A vending machine

Question

A vending machine takes only dimes and quarters. If it contains 100 coins with a total value of $14.50, how many of each type of the coin are in the machine?

Answer 1

First, define your number of dimes, and number of quarters in terms of x.

 

number of dimes = x;

number of quarters = 100 - x;  

 

Since the sum of dimes and quarters is 100, if there are x dimes, then 100 - x must be the amount of quarters. For example, if there are 35 dimes, then 100 - 35 = 65 must be the amount of quarters.

 

Next, we need to set up the equation.

 

0.10(x) + 0.25(100-x) = 14.50

 

If you had 27 nickels in your pocket and wanted to know how much money that is, what would you do? You would take 27 nickels times 0.05 cents each to get $1.35, right?

 

That is what we do here. We are gonna take our number of dimes, x, times 0.10 cents each to get our first term. We do the same for the second term, where we take our number of quarters, 100 - x, and multiply that by 0.25 cents each. The sum of both of these terms must equal $14.50. It is important to remember that 0.10 and 0.25 must be decimals. If you do 10(x) + 25(100 - x) = 14.50, you're incorrectly saying a dime is worth 10 dollars, and a quarter is worth 25 dollars.

 

Simplify:

 

0.10(x) + 0.25(100-x) = 14.50      Distribute;

 

0.10x + 25 - 0.25x = 14.50         Combine like x terms;

 

-0.15x + 25 = 14.50                   Subtract 25 from both sides;

 

-0.15x = -10.5                           Divide both sides by -0.15 to isolate x;

 

x = 70

 

X represents the number of dimes, and 100 - x represents number of quarters. The final answer is that you have 70 dimes, and 30 quarters.