Question

Two teachers, Mr. Donahue and Mrs. Ortiz, decide to buy mechanical pencilsand rulers for their classrooms. Mrs. Ortizbuys 50 mechanical pencils and 30 rulers for $55. Mr. Donahue buys 30 of each for $45. What was the price per mechanical pencil?What was the price per ruler?a.

Answer 1

Price per mechanical pencil:
`\$ 0.5`.

Price per ruler:
`\$ 1`.

Explanation:

Let the cost per mechanical pencil be
`\$ p`, and let the cost per ruler be
`\$ r`.

Set up two equations based on the amount of money that each teacher paid.

The cost of
`50` mechanical pencils (
`\$ p` each) and
`30` rulers (
`\$ r` each) would be
`(50 \, p + 30\, r)`. That should be equal to the amount of money that Mrs. Ortiz paid. In other words,
`50 \, p + 30\, r = 55`.

Similarly, the cost of
`30` mechanical pencils (
`\$ p` each) and
`30 \!` rulers (
`\$ r` each) would be
`(50 \, p + 30\, r)`. That should be equal to the amount of money that Mr. Donahue paid. In other words,
`30 \, p + 30\, r = 45`.

That gives a system of two equations:

`\left\lbrace \begin{aligned}& 50 \, p + 30\, r = 55 \\ & 30 \, p + 30\, r = 45 \end{aligned}\right.`.

Subtract the second equation for the first:

`20\, p = 10`.

`\displaystyle p = (1)/(2) = 0.5`.

In other words, the cost of one mechanical pencil would be
`\$0.5`.

Substitute
`p = 0.5` back to either equation and solve for
`r`, the cost of one ruler.

`50 * 0.5 + 30\, r = 55`.

`30\, r = 55 - 25 = 30`.

`r = 1`.

In other words, the cost of one ruler would be
`\$ 1`.