Peter has some twenty-cent coins and fifty-cent coins which amount to more than $45 but less than $50. The number of twenty-cent coins is
112 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 7 : 1.
- What is the largest possible amount of money Peter 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)x7 = (3) |
Before |
1 u |
11 u |
77 u |
Change |
+ 5 p |
- 2 p |
- 14 p |
After |
7 b |
1 b |
7 b
|
(a)
Total value of coins per set
= 1 x 20 + 11 x 50
= 20 + 550
= 570¢
$1 = 100¢
$50 = 5000¢
Number of sets of 570¢
= 5000 ÷ 570
= 8 r 440
Largest possible amount that Peter has
= 8 x 570
= 4560¢
= $45.60
(b)
Twenty-cent coins : Fifty-cent coins = 1 : 11
1 u + 5 p =
7 b --- (1)
11 u - 2 p = 1 b --- (2)
Make b the same.
(2)
x7 = (3)
77 u - 14 p =
7 b --- (3)
(3) = (1)
77 u - 14 p = 1 u + 5 p
77 u - 1 u = 5 p + 14 p
76 u = 19 p
19 p = 76 u
1 p = 76 u ÷ 19 = 4 u
Since the number of sets of 570¢ 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 4 u
= 8 u
= 8 x 8
= 64
Total value of fifty-cent coins that have been exchanged for twenty-cent coins
= 64 x 50
= 3200¢
= $32
Answer(s): (a) $45.60; (b) $32