Least Common Denominator: Compare and Order Fractions | Grade 7 Math | LevelKite

Which is more of the same whole: $\frac{2}{3}$ or $\frac{5}{8}$ ? It is hard to tell at a glance. This lesson shows you how to decide.

Start with what a fraction means. The denominator (the bottom number) tells how many equal parts the whole is cut into. The numerator (the top number) counts how many of those parts you have. So $\frac{2}{3}$ is 2 parts when a whole is cut into 3 equal parts, and $\frac{5}{8}$ is 5 parts when the same whole is cut into 8 equal parts.

Thirds and eighths are different sizes, so 2 parts against 5 parts is not a fair comparison. To compare fairly, both fractions must count parts of the same size from the same whole.

In the last lesson you made equivalent fractions: fractions that name the same amount of the same whole. You multiplied the numerator and the denominator by the same counting number. Multiplying both by 2, for example, cuts every part into 2 equal smaller parts, so the amount does not change.

Look at what that does to the denominator: it becomes a multiple of the old one. A multiple of a number is that number times a counting number, so the multiples of 3 are 3, 6, 9, 12, and so on.

That tells you which denominators two or more fractions of the same whole can share. A denominator that all of them can reach in this way must be a multiple of every original denominator. Any such number is a common denominator. The smallest one is the least common denominator, or LCD. Using the LCD keeps the numbers you work with small.

The table below lists the first six multiples of 4 and of 6. Look for numbers that appear in both lists.

Multiples of 4 and multiples of 6

Multiply by	Multiple of 4	Multiple of 6
$1$	$4$	$6$
$2$	$8$	$12$
$3$	$12$	$18$
$4$	$16$	$24$
$5$	$20$	$30$
$6$	$24$	$36$

Suppose two fractions of the same whole have denominators 4 and 6, such as $\frac{3}{4}$ and $\frac{5}{6}$ . In the table, the numbers 12 and 24 appear in both lists of multiples, so both are common denominators. The least is 12, so the LCD is 12.

You do not have to write full lists every time. For fractions of the same whole whose denominators are counting numbers, this method always finds the LCD:

List the multiples of the largest denominator in order, starting with the denominator itself.
Test each one: is it also a multiple of every other denominator?
Stop at the first one that passes. That is the LCD.

The first one that passes is the least, because every common denominator is a multiple of the largest denominator and you test those multiples from smallest to largest.

One caution: multiplying the denominators together always gives a common denominator, but not always the least one. Here $4 \times 6 = 24$ works, yet 12 is smaller.

Worked example

Find the LCD of two fractions

The fractions $\frac{5}{6}$ and $\frac{7}{10}$ are parts of the same whole, 1 unit. Find their least common denominator, and rename both fractions with it.

Read the denominators: 6 and 10. Both fractions describe the same whole, so they can be renamed with one shared denominator. The LCD must be a multiple of both 6 and 10. The larger denominator is 10, so list its multiples in order and test each one against 6.
$10, 20, 30, 40, \dots$
Test 10. Count by sixes: 6, then 12. The count skips 10, so 10 is not a multiple of 6. It fails.
$6 \times 1 = 6, \quad 6 \times 2 = 12$
Test 20. Keep counting by sixes: 18, then 24. The count skips 20, so 20 is not a multiple of 6. It fails as well.
$6 \times 3 = 18, \quad 6 \times 4 = 24$
Test 30. It equals 6 times 5 and 10 times 3, so it is a multiple of both denominators. It is the first multiple of 10 that passes, so 30 is the LCD.
$6 \times 5 = 30, \quad 10 \times 3 = 30$
Rename each fraction with denominator 30. Multiply the numerator and the denominator by the same counting number, so each fraction still names the same amount of the same whole.
$\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}, \quad \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30}$

Answer

The least common denominator of $\frac{5}{6}$ and $\frac{7}{10}$ is $30$ . With it, $\frac{5}{6} = \frac{25}{30}$ and $\frac{7}{10} = \frac{21}{30}$ .

Key concept

Same whole, same-size parts, then compare numerators

When fractions count parts of the same size, cut from the same whole, the numerators decide which fraction is greater. If the denominators are different, the parts are different sizes, and the numerators alone cannot be trusted. So first rename the fractions as equivalent fractions with a common denominator. The least common denominator, the smallest number that is a multiple of every denominator, keeps the numbers small. Then the greater numerator marks the greater fraction.

In the formula, $a$ and $b$ are the numerators of two fractions of the same whole, and $c$ is their shared denominator. Because $c$ is a counting number, $c > 0$ .

\text{If } a > b \text{ and } c > 0, \text{ then } \frac{a}{c} > \frac{b}{c}

To compare two numbers is to decide which one is greater, or whether they are equal. You record the result with a symbol: $3 < 5$ reads 3 is less than 5, and $5 > 3$ reads 5 is greater than 3. To order a list of numbers is to arrange it from least to greatest, or from greatest to least.

To compare or order positive fractions of the same whole:

Find the LCD of all the denominators.
Rename each fraction as an equivalent fraction with that denominator.
Compare the numerators. A greater numerator means a greater fraction, and equal numerators mean equal fractions.

Step 3 is fair only because of steps 1 and 2: every fraction now counts parts of one size, cut from the same whole.

