Teach Me Maths

Simplifying and Ordering Fractions

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How to find equivalent fractions, write a fraction in its simplest form, work out what fraction of a shape is shaded, change between mixed numbers and improper fractions, and put fractions in order of size.

What a fraction means

A fraction is part of a whole. The whole is split into equal parts. The denominator, on the bottom, says how many equal parts there are. The numerator, on the top, says how many of those parts you have.

So \frac{3}{8} of the rectangle is shaded: 3 parts out of 8 equal parts. The parts must be the same size. If they are not, you can't just count them.

In the exam you are usually asked to write a fraction in its simplest form, to say what fraction of a shape is shaded, or to write fractions in order of size, for 1 or 2 marks. Many longer questions also end by asking for an answer as a fraction in its simplest form.

Equivalent fractions

Equivalent fractions are fractions that are the same size. Split each quarter of a bar in half and you get eighths: 3 quarters become 6 eighths, so \frac{3}{4}=\frac{6}{8}

To make an equivalent fraction, multiply the numerator and the denominator by the same number, or divide them both by the same number:

\frac{2}{7}=\frac{4}{14}=\frac{6}{21}=\frac{8}{28}

Never add or subtract the same number on the top and bottom: that changes the size of the fraction.

Sometimes you know what the numerator and denominator add up to. Say \frac{a}{b}=\frac{2}{7} and a+b=36. In \frac{2}{7} the top and bottom add up to 2+7=9, and 36 \div 9 = 4. So multiply the top and bottom by 4: \frac{2}{7}=\frac{8}{28}. Check: 8+28=36

Worked example 1: Finding a missing number

Find the missing number. \frac{2}{5}=\frac{\square}{35}

  1. The denominator has changed from 5 to 35. Work out what it was multiplied by.

  2. Your turn. What is 35 \div 5?

    35 \div 5 = 7, so the denominator was multiplied by 7

  3. Multiply the numerator by 7 too: 2 \times 7 = 14
  4. Answer: 14, so \frac{2}{5}=\frac{14}{35}

Simplest form

To simplify a fraction, divide the numerator and the denominator by the same number. That number must be a common factor: a number that goes into both exactly.

A fraction is in its simplest form when no whole number except 1 goes into both the top and the bottom. For example, \frac{10}{15}=\frac{2}{3} (divide both by 5), and 2 and 3 have no other common factor.

You can simplify in small steps, dividing by 2, then by 3 and so on, but keep going until you can't go any further. Dividing by the highest common factor (HCF) does it in one step.

Worked example 2: Simplifying in steps

Write \frac{36}{48} in its simplest form.

  1. 36 and 48 are both even. Divide them both by 2: \frac{36}{48}=\frac{18}{24}
  2. 18 and 24 are both even too. Divide by 2 again: \frac{18}{24}=\frac{9}{12}

  3. Your turn. 9 and 12 are both in the 3 times table. Divide them both by 3. What is the new denominator?

    9 \div 3 = 3 and 12 \div 3 = 4, so \frac{9}{12}=\frac{3}{4}

  4. No whole number except 1 goes into both 3 and 4, so you can stop.
  5. In one step: the HCF of 36 and 48 is 12, and 36 \div 12 = 3, 48 \div 12 = 4
  6. Answer: \frac{3}{4}

Worked example 3: The fraction of a shape that is shaded

The rectangle is made of equal squares. What fraction of the rectangle is shaded? Give your answer in its simplest form.

  1. The rectangle has 3 rows of 6 squares: 3 \times 6 = 18 equal squares.
  2. 10 whole squares are shaded, and 4 more squares are half shaded.

  3. Your turn. How many whole squares do the 4 halves make?

    4 halves make 2 whole squares, so 10 + 2 = 12 squares are shaded

  4. 12 squares out of 18 are shaded: \frac{12}{18}
  5. The HCF of 12 and 18 is 6. 12 \div 6 = 2 and 18 \div 6 = 3
  6. Answer: \frac{2}{3}

Comparing fractions

If two fractions have the same denominator, the parts are the same size, so the bigger numerator gives the bigger fraction: \frac{5}{9} is bigger than \frac{4}{9}

If they have the same numerator, look at the size of the parts. A bigger denominator means the whole is split into more parts, so each part is smaller.

Otherwise, write the fractions with a common denominator: a number in the times table of every denominator. Then compare the numerators. For \frac{3}{4} and \frac{5}{6}, use 12: \frac{9}{12} and \frac{10}{12}, so \frac{5}{6} is bigger.

Worked example 4: Ordering with a common denominator

Write \frac{2}{3}, \frac{5}{6}, \frac{7}{9} and \frac{13}{18} in order of size. Start with the smallest fraction.

