Compare, Order and Connect Fractions to Division | Grade 6 Math | LevelKite

In this lesson you will compare fractions, put them in order and see why every fraction is also a division.

A fraction is a number that counts equal parts of a whole. The whole is the one complete thing, or the one unit, that the parts are cut from. In $\frac{3}{4}$ , the denominator $4$ says the whole is cut into 4 equal parts. The numerator $3$ says how many of those parts you have. Here every numerator and denominator is a whole number ( $0$ , $1$ , $2$ , $3$ and so on), and no denominator is $0$ .

To compare two fractions means to decide which one is the greater amount, or whether they are equal. The symbol $<$ means is less than. The symbol $>$ means is greater than.

A comparison of amounts only makes sense when both fractions refer to the same whole. Half of a large pizza is more food than half of a small pizza, even though both are written $\frac{1}{2}$ . So before you compare, check that both fractions measure the same whole, such as bars of the same size or the same unit of length.

Start with two fractions of the same whole that have the same denominator, such as $\frac{3}{8}$ and $\frac{5}{8}$ . Both count eighths, so every part is the same size. Five eighths is two more eighths than three eighths, so $\frac{3}{8} < \frac{5}{8}$ .

A number line shows the same result. On a number line, the whole is the distance from $0$ to $1$ . Cut that distance into 8 equal parts. Then $\frac{3}{8}$ is 3 parts from $0$ , and $\frac{5}{8}$ is 5 parts from $0$ . When the numbers increase to the right, the lesser number is farther to the left. So $\frac{3}{8}$ is to the left of $\frac{5}{8}$ .

Key concept

Same whole, same-size parts, then compare the counts

You can compare two fractions as amounts only when they refer to the same whole.

If they also have the same denominator, every part is the same size. The fraction with the greater numerator is the greater amount.
If the denominators are different, the parts are different sizes, and you cannot compare the numerators yet. First rewrite the fractions as equal amounts with one shared denominator. Then compare the numerators.

Now take two fractions of the same whole with different denominators, such as $\frac{3}{4}$ and $\frac{5}{6}$ . Fourths and sixths are parts of different sizes, so you cannot compare the counts $3$ and $5$ yet.

You already know how to find the least common denominator (LCD). It is the smallest number, other than $0$ , that is a multiple of each denominator. Use it to write equivalent fractions. These are fractions that name the same amount of the same whole with parts of a different size.

Here is why the amount stays the same. Multiplying a fraction's numerator and denominator by $3$ cuts every part into 3 smaller equal parts. You have 3 times as many parts, and each part is one third of the size. It is the same as multiplying by $\frac{3}{3}$ , which equals $1$ . The same reasoning holds for any whole-number multiplier other than $0$ .

So the new fractions compare in the same way as the original ones. Once both fractions count parts of one size in the same whole, the greater numerator shows the greater amount.

Worked example

Compare two fractions with different denominators

Both fractions describe the same whole. Which is greater, $\frac{3}{4}$ or $\frac{5}{6}$ ?

Check the parts. Both fractions refer to the same whole, but one counts fourths and the other counts sixths. The parts are different sizes, so you cannot compare the numerators 3 and 5 yet.
Find the least common denominator of 4 and 6. List the multiples of each number until the same number appears in both lists. The first shared multiple is 12.
$\text{multiples of 4: } 4,\ 8,\ \mathbf{12} \qquad \text{multiples of 6: } 6,\ \mathbf{12}$
Rewrite three fourths in twelfths. Because 4 times 3 is 12, multiply the numerator and the denominator by 3. This cuts each fourth into 3 twelfths, so the amount stays the same.
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
Rewrite five sixths in twelfths. Because 6 times 2 is 12, multiply the numerator and the denominator by 2. This cuts each sixth into 2 twelfths, so the amount stays the same.
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
Compare. Both fractions now count twelfths of the same whole, so the parts are the same size. Nine parts is fewer than ten parts. The original fractions name these same amounts, so they compare in the same way.
$\frac{9}{12} < \frac{10}{12} \quad \text{so} \quad \frac{3}{4} < \frac{5}{6}$

Answer

The greater fraction is $\frac{5}{6}$ , because $\frac{3}{4} < \frac{5}{6}$ .

