Setting Up a Ratio

Today I would like to share the method I teach for solving problems involving ratios. I have a simple 4 step system.

Step 1: Identify two key words and write them down one over the other.

Step 2: Next to each of these key words write down the numbers, variables or expressions that correspond to each key word in two columns.

Step 3: Draw in 2 division symbols and an equal sign.

Step 4: Cross multiply and divide.

This procedure is best understood with a simple example:

Example: The sales tax on an $8.00 shirt is $0.60. At this rate what would be the sales tax on a $12.00 shirt?

Try to solve the problem yourself before checking the solution below.

Solution: We begin by identifying 2 key words. In this case, such a pair of key words is “shirt” and “tax.”

shirt            8        12
tax             0.60      x

Choose the words that are most helpful to you. Notice that we wrote in the shirt prices next to the word shirt, and the tax prices next to the word tax. Also notice that the tax for an $8 shirt is written under the number 8, and the (unknown) tax for a $12 shirt is written under the 12.

Now draw in the division symbols and equal sign, cross multiply and divide the corresponding ratio to find the unknown quantity x.

8/.060  =  12/x
8x = 12(0.60)
8x = 7.2
x = 0.90

So the tax on a $12 shirt is $0.90.

I’d also like to provide a quicker solution for the more advanced student.

* Mental math: If the tax on an $8.00 shirt is $0.60, then the tax on a $4.00 shirt would be $0.30 at this rate. Thus, the tax on a $12.00 shirt would be $0.90.

Here are a few more problems for you to try. Try to use my 4 step method each time. I will provide solutions to these throughout the week.

1. Running at a constant speed, a cheetah traveled 200 miles in 5 hours. At this rate, how many miles did the cheetah travel in 4 hours?

2. A copy machine makes 4800 copies per hour. At this rate, in how many minutes can the copy machine produce 920 copies?

3. The height of a solid cone is 22 centimeters and the radius of the base is 15 centimeters. A cut parallel to the circular base is made completely through the cone so that one of the two resulting solids is a smaller cone. If the radius of the base of the small cone is 5 centimeters, what is the height of the small cone, in centimeters?

More Problems with Explanations

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