  1. 18 is in the 3, 6, 9 and 18 times tables, so use 18 as the common denominator.
  2. Multiply the top and bottom of \frac{2}{3} by 6, and of \frac{5}{6} by 3: \frac{12}{18} and \frac{15}{18}

  3. Your turn. Write \frac{7}{9} with a denominator of 18. What is the numerator?

    Multiply the top and bottom by 2: \frac{7}{9}=\frac{14}{18}

  4. \frac{13}{18} stays as it is. The numerators in order are 12, 13, 14, 15
  5. Write the fractions as they were given, not the eighteenths.
  6. Answer: \frac{2}{3}, \frac{13}{18}, \frac{7}{9}, \frac{5}{6}

Mixed numbers and improper fractions

A mixed number, such as 2\frac{4}{7}, is a whole number and a fraction. An improper fraction, such as \frac{18}{7}, has a numerator bigger than its denominator. These two are the same amount.

To change a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Keep the denominator:

2\frac{4}{7}=\frac{2\times7+4}{7}=\frac{18}{7}

Each whole is 7 sevenths, so 2 wholes are 14 sevenths, and 4 more make 18

To change back, divide the numerator by the denominator. 23 \div 5 = 4 remainder 3, so \frac{23}{5}=4\frac{3}{5}. The remainder is the new numerator.

To compare mixed numbers and improper fractions, first change any improper fractions to mixed numbers. Then compare the whole numbers. If they are the same, compare the fraction parts.

Worked example 5: Ordering mixed numbers and improper fractions

Write \frac{13}{4}, 3\frac{1}{3} and \frac{19}{6} in order of size. Start with the smallest number.

  1. 13 \div 4 = 3 remainder 1, so \frac{13}{4}=3\frac{1}{4}

  2. Your turn. 19 \div 6 = 3 remainder what?

    19 \div 6 = 3 remainder 1, so \frac{19}{6}=3\frac{1}{6}

  3. All three numbers are 3 and a bit, so compare the fraction parts: \frac{1}{4}, \frac{1}{3} and \frac{1}{6}
  4. They have the same numerator, so the biggest denominator gives the smallest: \frac{1}{6} is less than \frac{1}{4}, which is less than \frac{1}{3}
  5. Answer: \frac{19}{6}, \frac{13}{4}, 3\frac{1}{3}

Other ways to compare

Equivalent fractions help with number lines. On a line from 0 to 1 split into 10 equal parts, each part is \frac{1}{10}. Since \frac{4}{5}=\frac{8}{10}, the fraction \frac{4}{5} is at the 8th mark.

On a calculator paper, you can change each fraction to a decimal by dividing the numerator by the denominator: \frac{3}{8}=3 \div 8=0.375. If two decimals are close, write them both to 3 or more decimal places before you compare them.

Fractions just below 1 can be compared by how far they are from 1. \frac{5}{6} is \frac{1}{6} short of 1, and \frac{7}{8} is only \frac{1}{8} short, so \frac{7}{8} is bigger.

To find a fraction between two others, write them with the same denominator and look for a numerator in between. \frac{1}{4}=\frac{3}{12} and \frac{1}{3}=\frac{4}{12} leave no gap, but written over 24 they are \frac{6}{24} and \frac{8}{24}, so \frac{7}{24} is between them.

Worked example 6: Ordering with a calculator

You may use a calculator. Write \frac{5}{8}, \frac{7}{11}, \frac{9}{14} and \frac{11}{18} in order of size. Start with the smallest fraction.

  1. There is no easy common denominator, so change each fraction to a decimal.
  2. 5 \div 8 = 0.625 and 7 \div 11 = 0.6363\ldots

  3. Your turn. Work out 9 \div 14. Give your answer correct to 3 decimal places.

    9 \div 14 = 0.6428\ldots, which is 0.643 to 3 decimal places

  4. 11 \div 18 = 0.6111\ldots
  5. To 3 decimal places the decimals are 0.611, 0.625, 0.636, 0.643. Write the fractions, not the decimals, in your answer.
  6. Answer: \frac{11}{18}, \frac{5}{8}, \frac{7}{11}, \frac{9}{14}

Common mistakes

  • Ordering by the denominator alone. A bigger denominator means smaller parts, so change to a common denominator or to decimals before you order.
  • Changing the denominator but not the numerator. Whatever you multiply or divide the bottom by, do the same to the top.
  • Stopping before the simplest form. Keep dividing until no whole number except 1 goes into both the top and the bottom.
  • Adding or subtracting on the top and bottom. Only multiplying or dividing both by the same number keeps a fraction the same size.
  • Counting parts that are not equal, or shaded over unshaded. The denominator is the total number of equal parts, shaded and unshaded together.
  • Writing the list the wrong way round. Check whether the question says start with the smallest, and write the numbers as they were given.

Practice questions

More on this topic: Simplifying and Ordering Fractions worksheet with full solutions