Ratios

A ratio is a comparison of two or more numbers.

For example, each human foot has 5 toes. The ratio 1:5 describes how feet and toes are related. The ratio 1:5 can also be written as 1/5.

To solve a practical problem involving ratios, we use cross-multiplication (an application of the properties of proportional ratios).

e.g., 1/5 = x/30, where x is the unknown:

Cross-multiplying:  5x = 30 x 1

Now divide both sides by 5:  5x/5 = 30/5

Answer:  x = 30/5 = 6

"Per cent" comes from the Latin for "per hundred."

A percentage is a ratio: 10% is 10 parts per hundred or the ratio 10/100

Three types of percentage problems are often encountered:

1. 20% of 30 = x (the unknown)

Try to say the question in English:

20 is to 100 as x is to 30

Next, write it as proportional ratios:

20/100 = x/30

Now, as in calculating a ratio, cross-multiply and solve:

100x = 20 x 30

x = (20 x 30)/100 = 600/100

x = 6

2. x% of 30 = 6

English:  x is to 100 as 6 is to 30

Proportional ratios:  x/100 = 6/30

Solution:

30x = 6 x 100

30x = 600

x = 600/30 = 20

3. 20% of x = 6

English:  20 is to 100 as 6 is to x

Proportional ratios:  20/100 = 6/x

Solution:

20x = 6 x 100

20x = 600

x = 600/20 = 30

Don’t be afraid to trust your instincts--they will probably be right.  Begin by verbalizing the problem, then write the ratios, cross-multiply, and solve.

If you have a calculator, read the instructions and learn the operations of calculating percentages.