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InequalitiesSimplifying fractions

Equivalent fractions Lowest common multipleThis topic is relevant for:

Here we will learn about comparing fractions, including how to compare fractions with different denominators to work out which of two fractions is the smaller fraction and which is the larger fraction.

There are also comparing fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Comparing fractions **allows us to compare the size of a fraction with other fractions.

To do this we can use equivalent fractions. We can also use inequality symbols < (less than) and > (greater than) to help us write about the comparison.

For example, which fraction is larger, \frac{2}{5} or \frac{3}{8}?

In the diagrams, the fractions are indicated by the white sections.

Using equivalent fractions we can convert both fractions so they have a common denominator. The least common denominator (lcd), or, the lowest common multiple (least common multiple) of the denominators.

The lowest common multiple of 5 and 8 is 40.

\frac{2}{5}=\frac{2\times 8}{5\times 8}=\frac{16}{40}and

\frac{3}{8}=\frac{3\times 5}{8\times 5}=\frac{15}{40}\frac{16}{40} has a larger numerator than \frac{15}{40}.

Therefore \frac{2}{5} is the larger number.

We can write this as

\frac{2}{5}>\frac{3}{8}.In order to compare fractions:

**See if the fractions have the same denominators.****Make equivalent fractions if needed.****Write the answer using the original fractions.**

Get your free comparing fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free comparing fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONWhich is the smallest fraction \frac{3}{8} or \frac{3}{10}?

**See if the fractions have the same denominators.**

The fractions do not have the same denominators (bottom numbers), but they have the same numerators (top numbers).

We can use the fact that eighths are bigger than tenths.

So the \frac{3}{8} will be larger than \frac{3}{10}.

2**Make equivalent fractions if needed.**

We do not need to use equivalent fractions. The fraction with the smaller denominator will be the larger number.

3**Write the answer using the original fractions.**

\frac{3}{10} is the smaller number.

We can write this as

\frac{3}{8}>\frac{3}{10}.Which is the smallest fraction \frac{2}{7} or \frac{5}{7}?

**See if the fractions have the same denominators.**

The fractions have the same denominators (bottom numbers). We can then compare the numerators.

**Make equivalent fractions if needed.**

We do not need to use equivalent fractions. The fraction with the smaller numerator will be the smaller number.

**Write the answer using the original fractions.**

\frac{2}{7} is the smaller number.

We can write this as

\frac{2}{7}<\frac{5}{7}.

Which is the smallest fraction \frac{2}{5} or \frac{3}{4}?

**See if the fractions have the same denominators.**

The fractions do not have the same denominators (bottom numbers).

**Make equivalent fractions if needed.**

We need to use equivalent fractions.

The lowest common denominator will be 5 \times 4=20.

When the fractions have the same denominator, we can then compare the numerators.

\frac{2}{5}=\frac{2\times 4}{5\times 4}=\frac{8}{20}

\frac{3}{4}=\frac{3\times 5}{4\times 5}=\frac{15}{20}

**Write the answer using the original fractions.**

\frac{2}{5} is the smaller number.

We can write this as

\frac{2}{5}<\frac{3}{4}.

Alternatively this question could have been answered by knowing that \frac{2}{5} is less than a half and \frac{3}{4} is greater than a half.

Which is the largest fraction \frac{4}{9} or \frac{5}{11}?

**See if the fractions have the same denominators.**

The fractions do not have the same denominators (bottom numbers).

**Make equivalent fractions if needed.**

We need to use equivalent fractions.

The lowest common denominator will be 9 \times 11=99.

When the fractions have the same denominator, we can then compare the numerators.

\frac{4}{9}=\frac{4\times 11}{9\times 11}=\frac{44}{99}

\frac{5}{11}=\frac{5\times 9}{11\times 9}=\frac{45}{99}

**Write the answer using the original fractions.**

\frac{5}{11} is the larger number.

We can write this as

\frac{4}{9}<\frac{5}{11}.

Which is the largest value \frac{14}{3} or \frac{7}{2}?

**See if the fractions have the same denominators.**

The fractions do not have the same denominators (bottom numbers).

**Make equivalent fractions if needed.**

We can use equivalent fractions.

First it might be worth writing the improper fractions as mixed numbers.

\frac{14}{3}=4\frac{2}{3}

\frac{7}{2}=3\frac{1}{2}

We can then compare the whole number part of the mixed number.

**Write the answer using the original fractions.**

\frac{14}{3} is the larger number.

We can write this as

\frac{14}{3}>\frac{7}{2}.

Which is the largest value 2\frac{1}{3} or 2\frac{2}{7}?

**See if the fractions have the same denominators.**

The fractions do not have the same denominators (bottom numbers).

**Make equivalent fractions if needed.**

The mixed numbers have the same whole number part. We can use equivalent fractions. The common denominator is 21.

2\frac{1}{3}=\frac{7}{3}=\frac{7\times 7}{3\times 7}=\frac{49}{21}

2\frac{2}{7}=\frac{16}{7}=\frac{16\times 3}{7\times 3}=\frac{48}{21}

We can then compare the numerators.

Alternatively we could just change the fraction part of the mixed number.

2\frac{1}{3}=2\frac{7}{21}

2\frac{2}{7}=2\frac{6}{21}

**Write the answer using the original fractions.**

2\frac{1}{3} is the larger number.

We can write this as

2\frac{1}{3} > 2\frac{2}{7}.

