Ethan 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.
- What is the largest possible amount of money Ethan has?
- 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 Ethan 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