A collection of 25 coins consists of nickels, dimes, and quarters. There are three times as many dimes as nickels and three more dimes than quarters. What is the total value of the collection in dollars and cents?

1 Answer
Jan 4, 2018

$3.65

Explanation:

Let #x# be nickels, #y# be dimes, and #z# be quarters.

We know that #x+y+z=25# for our first equation.

We know that there are three times as many dimes as nickels, which means our second equation is:

#y=3x#

(Think: If there was 1 nickel there would be 3 dimes.)

We know that there are three more dimes than quarters so our third equation is:

#y = z+3#

(Think: If there were 3 quarters we would have 6 dimes.)

So we have three equations:
(1) #x+y+z=25#
(2) #y=3x#
(3) #y = z+3#

Solving (2) for #x# gives: #x = y/3#
Solving (3) for #z# gives: #z = y-3#

Taking these and substituting into (1) gives:

#x+y+z=25 \rightarrow(y/3) + y + (y-3) = 25#

Adding 3 to both sides and doing the easy combination on the left:

#y/3 + 2y =28#

Multiply through by 3:

#y+6y = 84#

Collect like terms:

#7y = 84#

Divide both sides by 7:

#y=12#

Know that #y=12# and going back to equations (2) and (3) we have:

from (2): #12=3x\rightarrow x =4#
from (3): #12=z+3\rightarrow z = 9#

So 4 nickels, 12 dimes, and 9 quarters.

The value of the collection is #.05x + .10y +.25z#, so

#.05(4) + .10(12) +.25(9) = 3.65#

so $3.65.