Question

A student makes a 9.25 purchase at the bookstore with a $20 bill. The store has no bills and gives the change in quarters and fifty-cent pieces. There are 30 coins in all. How many of each kind are there?

Answer 1

`"Quarters"=17, "Fifty-cent pieces"=13`

Explanation:

The student is going to get $10.75 of change ($20-$9.25=$10.75) and we get Quarters and Fifty-cent pieces. I'll assign Q for the number of quarters and F for the number of fifty-cent pieces.

We know two things:

`Q+F=30` - there are 30 coins in total and
`Q(.25)+F(.5)=10.75` - the total change comes to 10.75.

Now let's solve for Q and F:

`Q=30-F` - let's substitute this into the other equation:

`(30-F)(.25)+F(.5)=10.75`

Before I distribute this out, I'm going to multiply both sides by 4 to get rid of the decimals:

`4((30-F)(.25)+F(.5))=4(10.75)`

`4(30-F)(.25)+F(4)(.5))=4(10.75)`

`(30-F)+2F=43`

`F=13`

which means

`Q+13=30`

`Q=17`

Does this work? Let's check:

`17(.25)+13(.5)=10.75`

`4.25+6.5=10.75`

`10.75=10.75`