Question

A man has 19 coins in his pocket, all of which are dimes and quarters. If the total value of his change is 355

cents, how many dimes and how many quarters does he have?
Let a be the number of dimes and y be the number of quarters.
Write an equation that expresses the information in the sentence "A man has 19 coins."
Write an equation that expresses the information in the sentence "The total value of change is 355 cents."
Solve the system you have written above.
The number of dimes, a is
The number of quarters, y is

Answer 1

• 8 dimes
• 11 quarters

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.