The top and bottom numbers, and mixed fractions

You meet frac­tions all the time: half a roti, a quar­ter of an hour, three out of four slices of a pizza. Every one of these is a frac­tion, and every frac­tion is writ­ten with two num­bers and a line. Once you know what each num­ber does, frac­tions become much eas­ier to read, com­pare and work with.

A frac­tion is made of two num­bers. One sits above the line. One sits below it. Each num­ber has its own job. This les­son gives them their names.

The two num­bers in a frac­tion

Pic­ture a cake cut into 7 equal slices. You take 4 of those slices. You now have $\displaystyle \frac{4}{7}$ of the cake.

The slices must all be the same size. Sup­pose one slice were much big­ger. Then 4 slices out of 7 would not tell you how much cake you got. Equal slices are what make a frac­tion mean some­thing.

The bot­tom num­ber counts the equal parts in the whole cake. One com­plete cake is called one whole. This cake was cut into 7 parts. So the bot­tom num­ber is 7.

The top num­ber counts the parts you have. You took 4 slices. So the top num­ber is 4.

These two num­bers have proper names. The top num­ber is the numer­a­tor. The bot­tom num­ber is the denom­i­na­tor.

Remem­ber. In $\displaystyle \frac{4}{7}$ the numer­a­tor is 4. The denom­i­na­tor is 7.

Which one is which

Those names are long, so here is a trick. The denom­i­na­tor is down. Both words start with the let­ter d. The other name, numer­a­tor, must be the one on top.

Remem­ber. Denom­i­na­tor is down, below the line. The numer­a­tor sits on top.

The bot­tom num­ber does one more thing. It tells you the size of each piece. Cut a cake into 4 parts and the pieces are big. Cut the same cake into 8 parts and they are small. A big­ger bot­tom num­ber means smaller pieces.

Proper frac­tions

In a proper frac­tion the top num­ber is smaller. It is less than the bot­tom num­ber. You hold some of the parts, but not all. So a proper frac­tion is less than one whole.

Take $\displaystyle \frac{3}{4}$. The cake was cut into 4 parts. You hold 3 of them. One part is still on the plate.

When a cake is cut into 4 equal parts, each part is called a quar­ter. The bot­tom num­ber gives the pieces their name. 5 parts are fifths. 6 parts are sixths. 8 parts are eighths.

Improper frac­tions

Some­times the top num­ber is big­ger than the bot­tom. Some­times the two num­bers are the same. Both of these are improper frac­tions. Both mean you have a whole cake, or more than one.

Improper does not mean wrong. It is just the name for a frac­tion that has reached a whole cake or gone past it.

Look at $\displaystyle \frac{4}{4}$ first. It means 4 quar­ter pieces. Four quar­ters make the whole cake. So $\displaystyle \frac{4}{4}$ is 1.

Now look at $\displaystyle \frac{5}{4}$. It means 5 quar­ter pieces. One cake gives you only 4 of them. The fifth piece must come from a sec­ond cake. So $\displaystyle \frac{5}{4}$ is more than one whole.

And $\displaystyle \frac{7}{4}$ means 7 quar­ter pieces. That is one whole cake and 3 quar­ters of the next one.

Remem­ber. A proper frac­tion is less than one whole. An improper frac­tion is one whole or more.

Remem­ber. From here on, the word frac­tion means one of these two. It is either a proper frac­tion or an improper frac­tion.

Mixed frac­tions

Go back to $\displaystyle \frac{5}{4}$. You have one whole cake and one quar­ter. There is a short way to write that. You write $\displaystyle 1\frac{1}{4}$. This is called a mixed frac­tion. It mixes a whole num­ber with a frac­tion.

Writ­ing $\displaystyle \frac{5}{4}$ as a mixed frac­tion

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

The 4 quar­ters became one whole cake. The quar­ter left over stays a quar­ter. Noth­ing was added and noth­ing was lost.

Chang­ing $\displaystyle \frac{9}{4}$ into a mixed frac­tion

You have 9 quar­ter pieces. Each whole cake needs 4 of them. Two cakes use up 8 pieces. That leaves 1 piece over. So $\displaystyle \frac{9}{4}$ is $\displaystyle 2\frac{1}{4}$.

Work­ing out $\displaystyle \frac{9}{4}$

$$\displaystyle \begin{aligned}\frac{9}{4} &= \frac{8}{4} + \frac{1}{4} \\ &= 2 + \frac{1}{4} \\ &= 2\frac{1}{4}\end{aligned}$$

Going back the other way

You can turn a mixed frac­tion back again. Start with $\displaystyle 2\frac{1}{4}$. It means 2 whole cakes and 1 quar­ter.

Each cake holds 4 quar­ters. Two cakes hold 8 quar­ters. Add the 1 spare quar­ter. That gives 9 quar­ters in all. So $\displaystyle 2\frac{1}{4}$ is $\displaystyle \frac{9}{4}$.

Chang­ing $\displaystyle 2\frac{1}{4}$ into a frac­tion

$$\displaystyle \begin{aligned}2\frac{1}{4} &= \frac{2 \times 4 + 1}{4} \\ &= \frac{8 + 1}{4} \\ &= \frac{9}{4}\end{aligned}$$

The 2 is the num­ber of cakes. The 4 is the quar­ters in each cake. So $2 \times 4$ is the quar­ters in both cakes. The 1 is the spare quar­ter.

