How many quarters and dimes does nick have?

Question

Nick has 12 coins in his pocket that are either quarters or dimes. He has $1.80 in his pocket in coins Let x represent the number of quarters and y represent the number of dimes. The following system of equations represents the situation:

x+y=12

0.25x+0.10y=1.80

Answer 1

Hi David, you have the correct equations. Keep in mind, everything needs to be reasonable. Without actually solving the two equations, you can deduce the correct answer. First, the number of quarters must be an even number, 2, 4, 6, ... The number of quarters cannot be odd, 1, 3, 5, ... The reason is odd number of quarters in end a 5, .25, .75, 1.25 ... and there is no way the total value of the coins can be 1.80 with an odd number of quarters.

 

Therefore, start with 2 quarters or .50. How many dimes does he have? Answer: 13. One possible answer. If he has 4 quarters, or 1.00. How many dimes does he have? Answer: 8. Second possible answer. If he has 6 quarters, or 1.50, how many dimes does he have? Answer: 3. Third possible answer.

 

OK, now let's solve the equations.

     x   +      y  = 12

.25x   + .10y  =   1.80

 

I do not like to work with decimals or fractions, so multiply the second equation by 100 giving:

 

    x   +      y   =   12

25x   +   10y   = 180

 

Multiply the first equation by -10 giving:

 

-10x   -  10y   = -120

 25x   + 10y   =   180

-------------------------

 

Add the two equations together giving:

 

15x = 60

 

Divide both sides by 15 giving:  x = 4, or Nick has 4 quarters. Does this match up with one of our three possible answers?

 

To finish, replace the x with 4 in the original first equation to find the number of dimes Nick has

 

x    + y    = 12

4    + y    = 12

 

Subtract 4 from both sides of the equation giving:

 

y = 8

 

Now we know the number of each coin Nick has in his pocket. He has 4 quarters and 8 dimes.

 

Questions?

Answer 2

I assume that you would like to know how to solve the system of equations:

 

x = # of quarters

y = # of dimes

 

First equation: Nick has 12 coins

         x + y = 12

 

Second equation: Nick has $1.80 in his pocket

   if he has x quarters, then he has 0.25*x dollars in quarters

   if he has y dimes, then he has 0.1*y dollars in dimes

   Total he has in his pocket is 0.25*x + 0.1*y= 1.80

 

If x + y = 12, then

   y=12-x

   

 

Substitute that  back into the above equation:

 

0.25*x + 0.1*(12-x) = 1.80

0.25*x + 1.2 - 0.1*x = 1.80

0.15*x = 1.80-1.2

0.15*x = 0.6

x =0.6/0.15 =4 quarters

Then the number of dimes is: 12 - 4 = 8

 

So Nick has 4 quarters and 8 dimes