Proper and improper fractions

Frac­tions are every­where at home. Amma cuts a roti in half. A pizza comes in eight slices. A bot­tle is three quar­ters full. Each time, some­thing whole has been cut or shared into equal parts, and a frac­tion tells you how much of it you are talk­ing about.

In this les­son you will learn what the two num­bers of a frac­tion mean, how to tell a proper frac­tion from an improper frac­tion, and why an improper frac­tion is noth­ing to be afraid of.

Think of a cake on a plate. You cut it so that every­one gets a piece. Frac­tions are how you name those pieces.

Every piece must be the same size

Cut a cake into two pieces. Make one piece huge and the other tiny. You can­not call the tiny piece a half.

A half means one of two equal pieces. Equal means the same size as the other. So cut with care, and check that the pieces match.

Remem­ber. A frac­tion only works when the pieces are equal. Same size, every time.

One half

Take one whole cake. Whole means all of it, before you cut any­thing away. Cut it into two equal parts. Each part is $\displaystyle \frac{1}{2}$ of the cake. You say it as one half, or as one by two.

One third

Now cut a whole cake into three equal parts. Each part is $\displaystyle \frac{1}{3}$. You say it as one third, or as one by three.

One quar­ter

Cut a whole cake into four equal parts. Each part is $\displaystyle \frac{1}{4}$. You say it as a quar­ter, or as one by four.

Look at the three cakes side by side. A half is big­ger than a third, and a third is big­ger than a quar­ter. When you cut the same cake into more pieces, each piece gets smaller. So $\displaystyle \frac{1}{4}$ of a cake is less than $\displaystyle \frac{1}{2}$ of the same cake, even though 4 is a big­ger num­ber than 2.

Tak­ing more than one piece

Pic­ture a bar of choco­late cut into three equal pieces. Two of them are coloured in. Each coloured piece is $\displaystyle \frac{1}{3}$ of the bar.

You have two of those pieces. So you add $\displaystyle \frac{1}{3}$ twice.

Two pieces out of three

$$\displaystyle \begin{aligned}&\frac{1}{3} + \frac{1}{3} \\ &= \frac{2}{3}\end{aligned}$$

Here is another bar. It is cut into four equal pieces and three are coloured in. The coloured part is $\displaystyle \frac{3}{4}$ of the bar.

Three pieces out of four

$$\displaystyle \begin{aligned}&\frac{1}{4} + \frac{1}{4} + \frac{1}{4} \\ &= \frac{3}{4}\end{aligned}$$

Exam­ple: a pizza in eight slices

A pizza is cut into 8 equal slices. Ravi eats 3 of them. Each slice is $\displaystyle \frac{1}{8}$ of the pizza, so Ravi ate

$$\displaystyle \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8}$$

of the pizza. The 8 slices are still there in the count; only 3 of them are Rav­i's.

All the pieces make the whole

Now colour the last quar­ter too. All four pieces are coloured in. You are back to the whole bar.

Four quar­ters

$$\displaystyle \begin{aligned}&\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} \\ &= \frac{4}{4} \\ &= 1\end{aligned}$$

The same thing hap­pens with thirds. Three thirds fill one whole cake.

Three thirds

$$\displaystyle \begin{aligned}&\frac{1}{3} + \frac{1}{3} + \frac{1}{3} \\ &= \frac{3}{3} \\ &= 1\end{aligned}$$

It works what­ever you cut. Cut a bar into five equal pieces. All five together are $\displaystyle \frac{5}{5}$, which is one whole. Cut a bar into eight equal pieces. All eight together are $\displaystyle \frac{8}{8}$, one whole again.

Remem­ber. When the top num­ber and the bot­tom num­ber match, you have one whole. So $\displaystyle \frac{4}{4} = 1$ and $\displaystyle \frac{3}{3} = 1$.

What the two num­bers tell you

A frac­tion has a num­ber on top and a num­ber below. The bot­tom num­ber counts the equal pieces in the whole. The top num­ber counts the pieces you have.