Now settle the opening question: which is greater, $\frac{2}{3}$ or $\frac{5}{8}$ of the same whole?

The denominators are 3 and 8. Test the multiples of 8 in order: 8 and 16 are not multiples of 3, but $24 = 3 \times 8$ is. So the LCD is 24. (This time the LCD is the same number you get by multiplying the denominators.)

Rename each fraction in twenty-fourths: $\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}$ and $\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$ .

Both fractions now count twenty-fourths of the same whole, and $16 > 15$ . So $\frac{2}{3} > \frac{5}{8}$ : two-thirds is the greater amount, even though its numerator and its denominator are both smaller.

Worked example

Order three fractions

Order $\frac{3}{4}$ , $\frac{5}{6}$ and $\frac{7}{9}$ from least to greatest. All three are parts of the same whole, 1 unit.

The denominators 4, 6 and 9 are different, so find the LCD. Test the multiples of the largest denominator, 9, in order. First, 9 is not a multiple of 4 or of 6. Next, 18 is a multiple of 6 but not of 4. Then 27 is not a multiple of 4 or of 6. Finally, 36 is a multiple of all three. It is the first to pass, so the LCD is 36.
$36 = 4 \times 9 = 6 \times 6 = 9 \times 4$
Rename three-fourths and five-sixths in thirty-sixths. Multiplying the numerator and the denominator by the same counting number keeps each amount of the whole unchanged.
$\frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36}, \quad \frac{5}{6} = \frac{5 \times 6}{6 \times 6} = \frac{30}{36}$
Rename seven-ninths in thirty-sixths the same way.
$\frac{7}{9} = \frac{7 \times 4}{9 \times 4} = \frac{28}{36}$
All three fractions now count thirty-sixths of the same whole, so the numerators decide. Put the numerators 27, 30 and 28 in order from least to greatest.
$27 < 28 < 30 \quad \text{so} \quad \frac{27}{36} < \frac{28}{36} < \frac{30}{36}$
Swap each renamed fraction back to the original fraction it equals.
$\frac{3}{4} < \frac{7}{9} < \frac{5}{6}$

Answer

From least to greatest, the order is $\frac{3}{4}$ , $\frac{7}{9}$ , $\frac{5}{6}$ .

A mixed number is a whole number (such as 1, 2 or 3) together with a fraction that is less than 1. For example, $2\frac{3}{4}$ means $2 + \frac{3}{4}$ . Its whole-number part is 2 and its fraction part is $\frac{3}{4}$ .

To compare positive mixed numbers that are measured against the same whole, work in two stages:

Compare the whole-number parts first. A fraction part that is less than 1 can never make up a gap of a full whole. For example, $3\frac{1}{8} > 2\frac{7}{8}$ , because $2\frac{7}{8}$ is less than 3 and $3\frac{1}{8}$ is more than 3.
If the whole-number parts are equal, compare the fraction parts with a common denominator. Adding the same whole number to two fractions does not change which one is greater.

A fraction that is less than 1 fits the same pattern: its whole-number part is 0.

Worked example

Order fractions and mixed numbers

Four trails in one park are measured in miles, so every length is counted against the same whole, 1 mile. The lengths are $2\frac{3}{4}$ , $\frac{7}{8}$ , $3\frac{1}{8}$ and $2\frac{7}{10}$ miles. Order the trails from shortest to longest.

Compare the whole-number parts first. They are 2, 0, 3 and 2, because the seven-eighths trail has 0 whole miles. Every fraction part here is less than 1, so fewer whole miles means a shorter trail. The trail with 0 whole miles is the shortest, the trail with 3 whole miles is the longest, and the two trails with 2 whole miles are tied so far.
$0 < 2 < 3$
Break the tie by comparing the fraction parts, three-fourths and seven-tenths. Their denominators are 4 and 10. Test the multiples of 10 in order: 10 is not a multiple of 4, but 20 is. So the LCD is 20.
$20 = 4 \times 5 = 10 \times 2$
Rename both fraction parts in twentieths of a mile. Each one keeps its value, because its numerator and denominator are multiplied by the same counting number.
$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}, \quad \frac{7}{10} = \frac{7 \times 2}{10 \times 2} = \frac{14}{20}$
Compare the numerators: 14 is less than 15, so seven-tenths is less than three-fourths. Adding the same 2 whole miles to both lengths does not change which one is greater.
$\frac{14}{20} < \frac{15}{20} \quad \text{so} \quad 2\frac{7}{10} < 2\frac{3}{4}$
Write all four lengths in one list, from shortest to longest.
$\frac{7}{8} < 2\frac{7}{10} < 2\frac{3}{4} < 3\frac{1}{8}$

Answer

From shortest to longest, the trails measure $\frac{7}{8}$ mile, $2\frac{7}{10}$ miles, $2\frac{3}{4}$ miles and $3\frac{1}{8}$ miles.

The takeaway: when fractions of the same whole have the same denominator, their numerators tell you which is greater. When the denominators are different, find the least common denominator by testing the multiples of the largest denominator, rename each fraction, and then compare or order the numerators. For positive mixed numbers whose fraction parts are less than 1, compare the whole-number parts first and use the fraction parts to break ties.

Next, you will convert between mixed numbers and improper fractions (fractions whose numerator is at least as large as the denominator), and you will interpret fractions from models, graphs and tables.