Question

Lucy has 34 coins consisting of nickels and quarters amounting to $2.90. How many coins of each kind does she have?

Answer 1

Lucy has `"6 quarters"` and `"28 nickels"`

Explanation:

I am going to give nickels and quarters their own variable. Nickels will be `n` and quarters will be `q`. Since Lucy has 34 coins total, we can make this equation:
`n + q = 34`

The second part is about the value of the coins. Since nickels are worth 5 cents (or `5/100` of a dollar) and quarters are worth 25 cents (or `25/100` of a dollar), we can make this equation:

`0.05n + 0.25q = 2.90`

I'm actually going to multiply this whole equation by `100`, to move the decimal points two places and make this easier to solve:

`5n + 25q = 290`

Now we will rearrange the first equation to solve for one variable:
`n + q = 34`
`n = 34 - q`

Substitute that into the second equation and solve for `q`

`5n + 25q = 290`
`5(34 - q) + 25q = 290`
`170 - 5q + 25q = 290`
`170 + 20q = 290`
`20q = 120`
`q = 6`

Lucy has `"6 quarters"`. Now substitute `q` into the first equation to solve for `n`.
`n = 34 - q`
`n = 34 - 6`
`n = 28`

She has `"28 nickels"`.