Look at $\displaystyle \frac{3}{4}$. The bot­tom 4 says the whole was cut into four equal pieces. The top 3 says you have three of them.

Remem­ber. Bot­tom num­ber: how many equal pieces the whole was cut into. Top num­ber: how many of those pieces you have.

Look back at $\displaystyle \frac{1}{2}$, $\displaystyle \frac{1}{3}$, $\displaystyle \frac{1}{4}$, $\displaystyle \frac{2}{3}$ and $\displaystyle \frac{3}{4}$. In each one the top num­ber is smaller than the bot­tom one. That is what makes it a proper frac­tion. You have some of the pieces, but not all of them. So a proper frac­tion is less than one whole.

$\displaystyle \frac{4}{4}$ and $\displaystyle \frac{3}{3}$ are not proper frac­tions. Their top and bot­tom num­bers match. So each one is a whole, not a part of a whole.

Exam­ple: which part is left?

Ravi ate $\displaystyle \frac{3}{8}$ of the pizza. The whole pizza is $\displaystyle \frac{8}{8}$. Five slices are left, so $\displaystyle \frac{5}{8}$ of the pizza is left. Both $\displaystyle \frac{3}{8}$ and $\displaystyle \frac{5}{8}$ are proper frac­tions, and together they make the whole: $\displaystyle \frac{3}{8} + \frac{5}{8} = \frac{8}{8} = 1$.

These two num­bers have names. You will meet the names in the next les­son.

Improper frac­tions

Improper does not mean wrong. It is just the name for a frac­tion that holds a whole, or a whole and a bit more.

Some­times you have more than one whole. A proper frac­tion can­not say that. It is always less than one whole, because you never have all the pieces.

Pic­ture one full bar of choco­late on the table. Beside it is a quar­ter of a sec­ond bar. You have a whole and a quar­ter.

The full bar is four quar­ters. Add the loose quar­ter to it. That gives five quar­ters in all.

A whole and a quar­ter

$$\displaystyle \begin{aligned}&1 + \frac{1}{4} \\ &= \frac{4}{4} + \frac{1}{4} \\ &= \frac{5}{4}\end{aligned}$$

So $\displaystyle \frac{5}{4}$ is an improper frac­tion. It is a whole and a proper frac­tion joined together.

Now put three quar­ters beside the full bar. The full bar is still four quar­ters. Four quar­ters and three quar­ters give seven quar­ters.

A whole and three quar­ters

$$\displaystyle \begin{aligned}&1 + \frac{3}{4} \\ &= \frac{4}{4} + \frac{3}{4} \\ &= \frac{7}{4}\end{aligned}$$

You can draw this. Draw one bar cut into four pieces, all coloured in. Beside it draw a sec­ond bar with three of its four pieces coloured in.

Exam­ple: a whole and four fifths

Meena has one full bar cut into 5 equal pieces, and 4 pieces of a sec­ond bar of the same size. The full bar is five fifths.

$$\displaystyle \begin{aligned}&1 + \frac{4}{5} \\ &= \frac{5}{5} + \frac{4}{5} \\ &= \frac{9}{5}\end{aligned}$$

Exam­ple: going the other way

You can also start from an improper frac­tion and find the whole inside it. Take $\displaystyle \frac{11}{8}$. Eight of the eleven eighths fill one whole bar. That leaves three eighths.

$$\displaystyle \begin{aligned}&\frac{11}{8} \\ &= \frac{8}{8} + \frac{3}{8} \\ &= 1 + \frac{3}{8}\end{aligned}$$

When noth­ing is left over

Some­times you have every piece there is and noth­ing more. Four quar­ters fill one whole bar. You can write that as $\displaystyle \frac{4}{4}$, or you can write it as 1.

