Year 4 Maths Fractions

Fractions greater than one (mixed numerals)

Q: Is 13/4 bigger than 3 1/4?

No. They are the same amount written two different ways. 13/4=3 1/4.

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Theory

A fraction greater than one can be written as an improper fraction like \(\dfrac{4}{3}\) or as a mixed numeral like \(1\dfrac{1}{3}\). This Year 4 maths page shows how to read them from shapes and number lines, how to change an improper fraction into a mixed numeral and back again, and how to compare amounts greater than one. In Western Australia this is taught in Year 5. Curriculum: VC2M4N04 / AC9M4N04.

An improper fraction has a top number (the numerator) that is the same as or larger than its bottom number (the denominator). It is worth one whole or more, like \(\dfrac{4}{3}\).

A mixed numeral names the same amount as a whole number and a fraction together, like \(1\dfrac{1}{3}\). The whole number counts the full groups and the fraction counts the leftover parts.

The two forms are equal: \(\dfrac{4}{3}=1\dfrac{1}{3}\). Remember that one whole is a fraction with the same top and bottom, so \(\dfrac{4}{4}=1\), \(\dfrac{3}{3}=1\) and \(\dfrac{5}{5}=1\).

Four thirds is one whole and one more third: \(\dfrac{4}{3}=1\dfrac{1}{3}\).

On a number line, \(\dfrac{4}{3}\) sits one third-step past \(1\), at \(1\dfrac{1}{3}\).

These facts help you swap between improper fractions and mixed numerals.

\[ \dfrac{4}{4}=1 \]

\frac{4}{4} = 1

\[ \dfrac{4}{3}=1\dfrac{1}{3} \]

\frac{4}{3} = 1 \frac{1}{3}

\[ 2=\dfrac{12}{6} \]

2 = \frac{12}{6}

\[ 1\dfrac{3}{5}=\dfrac{8}{5} \]

1 \frac{3}{5} = \frac{8}{5}

          One whole = same top and bottom. Any fraction with a matching numerator and denominator is one whole: \(\dfrac{2}{2}=\dfrac{3}{3}=\dfrac{5}{5}=1\). To make wholes from an improper fraction, count how many of those groups you have.
        

How to change an improper fraction to a mixed numeral

Find one whole. The denominator tells how many parts make one whole, so \(4\) quarters make a whole.
Take out the wholes. Count how many full groups of that many parts you have.
Write the leftover. Put the whole number and the leftover fraction side by side.

To go back the other way, change each whole into parts and add them on: \(2\dfrac{1}{5}=\dfrac{10}{5}+\dfrac{1}{5}=\dfrac{11}{5}\).

How to compare fractions greater than one

Same bottom number? Compare the top numbers: more parts is the larger amount.
Use the wholes. Find how many parts make each whole number. In sixths, \(1=\dfrac{6}{6}\), \(2=\dfrac{12}{6}\), \(3=\dfrac{18}{6}\). A fraction is more than that whole number when its top number is bigger.
Parts still needed. To reach the next whole number, take the parts you have away from the parts in that whole.

Example 1 — Picture to improper

Each bar is one whole cut into fifths. One whole bar is shaded and \(3\) more fifths are shaded. Write the total as an improper fraction.

Solution

\(\text{one whole}\)	\(=\)	\(\dfrac{5}{5}\)
\(\dfrac{5}{5}+\dfrac{3}{5}\)	\(=\)	\(\dfrac{8}{5}\)

\frac{5}{5} + \frac{3}{5} = \frac{8}{5}

The total is \(\dfrac{8}{5}\).

Example 2 — Improper to mixed

Write \(\dfrac{11}{3}\) as a mixed numeral.

Solution

\(\dfrac{11}{3}\)	\(=\)	\(\dfrac{9}{3}+\dfrac{2}{3}\)
\(\dfrac{9}{3}\)	\(=\)	\(3\)
\(\dfrac{11}{3}\)	\(=\)	\(3\dfrac{2}{3}\)

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

Three wholes and two thirds: \(3\dfrac{2}{3}\).

