A bag contained gold, white and red pencils. ⁴₅ of the pencils were gold and ⁴₅ of the remainder were white pencils.The rest were red pencils.

1. Find the ratio of the number of red pencils to the number of gold pencils.
2. There were 176 fewer white pencils than gold pencils. How many pencils were there in the bag?

Gold pencils	White pencils	Red pencils
4x5	1x5
	4x1	1x1
20 u	4 u	1 u

(a)
Fraction of gold pencils = ⁴₅
Fraction of pencils that are not gold = 1 - ⁴₅ = ¹₅

Remaining fraction of white pencils = ⁴₅
Remaining fraction of red pencils = 1 - ⁴₅ = ¹₅

The total number of white pencils and red pencils is the repeated identity.
Make the repeated identity the same by using LCM.
LCM of 1 and 5 = 5

Red pencils : Gold pencils
1 : 20

(b)
Difference in the number of gold pencils and white pencils
= 20 u - 4 u
= 16 u

16 u = 176
1 u = 176 ÷ 16 = 11

Total number of pencils
= 20 u + 4 u + 1 u
= 25 u
= 25 x 11
= 275

Answer(s): (a) 1 : 20; (b) 275