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Year 5 Maths Fractions

Mixed numerals to improper fractions

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Theory

A mixed numeral such as \(2\dfrac{2}{3}\) has a whole-number part and a fraction part. An improper fraction such as \(\dfrac{8}{3}\) has a numerator bigger than its denominator. This Year 5 maths page shows how to change a mixed numeral into an improper fraction: count the parts in the wholes, add the extra parts, and keep the denominator.

A mixed numeral is a whole number and a fraction together: \(2\dfrac{2}{3}\) means \(2\) wholes and \(\dfrac{2}{3}\) more.

An improper fraction has a numerator equal to or greater than its denominator, such as \(\dfrac{8}{3}\). It is \(1\) or more.

If each whole is cut into thirds, \(1\) whole is \(\dfrac{3}{3}\) and \(2\) wholes are \(\dfrac{6}{3}\). So \(2\dfrac{2}{3} = \dfrac{6}{3} + \dfrac{2}{3} = \dfrac{8}{3}\).

The denominator stays the same, because every part is still a third. Only the number of parts (the numerator) changes.

Two and two thirds as bars Three bars, each cut into 3 equal parts. The first two bars are fully shaded and the third has 2 of its 3 parts shaded: 8 thirds altogether. 1 whole 1 whole 23
\(2\dfrac{2}{3}\): \(3 + 3 + 2 = 8\) thirds, so \(\dfrac{8}{3}\).
One and five sixths as bars Two bars, each cut into 6 equal parts. The first bar is fully shaded and the second has 5 of its 6 parts shaded: 11 sixths altogether. 1 whole 56
\(1\dfrac{5}{6}\): \(6 + 5 = 11\) sixths, so \(\dfrac{11}{6}\).

Whole times denominator, plus the numerator, over the same denominator:

\[ 2\dfrac{2}{3} = \dfrac{2 \times 3 + 2}{3} = \dfrac{8}{3} \]
22/3=(2×3+2)/3=8/3
\[ w\dfrac{a}{b} = \dfrac{w \times b + a}{b} \]
w a / b =( w × b + a )/ b
\[ 5 = \dfrac{15}{3} \]
5=15/3
Multiply, then add. Multiply the whole number by the denominator first, then add the numerator.
Keep the denominator. The size of each part does not change, so the denominator stays the same.

How to change a mixed numeral to an improper fraction

  1. Multiply the whole number by the denominator: this counts the parts in the wholes.
  2. Add the numerator: these are the extra parts.
  3. Write the total over the same denominator.
  4. Check with a diagram or by changing it back.
Example 1 — From a diagram
Use the first diagram. How many thirds are shaded? Write \(2\dfrac{2}{3}\) as an improper fraction.
Solution
\(3 + 3 + 2\)\(=\)\(8\)
22/3=8/3

\(8\) thirds, so \(2\dfrac{2}{3} = \dfrac{8}{3}\).

Example 2 — Parts in the wholes
How many fifths are there in \(2\) wholes?
Solution
\(2 \times 5\)\(=\)\(10\)
2×5=10

\(10\) fifths, so \(2 = \dfrac{10}{5}\).

Example 3 — Use the rule
Write \(3\dfrac{3}{4}\) as an improper fraction.
Solution
\(3 \times 4 + 3\)\(=\)\(15\)
33/4=15/4

\(3\dfrac{3}{4} = \dfrac{15}{4}\).

Example 4 — Fifths and eighths
Write each as an improper fraction.
(i) \(4\dfrac{2}{5}\)
(ii) \(2\dfrac{5}{8}\)
Solution
\(4 \times 5 + 2\)\(=\)\(22\)
\(2 \times 8 + 5\)\(=\)\(21\)
42/5=22/5,25/8=21/8

(i) \(\dfrac{22}{5}\)
(ii) \(\dfrac{21}{8}\)

Example 5 — A whole number
Write \(5\) as an improper fraction with denominator \(3\).
Solution
\(5 \times 3\)\(=\)\(15\)
5=15/3

\(5 = \dfrac{15}{3}\).

Example 6 — In context
Some cakes are cut into sixths. There are \(3\dfrac{1}{6}\) cakes left. How many sixths is that?
Solution
\(3 \times 6 + 1\)\(=\)\(19\)
31/6=19/6

\(19\) sixths, or \(\dfrac{19}{6}\) of a cake.

Example 7 — Find the mistake
Lily wrote \(4\dfrac{3}{4} = \dfrac{7}{4}\). What did she do wrong?
Solution
\(4 \times 4 + 3\)\(=\)\(19\)
43/4=19/4

She added \(4 + 3\) instead of multiplying \(4 \times 4\) first. The answer is \(\dfrac{19}{4}\).

Common pitfalls

Adding the whole number to the numerator. \(4\dfrac{3}{4}\) is not \(\dfrac{7}{4}\). The \(4\) wholes are \(16\) quarters.
Changing the denominator. \(2\dfrac{5}{8}\) is \(\dfrac{21}{8}\), not \(\dfrac{21}{16}\). The parts are still eighths.
Multiplying by the numerator. Multiply the whole number by the denominator (the number of parts in each whole).
Forgetting the extra parts. \(3\dfrac{3}{4}\) is \(12 + 3 = 15\) quarters, not \(12\).

Frequently asked questions

What is a mixed numeral?

A whole number and a fraction written together, such as \(3\dfrac{3}{4}\).

What is an improper fraction?

A fraction whose numerator is equal to or bigger than its denominator, such as \(\dfrac{15}{4}\).

How do you change a mixed numeral to an improper fraction?

Multiply the whole number by the denominator, add the numerator, and keep the denominator. \(3\dfrac{3}{4} = \dfrac{3 \times 4 + 3}{4} = \dfrac{15}{4}\).

Why does the denominator stay the same?

Every whole is cut into the same size parts as the fraction, so the parts keep the same name.

How do you write a whole number as an improper fraction?

Multiply it by the denominator you want: \(5 = \dfrac{15}{3}\).

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