**Use inequality symbols correctly**

You may be asked to use inequality symbols. Make sure you get them the right way around, < is “less than” and > is “greater than”.

**Equivalent fractions is only one way to compare fractions**

You do not have to use equivalent fractions to compare fractions. It may be easier to use decimals instead. For example, comparing \frac{1}{2} and \frac{3}{10}. These fractions as decimals would be 0.5 and 0.3. Therefore \frac{1}{2} is the smallest fraction and \frac{3}{10} is the larger fraction.

1. Which is the smallest value?

\frac{3}{5} \hspace{1cm} \frac{3}{17} \hspace{1cm} \frac{3}{4} \hspace{1cm} \frac{3}{11}

\frac{3}{5}

\frac{3}{4}

\frac{3}{17}

\frac{3}{11}

All the fractions have the same numerator. We can then compare the denominators. 17 is the largest denominator, so \frac{3}{17} is the smallest.

2. Which is the largest fraction?

\frac{8}{15} \hspace{1cm} \frac{11}{15} \hspace{1cm} \frac{7}{15} \hspace{1cm} \frac{13}{15}

\frac{11}{15}

\frac{13}{15}

\frac{8}{15}

\frac{7}{15}

All the fractions have the same denominators. We can then compare the numerators. 13 is the largest numerator, so \frac{13}{15} is the largest.

3. Which is the largest fraction?

\frac{4}{5} \hspace{1cm} \frac{1}{2} \hspace{1cm} \frac{3}{4} \hspace{1cm} \frac{7}{10}

\frac{7}{10}

\frac{4}{5}

\frac{1}{2}

\frac{3}{4}

All the fractions have different denominators. We can use equivalent fractions so that all the fractions have the same denominators and we can compare the fractions more easily.

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

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

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

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

We can then compare the numerators. 16 is the largest numerator, so \frac{4}{5} is the largest.

4. Which is the smallest fraction?

\frac{2}{3} \hspace{1cm} \frac{5}{6} \hspace{1cm} \frac{3}{4} \hspace{1cm} \frac{11}{12}

\frac{5}{6}

\frac{11}{12}

\frac{3}{4}

\frac{2}{3}

All the fractions have different denominators. We can use equivalent fractions so that all the fractions have the same denominators and we can compare the fractions more easily.

\frac{2}{3}=\frac{8}{12}

\frac{5}{6}=\frac{10}{12}

\frac{3}{4}=\frac{9}{12}

\frac{11}{12}

We can then compare the numerators. 8 is the smallest numerator, so \frac{2}{3} is the smallest fraction.

5. Which is the smallest number?

\frac{10}{3} \hspace{1cm} \frac{17}{7} \hspace{1cm} \frac{9}{2} \hspace{1cm} \frac{11}{4}

\frac{9}{2}

\frac{11}{4}

\frac{17}{7}

\frac{10}{3}

We can rewrite these improper fractions as mixed numbers first.

\frac{10}{3}=3\frac{1}{3}

\frac{17}{7}=2\frac{3}{7}

\frac{9}{2}=4\frac{1}{2}

\frac{11}{4}=2\frac{3}{4}

We can see that we need to further look at the numbers with 2 as the whole number part. We could write them using equivalent fractions, but \frac{3}{4} is greater than a half and \frac{3}{7} is less than a half. So, \frac{17}{7} will be the smallest number.

6. Which is the largest number?

7\frac{1}{2} \hspace{1cm} 7\frac{7}{10} \hspace{1cm} 7\frac{13}{20} \hspace{1cm} 7\frac{3}{5}

7\frac{1}{2}

7\frac{3}{5}

7\frac{13}{20}

7\frac{7}{10}

We can rewrite these improper fractions so that the fraction parts have a common denominator of 20.

7\frac{1}{2}=7\frac{10}{20}

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

7\frac{13}{20}

7\frac{3}{5}=7\frac{12}{20}

The largest number will be 7\frac{7}{10}.

1. Compare these unit fractions.

\frac{1}{6} \hspace{1cm} \frac{1}{3} \hspace{1cm} \frac{1}{9} \hspace{1cm} \frac{1}{4}

(a) Write down the smallest value.

(b) Write down the largest value.

**(2 marks)**

Show answer

(a) \frac{1}{9}

**(1)**

(b) \frac{1}{3}

**(1)**

2. Complete these statements using the correct symbol

<, \ > or =.

(a)

(b)

(c)

**(3 marks)**

Show answer

(a) >

**(1)**

(b) =

**(1)**

(c) >

**(1)**

3. Compare these numbers.

3\frac{5}{8} \hspace{1cm} \frac{7}{2} \hspace{1cm} 3.4 \hspace{1cm} \frac{25}{7}

(a) Write down the smallest value.

(b) Write down the largest value.

**(2 marks)**

Show answer

(a) 3.4

**(1)**

(b) 3\frac{5}{8}

**(1)**

4. Robbie says that “two-thirds is smaller than seven-tenths”.

Is Robbie correct? Explain your answer.

**(2 marks)**

Show answer

For using equivalent fractions.

For example, \frac{2}{3} = \frac{20}{30} \ and \ \frac{7}{10} = \frac{21}{30}.

**(1)**

Yes Robbie is correct, with correct workings.

**(1)**

You have now learned how to:

- Compare fractions with the same numerators
- Compare fractions with the same denominators
- Compare fractions with different denominators

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