Remem­ber. The same steps work every time. Mul­ti­ply the whole num­ber by the bot­tom num­ber. Then add the top num­ber. The bot­tom num­ber does not change.

The bot­tom num­ber stays at 4 for a rea­son. You are still count­ing quar­ters. Only the num­ber of quar­ters changed.

Going back the other way

Now turn $\displaystyle \frac{11}{3}$ into a mixed frac­tion. You have 11 thirds. Each whole cake takes 3 of them. So ask how many times 3 goes into 11.

11 thirds

$$\displaystyle \begin{aligned}11 \div 3 &= 3 \text{ remainder } 2 \\ \frac{11}{3} &= 3\frac{2}{3}\end{aligned}$$

3 goes into 11 three times, with 2 left over. So you get 3 whole cakes and 2 thirds spare.

Remem­ber. To go the other way, divide the top num­ber by the bot­tom num­ber. The answer is the whole part. What is left over is the new top num­ber. The bot­tom num­ber does not change.

Two more worked exam­ples

Change $\displaystyle \frac{17}{5}$ into a mixed frac­tion

You have 17 fifths. Each whole needs 5 of them. Three wholes use 15 fifths, and 2 fifths are left.

$$\displaystyle \begin{aligned}17 \div 5 &= 3 \text{ remainder } 2 \\ \frac{17}{5} &= \frac{15}{5} + \frac{2}{5} = 3\frac{2}{5}\end{aligned}$$

Change $\displaystyle 4\frac{2}{3}$ into a frac­tion

Four wholes hold $4 \times 3 = 12$ thirds. Add the 2 spare thirds.

$$\displaystyle \begin{aligned}4\frac{2}{3} &= \frac{4 \times 3 + 2}{3} \\ &= \frac{12 + 2}{3} \\ &= \frac{14}{3}\end{aligned}$$

Check by going back: $14 \div 3 = 4$ remain­der $2$, which gives $\displaystyle 4\frac{2}{3}$ again.

Like and unlike frac­tions

Now look at a group of frac­tions together. Check the bot­tom num­bers. If they all match, they are like frac­tions.

Every one of these is cut into fifths. Only the top num­ber changes. The pieces are all the same size.

If the bot­tom num­bers do not match, the frac­tions are unlike.

One is cut into quar­ters. One is cut into sixths. One is cut into eighths. The pieces are all dif­fer­ent sizes.

Why this mat­ters

Like frac­tions have pieces of the same size. That makes them easy to work with. Unlike frac­tions have pieces of dif­fer­ent sizes. They need more care. The chap­ter comes back to this later.

Remem­ber. Like frac­tions have match­ing bot­tom num­bers. Unlike frac­tions do not.

Prac­tice

Say whether each frac­tion below is proper or improper.

Write each of these as a mixed frac­tion.

Write each of these as one frac­tion.

Com­mon mis­takes

• Mix­ing up the two num­bers. The denom­i­na­tor is down, and it counts the equal parts in one whole.
• Think­ing a big­ger denom­i­na­tor means a big­ger piece. More parts means smaller pieces.
• Call­ing $\displaystyle \frac{6}{6}$ proper. When the top equals the bot­tom, the frac­tion is one whole, and it is improper.
• Chang­ing the bot­tom num­ber when you make a mixed frac­tion. The pieces are still the same size, so the bot­tom stays the same.
• For $\displaystyle 2\frac{3}{4}$, adding $2 + 3$ instead of work­ing out $2 \times 4 + 3$.
• Cut­ting a whole into parts that are not equal. A frac­tion only makes sense with equal parts.

Key terms

Frac­tion

A num­ber that shows some equal parts of a whole.

Numer­a­tor

The top num­ber: how many parts you have.

Denom­i­na­tor

The bot­tom num­ber: how many equal parts make one whole.

Proper frac­tion

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

Improper frac­tion

A frac­tion whose top num­ber is equal to or big­ger than its bot­tom num­ber.

Mixed frac­tion

A whole num­ber and a proper frac­tion writ­ten together, such as $\displaystyle 1\frac{1}{4}$.

Like frac­tions

Frac­tions with the same denom­i­na­tor.

Unlike frac­tions

Frac­tions with dif­fer­ent denom­i­na­tors.

Answers

Proper or improper

1) proper 2) improper 3) improper, and equal to one whole 4) improper 5) proper 6) improper

Mixed frac­tions

1) $\displaystyle \frac{7}{4} = 1\frac{3}{4}$ 2) $\displaystyle \frac{11}{3} = 3\frac{2}{3}$ 3) $\displaystyle \frac{9}{2} = 4\frac{1}{2}$

One frac­tion

1) $\displaystyle 3\frac{1}{5} = \frac{16}{5}$ 2) $\displaystyle 2\frac{3}{4} = \frac{11}{4}$ 3) $\displaystyle 1\frac{5}{6} = \frac{11}{6}$

All the answers together, in a table:

Answers