A fraction as a division

Fractions, grade 6 Lesson 1 of 22Learning

A fraction as a division

Three fifths: parts and a division

Three fifths – two stories

This is the fraction

\displaystyle \frac{3}{5}

– three fifths. It is written as two numbers, one above the other.

The denominator is called that because it gives the parts their name (from the Latin nomen, “name”) – here, fifths.

The numerator, on top, counts the fifths (from the Latin numerare, “to count”).

Think of a cake, cut into 5 equal parts. Each part is

\displaystyle \frac{1}{5}

of the cake.

Now take 3 parts:

\displaystyle \frac{1}{5}

\displaystyle \frac{1}{5}

and

\displaystyle \frac{1}{5}

– that is

\displaystyle \frac{3}{5}

of the cake.

The line between the two numbers is called the fraction bar. Its other role is the sign for division – that is, 3 divided by 5.

Division tells the second story: not one cake cut into 5 parts, of which we take 3, but 3 cakes shared among 5 children!

How do we do that? There are fewer cakes than children – no child can get a whole cake.

Cut each cake into 5 equal parts.

The first child gets

\displaystyle \frac{1}{5}

from the first cake,

\displaystyle \frac{1}{5}

from the second and

\displaystyle \frac{1}{5}

from the third.

And so does every child: 3 pieces, each

\displaystyle \frac{1}{5}

of a cake –

\displaystyle \frac{3}{5}

of a cake for each.

Two different stories – the same fraction.

\displaystyle \frac{3}{5}

is a number: 3 parts, each a fifth. And it is also a division: 3 divided by 5,

\displaystyle {3:5}={\frac{3}{5}}

Step 1 of 12

Division as a fraction

Read: “three divided by five is three fifths”.

A fraction is also the result of a division: $\displaystyle {3\div5}={\frac{3}{5}}$ . Any division can be written as a fraction: the numerator is what is shared, the denominator is into how many parts.

7 cakes for 4 children

Four children share 7 cakes: each gets $\displaystyle {7\div4}={\frac{7}{4}}$ cakes – more than a whole cake.

Improper fraction

A fraction whose numerator equals its denominator, like

\displaystyle \frac{4}{4}

, is improper too – it equals 1.

In $\displaystyle \frac{7}{4}$ the numerator is bigger than the denominator. Such a fraction is called an improper fraction, and it is more than 1: four quarters are one whole, and there are three more quarters.

\displaystyle 0

\displaystyle 1

\displaystyle \frac{7}{4}

\displaystyle 2

Mixed number

$\displaystyle \frac{7}{4}$ can also be written as a mixed number – a whole number and a fraction together: $\displaystyle {\frac{7}{4}}={1\frac{3}{4}}$ . Read: “one and three quarters”.

Improper fraction to mixed number and back

From an improper fraction to a mixed number: how many wholes are in it? How many times does 4 go into 7? Once, with 3 left over – so $\displaystyle {\frac{7}{4}}={1\frac{3}{4}}$ .

From a mixed number to an improper fraction: every whole is 5 fifths, so 3 wholes are 15 fifths, and one more fifth: $\displaystyle {3\frac{1}{5}}={\frac{3\times5+1}{5}}={\frac{16}{5}}$ .

Exercise 1

Write the improper fraction as a mixed number, and the mixed number as an improper fraction.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Draw 2 cakes shared equally among 3 children.
How much does each child get?
Write it under the drawing.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Write each division as a fraction.
If the fraction is more than a whole, write it as a mixed number too.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A ribbon 5 m long is cut into 4 equal pieces.
How long is each piece?
Write the division and solve it.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power