Question

How many one-cent stamps did Maria buy?

Maria bought several post stamps and paid one dollar. There are stamps costing five cents, two cents, and one cent in the post office. She bought 10 times as many one-cent stamps as two-cent stamps, while the rest of the stamps she bought were five-cent stamps. How many one-cent stamps did Maria buy?

Answer 1

Maria bought 50 one-cent stamps.

Explanation:

The word problem gives us an expression that looks like this:

`1.00=0.05n+0.02t+0.01p`

where `n` is the number of five-cent stamps, `t` is the number of two-cent stamps, and `p` is the number of one-cent stamps.

We also know that Maria bought ten times as many one-cent stamps as two-cent stamps. If we write this out as another expression:

`color(blue)(p=10t)`

We then substitute it into the first expression:

`1.00=0.05n+0.02t+0.01color(blue)((10t))`

`1.00=0.05n+0.02t+0.10t`

`1.00=0.05n+0.12t`

Now, we need to figure out how many two and five-cent stamps were purchased. Assuming Maria spent EXACTLY $1, the number of two-cent stamps must give a total such that 0.12 times that number gives a 5 or zero in the remainder. This is so we have an integral value for `n`.

The only multiple of 0.12 that satisfies this, AND results in a value less than $1 is 5, proof below:

`1.00=0.05n+0.12(5)`

`1.00=0.05n+0.6color(red)(0)`

`0.40=0.05n`

`n=8`

Now we have a solution for `n` and `t`, but we really only need `p`. Luckily, we can use that relationship in the problem statement:

`color(blue)(p=10t)`

`p=10(5)`

`color(green)(p=50)`

Plugging in all of the values to check:

`1.00=0.05(8)+0.02(5)+0.01(50)`

`1.00=0.40+0.10+0.50`

`1.00=1.00`

The math checks out.