Howard has some ten-cent coins and fifty-cent coins which amount to more than $40 but less than $50. The number of ten-cent coins is ¹₈ of all the coins he has. When he exchanges some fifty-cent coins for ten-cent coins, the ratio of the number of ten-cent coins to fifty-cent coins he now has become 7 : 1.

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

	Ten-cent coins (1)	Fifty-cent coins (2)	Make b the same (2)x7 = (3)
Before	1 u	7 u	49 u
Change	+ 5 p	- 1 p	- 7 p
After	7 b	1 b	7 b

(a)
Total value of coins per set
= 1 x 10 + 7 x 50
= 10 + 350
= 360¢ 

$1 = 100¢ 
$50 = 5000¢ 

Number of sets of 360¢ 
= 5000 ÷ 360
= 13 r 320

Largest possible amount that Howard has
= 13 x 360
= 4680¢ 
= $46.80

(b)
Ten-cent coins : Fifty-cent coins = 1 : 7

1 u + 5 p = 7 b --- (1)

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

Make b the same.

(2)_x7 = (3)
49 u - 7 p = 7 b --- (3)

(3) = (1)
49 u - 7 p = 1 u + 5 p
49 u - 1 u = 5 p + 7 p
48 u = 12 p

12 p = 48 u

1 p = 48 u ÷ 12 = 4 u

Since the number of sets of 360¢ is 13, 1 u is equal to 13.

Number of fifty-cent coins that have been exchanged for ten-cent coins
= 1 p
= 1 x 4 u
= 4 u
= 4 x 13
= 52

Total value of fifty-cent coins that have been exchanged for ten-cent coins
= 52 x 50
= 2600¢
= $26

Answer(s): (a) $46.80; (b) $26