Algebra Know-It-ALL

86 Fractions Built of Integers

Here’s a challenge!

Suppose a car dealer tells you that a certain sports car has a top speed that’s “half again” as fast as the top

speed of a certain pickup truck. What is the car’s top speed compared to the truck’s top speed, as a ratio of

two integers? What is the truck’s top speed to the car’s top speed, as a ratio of two integers?

Solution

The term “half again” means “1-1/2 times as great.” The ratio of the car’s top speed to the truck’s top speed

is therefore 1-1/2 to 1, or 3/2 to 1. But that’s not a ratio of two integers! The correct way to express the

ratio is 3 to 2. You would write this as 3/2 or 3:2. When you want to express a ratio in the reverse sense,

switch the numerator and the denominator. The ratio of the truck’s top speed to the car’s top speed is 2 to 3,

which you can write as 2/3 or 2:3.

Technically, you can express both of these ratios in infinitely many other ways. In general, the ratio

of the car’s highest speed (call it c) to the truck’s highest speed (call it t), expressed as a ratio between two

integers, is

c/t= (3a)/(2a)

where a can be any integer except 0. The ratio of the truck’s highest speed to the car’s highest speed, in

general, is

t/c= (2a)/(3a)

What if you want to specify the actual number of miles per hour that each vehicle can travel? Suppose the

truck can go a maximum of 100 miles per hour on a straight, level road with no wind, and the car can

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2

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2

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0 Proper fractions

Figure 6-2 Proper fractions

correspond to points

between −1 and 1 on

the number line. The

open circles at −1 and

1 indicate that they

are not included in the

interval.