Math CentralQuandaries & Queries


Question from Maggie, a student:

A mechanical pencil costs 50 cents and a pen costs 25 cents. If I spend 5.00 on 17 pens and pencils, how many of each does that buy?

Hi Maggie,

I can show you two approaches to this problem. The first method uses some ingenuity and some arithmetic.

Suppose you spend the $5.00 and buy only pencils. Pencils cost $0.50 so you would be able to purchase 10 items, all pencils. But you want 17 items so return 1 pen and get 2 pens, and now you have 9 pencils and 2 pens for a total of 11 items. The total cost is still $5.00 and thus trading a pencil for two pens leaves the cost at $5.00 but increases the number of items by 1.

So start with 10 pencils. Trading a pencil for two pens increases the total number of items by 1. Thus to get to 17 items I have to trade in 7 pencils. This will leave you with 3 pencils and 14 pens.

Check: 3 pencils and 14 pens is 17 items costing 3 × $0.50 + 14 × $0.25 = $1.50 + $3.50 = $5.00.

The second approach is algebraic.

Suppose the number of pencils you buy is x and the number of pens you buy is y. Then

x + y = 17


$1/2 × x + $1/4 × y = $5.00

Solve these equations for x and y.

I hope this helps,

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