Suppose Tom has 23 coins totaling $3.65. If he has only dimes and quarters, how many of each type does he have?

Question

Tom has ______ dimes and _______ quarters. 

Answer 1

Hi Leila.

 

This is a pretty straightforward systems of equations problem. It's best to call some variables first:

 

D = number of dimes

Q = numbers of quarters

 

You know that Tom has 23 coins in total. That means the number of dimes and the number of quarters equals 23.

 

D + Q = 23

 

You also know that the total value of the coins is $3.65. A dime is worth $0.10 and a quarter is worth $0.25, so:

 

0.10*D + 0.25*Q = 3.65

 

That says that 10 cents per dime times the number of dimes, plus 25 cents per quarter times the number of quarters is three dollars and sixty-five cents.

 

So you have a system of linear equations:

 

D + Q = 23

0.10D + 0.25Q = 3.65

 

I'm going to multiply the bottom row by 100 to clear the decimals.

 

D + Q = 23

10D + 25Q = 365

 

Now I can multiply the top row by -10 to eliminate the D terms

 

-10D - 10Q = -230

10D + 25Q = 365

________________

0D + 15Q = 135

 

15Q = 135

Q = 9

 

Tom has 9 quarters.

 

If D + Q = 23, and Q = 9, then:

 

D + 9 = 23

 

D = 14

 

Tom has 14 dimes.

 

Always good to double check:

 

9 quarters is worth $2.25. 14 dimes is worth $1.40. These should sum to $3.65.

 

2.25 + 1.40 = 3.65. Check.

 

Hope this helps.