?
After	20 u	20 u
Comparing purple, grey and gold pencils in the end	1	1

(a)
The total number of grey pencils and gold pencils is repeated. Make the total number of grey pencils and gold pencils the same.  LCM of 5 and 4 is 20.

Total number of pencils at first
= 8 u + 5 u + 15 u
= 28 u

28 u = 280

1 u = 280 ÷ 28 = 10

Number of less grey pencils than purple pencils at first
= 8 u - 5 u
= 3 u
= 3 x 10
= 30

(b)
Number of more purple pencils that Linda had to buy
= 20 u - 8 u
= 12 u
= 12 x 10
= 120

Answer(s): (a) 30; (b) 120