Ryan has some twenty-cent coins and fifty-cent coins which amount to more than $70 but less than $80. The number of twenty-cent coins is
119 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 9 : 1.
- What is the largest possible amount of money Ryan has?
- 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)x9 = (3) |
Before |
1 u |
18 u |
162 u |
Change |
+ 5 p |
- 2 p |
- 18 p |
After |
9 b |
1 b |
9 b
|
(a)
Total value of coins per set
= 1 x 20 + 18 x 50
= 20 + 900
= 920¢
$1 = 100¢
$80 = 8000¢
Number of sets of 920¢
= 8000 ÷ 920
= 8 r 640
Largest possible amount that Ryan has
= 8 x 920
= 7360¢
= $73.60
(b)
Twenty-cent coins : Fifty-cent coins = 1 : 18
1 u + 5 p =
9 b --- (1)
18 u - 2 p = 1 b --- (2)
Make b the same.
(2)
x9 = (3)
162 u - 18 p =
9 b --- (3)
(3) = (1)
162 u - 18 p = 1 u + 5 p
162 u - 1 u = 5 p + 18 p
161 u = 23 p
23 p = 161 u
1 p = 161 u ÷ 23 = 7 u
Since the number of sets of 920¢ is 8, 1 u is equal to 8.
Number of fifty-cent coins that have been exchanged for twenty-cent coins
= 2 p
= 2 x 7 u
= 14 u
= 14 x 8
= 112
Total value of fifty-cent coins that have been exchanged for twenty-cent coins
= 112 x 50
= 5600¢
= $56
Answer(s): (a) $73.60; (b) $56