Comparing Fractions

Some situations require us to compare fractions, that is, to rank them in order of size.

For instance, suppose that of one airline’s flights arrive on time, in contrast to of another airline’s flights. To decide which airline has a better record for on-time arrivals, we need to compare the fractions.

Or to take another example, suppose that the drinking water in your home, according to a lab report, has 2 parts per million (ppm) of lead. Is the water safe to drink? If the federal limit on lead in drinking water is 15 parts per billion (ppb), again you need to compare fractions.

One way to handle such problems is to draw diagrams corresponding to the fractions in question. The larger fraction corresponds to the larger shaded region.

For instance, the diagrams show that is greater than . The symbol > stands for “greater than.”

Both and have the same denominator, so we can rank them simply by comparing their numerators.

Note that the symbols < and > always point to the smaller number.

For like fractions, the fraction with the larger numerator is the larger fraction.

Definitions

Like fractions are fractions with the same denominator.

Unlike fractions are fractions with different denominators.

To Compare Fractions

• compare the numerators of like fractions, and
• express unlike fractions as equivalent fractions having the same denominator and then compare their numerators.

EXAMPLE 1

Compare and .

Solution

These fractions are unlike because they have different denominators. Therefore we need to express them as equivalent fractions having the same denominator. But what should that denominator be?

One common denominator that we can use is the product of the denominators: 15 · 9 = 135

, 135 = 15 · 9 so the new numerator is 7 · 9, or 63.

, 135 = 9 · 15, so the new numerator is 4 · 15, or 60.

Next, we compare the numerators of the like fractions that we just found.

Because . Therefore .

Another common denominator that we can use is the least common multiple of the denominators.

The LCM is . We then compute the equivalent fractions.

Because , we know that .

Note that in Example 1 we computed the LCM of the two denominators. This type of computation is used frequently in working with fractions.

Definition

For any set of fractions, their least common denominator (LCD) is the least common multiple of their denominators.

In Example 2, pay particular attention to how we use the LCD.

EXAMPLE 2

Order from smallest to largest: .

Solution

Because these fractions are unlike, we need to find equivalent fractions with a common denominator. Let’s use their LCD as that denominator.

The LCD . Check: 4 and 10 are both factors of 40.

We write each fraction with a denominator of 40.

Then we order the fractions from smallest to largest. (The symbol < stands for “less than.”)

EXAMPLE 3

About of Earth’s surface is covered by water and is covered by desert. Does water or desert cover more of Earth?

Solution

We need to compare with .

Since . Therefore water covers more of Earth than desert does.