To order numbers means to list them from least to greatest, or from greatest to least. Ordering is comparing more than two numbers, so the same condition applies. Every number must be measured with the same whole.

Your list may mix two forms. A mixed number, such as $2\frac{1}{3}$ , is a whole number plus a fraction that is less than $1$ . It means $2 + \frac{1}{3}$ . An improper fraction, such as $\frac{9}{4}$ , has a numerator that is greater than or equal to its denominator. You already know how to change one form into the other.

Here is a plan for ordering numbers that are measured with the same whole:

First, change every improper fraction into a mixed number or a whole number.
Next, compare the whole-number parts. A fraction that is less than $1$ has $0$ wholes. Each fraction part is less than $1$ , so a greater whole-number part means a greater number.
Then, where the whole-number parts are equal, rewrite the fraction parts with a common denominator and compare the numerators.

Writing the numbers in one form does not make their parts the same size. It only lines up the whole-number parts. The fraction parts still need a common denominator.

Worked example

Order fractions and mixed numbers

Four ribbons are measured in meters. Their lengths are $2\frac{1}{3}$ , $\frac{9}{4}$ , $1\frac{5}{6}$ and $\frac{7}{2}$ meters. List the lengths from least to greatest.

Check the whole, then match the forms. All four lengths use the same whole, 1 meter, so you can compare them. Change the two improper fractions into mixed numbers. Since 9 divided by 4 is 2 with remainder 1, nine fourths is 2 and one fourth. Since 7 divided by 2 is 3 with remainder 1, seven halves is 3 and one half.
$\frac{9}{4} = 2\frac{1}{4} \qquad \frac{7}{2} = 3\frac{1}{2}$
Compare the whole-number parts: 1, 2, 2 and 3. Every fraction part is less than 1, so a length with fewer whole meters is shorter. The length with 1 whole meter is the least. The length with 3 whole meters is the greatest.
$1\frac{5}{6} \text{ is least} \qquad 3\frac{1}{2} \text{ is greatest}$
The two middle lengths both have 2 whole meters, so compare their fraction parts, one third and one fourth. Thirds and fourths are different sizes. The least common denominator of 3 and 4 is 12, so rewrite both parts in twelfths. Multiplying a numerator and its denominator by the same nonzero number keeps the amount the same.
$\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \qquad \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
Compare. Both fraction parts now count twelfths of a meter, so the parts are the same size. Three twelfths is less than four twelfths, so 2 and one fourth is less than 2 and one third.
$\frac{3}{12} < \frac{4}{12} \quad \text{so} \quad 2\frac{1}{4} < 2\frac{1}{3}$
Write all four lengths in order. Use the forms from the problem: 2 and one fourth was given as nine fourths, and 3 and one half was given as seven halves.
$1\frac{5}{6} < \frac{9}{4} < 2\frac{1}{3} < \frac{7}{2}$

Answer

From least to greatest, the lengths are $1\frac{5}{6}$ , $\frac{9}{4}$ , $2\frac{1}{3}$ and $\frac{7}{2}$ meters.

A fraction also answers a division question. Suppose 4 friends share 3 same-size granola bars equally. The division is $3 \div 4$ . It asks what share, taken 4 times, makes 3 bars. The whole here is one bar.

Cut each bar into 4 equal pieces. Each piece is $\frac{1}{4}$ of a bar. Each friend takes one piece from each of the 3 bars. That is 3 pieces of size $\frac{1}{4}$ , which is $\frac{3}{4}$ of a bar. So $3 \div 4 = \frac{3}{4}$ .

Multiplying by 4 undoes dividing by 4, so you can check by putting the 4 shares back together: $\frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4} = \frac{12}{4} = 3$ . You can add the numerators because all four shares count fourths of the same-size bar. Since 4 fourths make 1 bar, 12 fourths make the 3 bars you started with.

