Jamie has 25 nickels, 14 dimes and 39 quarters in his collection
Step-by-step explanation:
The given is:
1. Jamie has a collection of 78 nickels, dimes, and quarters
2. They worth $12.40
3. If the number of quarters is doubled, the value becomes $22.15
Assume that there are n nickels, d dimes and q quarters
∵ The collection of Jamie has 78 coins
∴ n + d + q = 78 ⇒ (1)
∵ 1 nickels = 5 cents
∵ 1 dimes = 10 cents
∵ 1 quarter = 25 cents
∵ The collection of Jamie worth $12.40
- Change the value of the collection to cents
∵ $1 = 100 cents
∴ The collection of Jamie worth = 12.40 × 100 = 1240 cents
∴ 5n + 10d + 25q = 1240
- Divide all terms by 5 to simplify it
∴ n + 2d + 5q = 248 ⇒ (2)
∵ When the number of quarters is doubled, the value becomes
$22.15
∵ The number of quarters is q
∴ The new number of quarters is 2q
∵ The value of coins is $22.15
∵ $22.15 = 22.15 × 100 = 2215 cents
∴ 5n + 10d + 25(2q) = 2215
∴ 5n + 10d + 50q = 2215
- Divide each term by 5 to simplify it
∴ n + 2 d + 10q = 443 ⇒ (3)
Subtract equation (2) from equation (3)
∵ n + 2d + 5q = 248 ⇒ (2)
∵ n + 2 d + 10q = 443 ⇒ (3)
∴ 5q = 195
- Divide both sides by 5
∴ q = 39
Substitute the value of q in equations (1) and (2)
∵ n + d + q = 78 ⇒ (1)
∴ n + d + 39 = 78
- Subtract 39 from both sides
∴ n + d = 39 ⇒ (4)
∵ n + 2d + 5q = 248 ⇒ (2)
∴ n + 2d + 5(39) = 248
∴ n + 2d + 195 = 248
- Subtract 195 from both sides
∴ n + 2d = 53 ⇒ (5)
Subtract equation (4) from equation (5)
∵ n + d = 39 ⇒ (4)
∵ n + 2d = 53 ⇒ (5)
∴ d = 14
Substitute the value of d in equation (4)
∵ n + d = 39 ⇒ (4)
∴ n + 14 = 39
- Subtract 14 from both sides
∴ n = 25
Jamie has 25 nickels, 14 dimes and 39 quarters in his collection
Learn more:
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