Question

There are two more quarters than dimes, and as many nickels, as quarters and dimes together. the total amount of money is $3.75. how many quarters, dimes, and nickels are there?

Answer 1

9 quarters, 7 dimes, and 16 nickels.

Answer 2

There are 7 dimes, 9 quarters and 16 nickels.

Explanation:

Let's start by defining the variables for this problem :

Q : ''number of quarters''

D : ''number of dimes''

N : ''number of nickels''

''There are two more quarters than dimes'' so we can write the following equation :

`Q=D+2`

''And as many nickels, as quarters and dimes together'' so we can write the following equation :

`N=Q+D`

''The total amount of money is $3.75'' so we can write the following equation :

`0.25Q+0.10D+0.05N=3.75`

We must solve the following system :

`Q=D+2` (I)

`N=Q+D` (II)

`0.25Q+0.10D+0.05N=3.75` (III)

In the equation (II) we can find Q in terms of N and D :

`N=Q+D`

`Q=N-D` (IV)

If we use (I) in (IV) :

`Q=N-D`

`D+2=N-D`

`2D=N-2`

`N=2D+2` (V)

If we use (I) and (V) in (III)

`0.25Q+0.10D+0.05N=3.75`

`(0.25).(D+2)+0.10D+(0.05).(2D+2)=3.75`

`0.25D+0.50+0.10D+0.1D+0.1=3.75`

`0.45D=3.15`

`D=(3.15)/(0.45)=7`

`D=7`

If
`D=7` ⇒ replacing in (I)

`Q=D+2`

`Q=7+2=9`

`Q=9`

If we replace this values in (II) ⇒

`N=Q+D`

`N=9+7=16`

`N=16`

We find that there are 7 dimes, 9 quarters and 16 nickels