Example 3 — Mixed to improper

Write \(2\dfrac{1}{5}\) as an improper fraction in fifths.

Solution

\(2\)	\(=\)	\(\dfrac{10}{5}\)
\(2\dfrac{1}{5}\)	\(=\)	\(\dfrac{10}{5}+\dfrac{1}{5}\)
\(2\dfrac{1}{5}\)	\(=\)	\(\dfrac{11}{5}\)

2 \frac{1}{5} = \frac{11}{5}

The improper form is \(\dfrac{11}{5}\).

Example 4 — Real world

Jelly is set in half-cup moulds. Priya pours \(7\) half-cups into a bowl. How many cups is that? Give a mixed numeral.

Solution

\(7\text{ halves}\)	\(=\)	\(\dfrac{7}{2}\)
\(\dfrac{7}{2}\)	\(=\)	\(\dfrac{6}{2}+\dfrac{1}{2}\)
\(\dfrac{7}{2}\)	\(=\)	\(3\dfrac{1}{2}\)

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

She has \(3\dfrac{1}{2}\) cups of jelly.

Example 5 — More than 3 wholes?

Which of these is more than \(3\) wholes: \(\dfrac{14}{5}\), \(\dfrac{15}{5}\) or \(\dfrac{17}{5}\)?

Solution

\(3\text{ wholes}\)	\(=\)	\(3 \times 5 = 15\text{ fifths}\)
\(14 < 15,\ \ 15 = 15\)	\(,\)	\(17 > 15\)

\frac{17}{5} > 3

Only \(\dfrac{17}{5}\) is more than 3 wholes. \(\dfrac{15}{5}\) is exactly 3 and \(\dfrac{14}{5}\) is less.

Example 6 — Parts still needed

Ben has \(2\dfrac{3}{5}\) bars of chocolate, each cut into fifths. How many more fifths does he need to have \(3\) whole bars?

Solution

\(2\dfrac{3}{5}\)	\(=\)	\(\dfrac{10}{5}+\dfrac{3}{5}=\dfrac{13}{5}\)
\(3\)	\(=\)	\(\dfrac{15}{5}\)
\(15 - 13\)	\(=\)	\(2\)

\frac{15}{5} - \frac{13}{5} = \frac{2}{5}

He needs 2 more fifths.

Common pitfalls

Do not flip the fraction. Watch out for turning it upside down. Six fifths is \(\dfrac{6}{5}\), never \(\dfrac{5}{6}\). The number of parts you have goes on top.

Count the whole groups too. In \(3\dfrac{1}{4}\) the \(3\) stands for \(\dfrac{12}{4}\), so the improper form is \(\dfrac{13}{4}\), not \(\dfrac{4}{4}\) or \(\dfrac{1}{4}\).

Same amount, two names. \(\dfrac{13}{4}\) and \(3\dfrac{1}{4}\) are equal. One form is not bigger than the other.

Frequently asked questions

What is an improper fraction?

An improper fraction has a top number the same as or bigger than the bottom number, so it is worth one whole or more. For example, \(\dfrac{13}{4}\) is thirteen quarters, which is more than three wholes.

What is a mixed numeral?

A mixed numeral is a whole number and a fraction written together, like \(3\dfrac{1}{4}\). It shows the same amount as an improper fraction.

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

Find how many whole groups you can make, then write the leftover as a fraction. For \(\dfrac{13}{4}\), twelve quarters make three wholes and one quarter is left over, so \(\dfrac{13}{4}=3\dfrac{1}{4}\).

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

Change each whole into parts and add on the extra parts. For \(2\dfrac{1}{5}\), two wholes are \(\dfrac{10}{5}\), plus \(\dfrac{1}{5}\) is \(\dfrac{11}{5}\).

Is 13/4 bigger than 3 1/4?

No. They are the same amount written two different ways. \(\dfrac{13}{4}=3\dfrac{1}{4}\).

How many sixths make 2 wholes?

Six sixths make one whole, so two wholes need twelve sixths. That is \(\dfrac{12}{6}\).

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