How to solve this problem?

Question

My teacher is giving out the answers for the study guide but I want to know how to do the work. Please help:

 

Mark has a pocket full of change with 12 dimes and quarters. The total value of coins in his pocket is $1.95. Write a system of linear equations and use it to find out how many dimes and how many quarters Mark has in his pocket. 

Answer 1

The first step is to define our variables.  What are we looking for?  The number of dimes and the number of quarters.

 

So lets say D is the number of dimes and Q is the number of quarters.

 

Now lets write down the relationships in the problem statement using these variables.

 

The first sentence tells us that the total number of coins (number of dimes and quarters) is 12.  So our first statement is

 

D+Q=12

 

To build the second relationship you have to remember that D is equal to the number of dimes NOT the dollar value of the number of dimes.

 

The dollar value of the number of dimes is $0.10D (ie 10 cents times every dime)

The dollar value of the number of quarters is $0.25Q (ie 25 cents times every quarter)

 

So the total dollar value of Mark's change is .10D+.25Q.  We know the total value of the coins is $1.95 so we can write down

 

.10D+.25Q=1.95

 

This is the second equation in our system of linear equations.

 

(1) D+Q=12

(2) .1D+.25Q=1.95

 

A simple method to solve this system is to isolate a variable in equation (1) then substitute it into equation (2)

 

D+Q=12

D=12-Q

 

We can substitute into (2)

 

.1D+.25Q=1.95

.1(12-Q)+.25Q=1.95

1.2-.1Q+.25Q=1.95

1.2+.15Q=1.95

.15Q=.75

Q=5

 

Substituting this back into D=12-Q=12-5=7

 

So we have 5 quarters and 7 dimes.

 

Note you should substitute Q=5 and D=7 into your system of linear equations to verify those values are a solution.

 

Good Luck

 

Cheers

 

-Chris