Here the top num­ber and the bot­tom num­ber are the same. So $\displaystyle \frac{4}{4}$ is counted with the improper frac­tions, even though it is exactly one whole. It is not a part of a whole, so it can­not be a proper frac­tion.

Remem­ber. An improper frac­tion has a top num­ber big­ger than its bot­tom num­ber, or equal to it. When it is big­ger, it is a whole and a proper frac­tion together, like $\displaystyle \frac{5}{4}$. When the two num­bers match, like $\displaystyle \frac{4}{4}$, it is exactly one whole.

Remem­ber. From here on, a frac­tion means either a proper frac­tion or an improper frac­tion.

Prac­tice

Add these equal pieces. Keep the bot­tom num­ber the same, because the pieces are still the same size. Only the num­ber of pieces you hold goes up. So only the top num­ber changes.

1. $$\displaystyle \frac{1}{5} + \frac{1}{5}$$
2. $$\displaystyle \frac{1}{8} + \frac{1}{8} + \frac{1}{8}$$
3. $$\displaystyle \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}$$
4. $$\displaystyle 1 + \frac{1}{3}$$
5. $$\displaystyle 1 + \frac{2}{5}$$
6. $$\displaystyle 1 + \frac{3}{8}$$

Proper or improper

Say which kind each one is. Com­pare the top num­ber with the bot­tom num­ber.

1. $$\displaystyle \frac{2}{3}$$
2. $$\displaystyle \frac{9}{8}$$
3. $$\displaystyle \frac{5}{5}$$
4. $$\displaystyle \frac{1}{10}$$

Com­mon mis­takes

• Call­ing unequal pieces frac­tions. A cake cut into one big and one small piece is not cut into halves.
• Think­ing a big­ger bot­tom num­ber means a big­ger piece. $\displaystyle \frac{1}{4}$ is smaller than $\displaystyle \frac{1}{2}$, because the whole was cut into more pieces.
• Adding the bot­tom num­bers. $\displaystyle \frac{1}{5} + \frac{1}{5}$ is $\displaystyle \frac{2}{5}$, not $\displaystyle \frac{2}{10}$. The pieces stay the same size.
• Call­ing $\displaystyle \frac{5}{5}$ a proper frac­tion. When the top and bot­tom match you have the whole, so it is counted as improper.
• Think­ing improper means wrong. $\displaystyle \frac{7}{4}$ is a per­fectly good frac­tion; it just holds more than one whole.

Key terms

Whole

All of some­thing, before any of it is cut away.

Equal parts

Pieces that are all exactly the same size.

Frac­tion

A num­ber that names some of the equal parts of a whole, such as $\displaystyle \frac{3}{4}$.

Bot­tom num­ber

How many equal pieces the whole was cut into.

Top num­ber

How many of those pieces you have.

Proper frac­tion

A frac­tion whose top num­ber is smaller than its bot­tom num­ber, so it is less than one whole.

Improper frac­tion

A frac­tion whose top num­ber is equal to or big­ger than its bot­tom num­ber, so it is one whole or more.

Answers

Add these equal pieces

1. $\displaystyle \frac{1}{5} + \frac{1}{5} = \frac{2}{5}$
2. $\displaystyle \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8}$
3. Six sixths: $\displaystyle \frac{6}{6} = 1$
4. $\displaystyle 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}$
5. $\displaystyle 1 + \frac{2}{5} = \frac{5}{5} + \frac{2}{5} = \frac{7}{5}$
6. $\displaystyle 1 + \frac{3}{8} = \frac{8}{8} + \frac{3}{8} = \frac{11}{8}$

Proper or improper

1. $\displaystyle \frac{2}{3}$: proper, because 2 is smaller than 3.
2. $\displaystyle \frac{9}{8}$: improper, because 9 is big­ger than 8. It is one whole and one eighth.
3. $\displaystyle \frac{5}{5}$: improper, because the num­bers match. It is exactly one whole.
4. $\displaystyle \frac{1}{10}$: proper, because 1 is smaller than 10.