Answer:
Step-by-step explanation:
You want to find the number of dimes (x) and the number of quarters (y) in 19 coins that have a total value of $3.55.
Setup
Using the given variables, we can write two equations. One is for the number of coins; the other is for their value.
x + y = 19 . . . . . . . . . . . number of coins
10x +25y = 355 . . . . . value in cents
Solution
It usually works well to solve an equation involving the highest-value coin. That means we want to eliminate the x-variable. Using the first equation, we can write ...
x = 19 -y
Substituting into the second equation, we get ...
10(19 -y) +25y = 355
190 +15y = 355 . . . simplify
15y = 165 . . . . . . . subtract 190
y = 11 . . . . . . . . . . divide by 15
x = 19 -11 = 8
The number of dimes is 8.
The number of quarters is 11.
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Alternate solution
The average coin value is 355/19 = 18 13/19 cents. The proportion of the highest-value coins is the ratio of the difference between this value and the lowest value, divided by the difference in values.
proportion of quarters = (18 13/19 -10)/(25 -10) = (8 13/19)/15 = (165/19)/15
proportion of quarters = 11/19
Multiplying this by the number of coins (19) gives the number of quarters:
(11/19)×19 = 11 . . . . quarters
You may recognize some similarities to the above solution.
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Additional comment
Once you understand this generic solution to a mixture problem, you can use it to write down the answer without bothering with variables and equations.