This works whenever a whole number of same-size wholes is shared equally. Let $a$ be the number of wholes being shared. Let $b$ be the number of equal shares, where $b$ is a whole number and not $0$ . Each share gets $a$ pieces of size $\frac{1}{b}$ of one whole. That amount is the fraction $\frac{a}{b}$ .

\frac{a}{b} = a \div b \qquad (b \neq 0)

Sharing same-size bars equally: each division written as a fraction

Sharing situation	Division	Each share, in bars	Check by multiplying
1 bar shared equally by 5 people	$1 \div 5$	$\frac{1}{5}$	$5 \times \frac{1}{5} = \frac{5}{5} = 1$
3 bars shared equally by 4 people	$3 \div 4$	$\frac{3}{4}$	$4 \times \frac{3}{4} = \frac{12}{4} = 3$
5 bars shared equally by 2 people	$5 \div 2$	$\frac{5}{2} = 2\frac{1}{2}$	$2 \times \frac{5}{2} = \frac{10}{2} = 5$
6 bars shared equally by 3 people	$6 \div 3$	$\frac{6}{3} = 2$	$3 \times \frac{6}{3} = \frac{18}{3} = 6$

Read across each row of the table. The dividend, the number being shared, becomes the numerator. The divisor, the number of equal shares, becomes the denominator. The divisor cannot be $0$ . The last column multiplies one share by the number of shares, which puts the shares back together.

The third row shows why the mixed-number conversion you already know works. When 2 people share 5 bars, each person gets 2 whole bars, and 1 bar is left over. They also split that last bar, so each person gets $\frac{1}{2}$ of a bar more. So $5 \div 2 = \frac{5}{2} = 2\frac{1}{2}$ .

The fourth row shows that a fraction can equal a whole number. Six thirds make exactly 2 wholes, so $6 \div 3 = \frac{6}{3} = 2$ .

Worked example

Use division to compare two shares

At table A, 3 students share 5 blocks of clay equally. At table B, 4 students share 7 blocks of clay equally. Every block is the same size. At which table does each student get more clay?

Write each share as a division, then as a fraction. Sharing equally is division. The number of blocks is the dividend, so it becomes the numerator. The number of students is the divisor, so it becomes the denominator. The whole for both fractions is one block of the same size.
$5 \div 3 = \frac{5}{3} \qquad 7 \div 4 = \frac{7}{4}$
Thirds and fourths are parts of different sizes, so find the least common denominator of 3 and 4. The first multiple they share is 12.
$\text{multiples of 3: } 3,\ 6,\ 9,\ \mathbf{12} \qquad \text{multiples of 4: } 4,\ 8,\ \mathbf{12}$
Rewrite both shares in twelfths. Multiplying a numerator and its denominator by the same nonzero number keeps the amount the same.
$\frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12} \qquad \frac{7}{4} = \frac{7 \times 3}{4 \times 3} = \frac{21}{12}$
Compare. Both shares now count twelfths of the same-size block, and 20 parts is fewer than 21 parts.
$\frac{20}{12} < \frac{21}{12} \quad \text{so} \quad \frac{5}{3} < \frac{7}{4}$
Write each share as a mixed number to see how much clay it is. Since 5 divided by 3 is 1 with remainder 2, and 7 divided by 4 is 1 with remainder 3, each student gets 1 whole block and part of another.
$\frac{5}{3} = 1\frac{2}{3} \qquad \frac{7}{4} = 1\frac{3}{4}$

Answer

Each student at table B gets more clay. A student at table B gets $\frac{7}{4} = 1\frac{3}{4}$ blocks, and a student at table A gets $\frac{5}{3} = 1\frac{2}{3}$ blocks.

Takeaway. A fraction is a number. You can compare it, put it in order and get it from a division.

Compare fractions only when they refer to the same whole. Give them a common denominator so that both count parts of one size, then compare the numerators.
To order fractions and mixed numbers of the same whole, write improper fractions as mixed numbers. Compare the whole-number parts first. Where those are equal, compare the fraction parts with a common denominator.
A fraction is the answer to a division: the numerator divided by the denominator, when the denominator is not $0$ . Multiplying that answer by the denominator gives back the numerator.