RATIO AND PROPORTION

There is no topic that is more important than that of ratio and proportion. Many problems that can be solved with algebra can also be solved by setting up a proportion. As we'll see, using equivalent fractions to solve proportion problems is often faster than using a calculator. Perhaps more importantly, because the relationships are easier to visualize you are also more likely to feel confident about your answers.

Definition: A "ratio" is a comparison of two quantities.

For example, assume that at a gathering there were 8 boys for every 3 girls. Perhaps there were 8 boys and 3 girls at the beginning of the evening and another 8 boys and 3 girls arrived later, bringing the total number of boys to 16 and the total number of girls to 6. The point is, we don't know if there were 8 boys and 3 girls, or 16 boys and 6 girls, or even 80 boys and 30 girls. But the relationship of the number of boys to the number of girls is constant. That is, the ratio of boys to girls is 8 to 3, regardless of the total numbers. The ratio of 8 boys to 3 girls may be written as

8 to 3 or 8 : 3 or

Notice that we can use equivalent fractions to see that the relationship of boys to girls is 8 to 3, regardless of the total numbers:

Definition: A "proportion" is a statement that two ratios are equal.

For example, in the example above, we could have 16 boys and 6 girls or even 80 boys and 30 girls. The ratio of boys to girls is constant. We can write

How do we know if two ratios are equal? We can reduce both 16/6 and 80/30 to 8/3 or we can "cross-multiply." That is, we can multiply the numerator of the first term by the denominator of the second term and multiply the numerator of the second term by the denominator of the first term.

16 x 30 = 6 x 80

480 = 480

Therefore, we know the ratios are equal.

SOLVING PROBLEMS USING PROPORTIONS

The classic proportion problem is that of the tree, the man, and the shadows:

Example:

If a tree casts a 12 foot shadow and a 6 foot man standing near the tree casts a 4 foot shadow, find the height of the tree.

We can visualize the problem proportionally as follows

If we let H represent the height of the tree, then

We can cancel the "ft." in the numerator and denominator on the right and get

Cross-multiplying yields

H x 4 = 6 x 12 ft.

4 H = 72 ft.

H = 18 ft.

Shortcut: Rather than cross multiplying and solving, in this case it might have been easier to think of the problem in terms of equivalent fractions. Note that 12, the denominator on the left, is evenly divisible by 4, the denominator on the right. Therefore, we can write

INVERSE VARIATION

So far we have been talking about "direct variation." If the number of boys were to increase and if the ratio of boys to girls were to remain constant, then the number of girls would also increase. However, some things vary "inversely." For instance, if I can do a job in 4 hours and two other people come to help, the job will get done a lot faster. That is, if the number of people working increases, then the time needed to do the job decreases. In this case, the problem would be set up like this

For example, if it takes me 4 hours to finish a job and two other people arrive who work at the same rate that I do, then it will take one third as long to do the job as it would have alone.

T x 3 workers = 1 worker x 4 hours

Another example of inverse variation is the relationship between rate and time. The faster one drives, the less time it takes to get from Point A to Point B.

Example: If a person drives from Allenville to Belltown at the rate of 50 mph, the drive takes two hours. How long would it take if the person drove at the rate of 75 mph?

Solution:

If we cross multiply we get

T x 75 mph = 2 hrs. x 50 mph

We can think of the mph in the numerator and the denominator as "canceling", so that

T = hours or 1 hour and 20 minutes.