Question

Each are of quarters and nickels is worth $3.70. There are 22 coins in all How many of there?

Answer 1

First, let's call the number of nickels we have `n` and the number of quarters `q`.

Now we can write two equations which we can use to solve the problem through substitution:

First we know there are 22 coins in total, therefore:

`n + q = 22`

And we know their value is $3.70 so we can write:

`0.05n + 0.25q = 3.70`

Step 1) solve the first equation for `n`

`n + q = 22`

`n + q - color(red)(q) = 22 - color(red)(q)`

`n + 0 = 22 - color(red)(q)`

`n = 22 - q`

Step 2) Substitute `22 - q` for `n` in the second equation and solve for `q`.

`0.05(22 - q) + 0.25q = 3.70`

`(0.05 xx22) - (0.05 xx q) + 0.25q = 3.70`

`1.1 - 0.05q + 0.25q = 3.70`

`1.1 + 0.25q - 0.05q = 3.70`

`1.1 + (0.25 - 0.05)q = 3.70`

`1.1 + 0.20q = 3.70`

`1.1 - color(red)(1.1) + 0.20q = 3.70 - color(red)(1.1)`

`0 + 0.20q = 2.60`

`0.20q = 2.60`

`(0.20q)/color(red)(0.20) = 2.60/color(red)(0.20)`

`(color(red)(cancel(color(black)(0.20)))q)/cancel(color(red)(0.20)) = 13`

`q = 13`

Step 3) Substitute `13` for `q` in the solution to the first equation in Step 1.

`n = 22 - 13`

`n = 9`

Solution:

There are 9 nickels and 13 quarters