 What you need to know

Things to remember:

• When comparing fractions, the denominators have to be the same.
• When comparing fractions with the same denominator, we look at the numerators and compare them like whole numbers.
• When ordering fractions, the denominators have to be the same.
• When ordering fractions with the same denominator, we look at the numerators and order them like whole numbers.

Which fraction is bigger, $\dfrac{3}{5}$ or $\dfrac{4}{7}$? Well, this is kind of tricky. We can’t really compare fractions with different denominators. How about if we look at this using some pictures? This still doesn’t really help us too much. $\dfrac{4}{7}$ has more pieces shaded in, but the shaded pieces in $\dfrac{3}{5}$ are bigger. So, we need to make the pieces the same size. We do this by making the denominators the same, which we saw in Use Common Multiples to Express Fractions with the Same Denominator. This will be our first step in comparing fractions.

Which fraction is bigger $\dfrac{3}{5}$ or $\dfrac{4}{7}$?

Step 1: Find a common multiple of the denominators.

HINT: The easiest way to do this is multiply the denominators together.

$$5\times7=35$$

Step 2: Multiply the fractions so that the denominators the same.

$$\frac{3}{5}=\frac{3\times7}{5\times7}=\frac{21}{35}$$

$$\frac{4}{7}=\frac{4\times5}{7\times5}=\frac{20}{35}$$

Step 3: Compare the fractions by looking at the numerators.

So, our two fractions become:

$\dfrac{21}{35}$     and      $\dfrac{20}{35}$

Looking at the numerators, we can see that $\dfrac{21}{35}$ is biggest. Remember though, this fraction is actually $\dfrac{3}{5}$, which means that $\dfrac{3}{5}$ is bigger than $\dfrac{4}{7}$

To order fractions, we follow the same first two steps as determining which fractions is bigger or smaller.

Order the following fraction from biggest to smallest.

$\dfrac{1}{2}$         $\dfrac{3}{4}$         $\dfrac{7}{10}$       $\dfrac{4}{5}$

Step 1: Find a common multiple of the denominators.

HINT: Multiplying to find the common denominator can be difficult here, as they will get super big. It can be easier to look for a number that is in all of these times tables.

Multiples of 2:              2          4          6          8          10        12        14        16        18        20

Multiples of 4:              4          8          12        16        20        24        28        32        36        40

Multiples of 5:              5          10        15        20        25        30        35        40        45        50

Multiples of 10:            10        20        30        40        50        60        70        80        90        100

So, our common multiple here is 20.

Step 2: Multiply the fractions so that the denominators the same.

$$\frac{1}{2}=\frac{1\times10}{2\times10}=\frac{10}{20}$$

$$\frac{3}{4}=\frac{3\times5}{4\times5}=\frac{15}{20}$$

$$\frac{7}{10}=\frac{7\times2}{10\times2}=\frac{14}{20}$$

$$\frac{4}{5}=\frac{4\times4}{5\times4}=\frac{16}{20}$$

Step 3: Look at the numerators to see how we order the fractions.

If we look at the numerators and order them from biggest to smallest, we get:

16                    15                    14                    10

So, this is how our fractions will go.

$\dfrac{16}{20}$                  $\dfrac{15}{20}$                  $\dfrac{14}{20}$                      $\dfrac{10}{20}$

And now we just need to remember to put in the original fractions.

$\dfrac{4}{5}$                      $\dfrac{3}{4}$                      $\dfrac{7}{10}$                        $\dfrac{1}{2}$

Example Questions

Step 1: Find a common multiple of the denominators.

HINT: The easiest way to do this is multiply the denominators together.

$$8\times9=72$$

Step 2: Multiply the fractions so that the denominators the same.

$$\frac{3}{8}=\frac{3\times9}{8\times9}=\frac{27}{72}$$

$$\frac{4}{9}=\frac{4\times8}{9\times8}=\frac{32}{72}$$

Step 3: Compare the fractions by looking at the numerators.

So, our two fractions become:

$\dfrac{27}{72}$     and      $\dfrac{32}{72}$

Looking at the numerators, we can see that $\dfrac{27}{72}$ is smallest. Remember though, this fraction is actually $\dfrac{3}{8}$, which means that $\dfrac{3}{8}$ is smaller than $\dfrac{4}{9}$

$\dfrac{2}{6}$         $\dfrac{9}{10}$       $\dfrac{3}{12}$       $\dfrac{8}{15}$

Step 1: Find a common multiple of the denominators.

HINT: Multiplying to find the common denominator can be difficult here, as they will get super big. It can be easier to look for a number that is in all of these times tables.

Multiples of 6:              6          12        18        24        30        36        42        48        54        60

Multiples of 10:            10        20        30        40        50        60        70        80        90        100

Multiples of 12:            12        24        36        48        60        72        84        96        108      120

Multiples of 15:            15        30        45        60        75        90        105      120      135      150

So, our common multiple here is 60.

Step 2: Multiply the fractions so that the denominators the same.

$$\frac{2}{6}=\frac{2\times10}{6\times10}=\frac{20}{60}$$

$$\frac{9}{10}=\frac{9\times6}{10\times6}=\frac{54}{60}$$

$$\frac{3}{12}=\frac{3\times5}{12\times5}=\frac{15}{60}$$

$$\frac{8}{15}=\frac{8\times4}{15\times4}=\frac{32}{60}$$

Step 3: Look at the numerators to see how we order the fractions.

If we look at the numerators and order them from smallest to biggest, we get:

15                    20                    32                    54

So, this is how our fractions will go.

$\dfrac{15}{60}$                  $\dfrac{20}{60}$                  $\dfrac{32}{60}$                      $\dfrac{54}{60}$

And now we just need to remember to put in the original fractions.

$\dfrac{3}{12}$                    $\dfrac{2}{6}$                      $\dfrac{8}{15}$                        $\dfrac{9}{10}$