Julian has some twenty-cent coins and fifty-cent coins which amount to more than $50 but less than $60. The number of twenty-cent coins is ¹₉ of all the coins he has. When he exchanges some fifty-cent coins for twenty-cent coins, the ratio of the number of twenty-cent coins to fifty-cent coins he now has become 8 : 1.

1. What is the largest possible amount of money Julian has?
2. What is the total value of fifty-cent coins that has been exchanged for twenty-cent coins?

	Twenty-cent coins (1)	Fifty-cent coins (2)	Make b the same (2)x8 = (3)
Before	1 u	8 u	64 u
Change	+ 5 p	- 2 p	- 16 p
After	8 b	1 b	8 b

(a)
Total value of coins per set
= 1 x 20 + 8 x 50
= 20 + 400
= 420¢ 

$1 = 100¢ 
$60 = 6000¢ 

Number of sets of 420¢ 
= 6000 ÷ 420
= 14 r 120

Largest possible amount that Julian has
= 14 x 420
= 5880¢ 
= $58.80

(b)
Twenty-cent coins : Fifty-cent coins = 1 : 8

1 u + 5 p = 8 b --- (1)

8 u - 2 p = 1 b --- (2)

Make b the same.

(2)_x8 = (3)
64 u - 16 p = 8 b --- (3)

(3) = (1)
64 u - 16 p = 1 u + 5 p
64 u - 1 u = 5 p + 16 p
63 u = 21 p

21 p = 63 u

1 p = 63 u ÷ 21 = 3 u

Since the number of sets of 420¢ is 14, 1 u is equal to 14.

Number of fifty-cent coins that have been exchanged for twenty-cent coins
= 2 p
= 2 x 3 u
= 6 u
= 6 x 14
= 84

Total value of fifty-cent coins that have been exchanged for twenty-cent coins
= 84 x 50
= 4200¢
= $42

Answer(s): (a) $58.80; (b) $42