Grade 5 · 5.NF.B.7 · 5.NF.B.7A

Dividing fractions: sharing into equal parts

One third of a garden bed is still bare, and four children share it equally. How much of the whole bed does each child get?

You need

Unit fractionsGrade 3 What division meansGrade 3

Unlocks

Dividing fractions: how many fitGrade 5 Word problems that divide fractionsMath 6

Where you are 6 in Dividing fractions

Fractions, ratio and percent3

Fractions15

Dividing fractions6

Dividing fractions: sharing into equal parts

Dividing fractions: how many fit Why flipping the second fraction works Word problems that divide fractions Dividing by multiplying by the reciprocal Dividing mixed numbers

Why this exists

The school garden has raised beds: long wooden boxes filled with soil, planted in sections. Ayasha’s class has one bed, and \displaystyle \frac{1}{3} of it is still bare. Four children want to plant the bare part, and the fair way is for each child to get the same amount. How much of the whole bed does each child get?

That is a division. You already know division as sharing: 12 \div 4 is how many each of 4 people gets when 12 things are shared out equally (What division means). You also know that sharing whole things among more people than there are things gives each person a fraction (A fraction as division). What is new here is that the amount being shared is already a fraction. The bare third is cut into 4 shares, and the answer has to say how big one share is compared with the whole bed, not just with the third.

The lesson takes four days. Part 1 shares a unit fraction by drawing it. Part 2 finds the rule for any unit fraction and checks a share by multiplying back. Part 3 tells the same idea as a story, a tape diagram and a number line. Part 4 shares fractions that are not unit fractions, and shows that sharing among n is the same as taking \displaystyle \frac{1}{n} of the amount.

Definition

A unit fraction has 1 on top, like \displaystyle \frac{1}{3}: one of 3 equal parts of a whole (Unit fractions).

Let \displaystyle \frac{a}{b} be a fraction and n a counting number (1, 2, 3 and so on). Then \displaystyle \frac{a}{b} \div n is the size of one share when the amount \displaystyle \frac{a}{b} is split into n equal shares. The share is measured in the same whole as \displaystyle \frac{a}{b}. If \displaystyle \frac{a}{b} is a fraction of a garden bed, the share is a fraction of that garden bed too.

Put the n equal shares back together and you must get the whole amount again. So the share is the number that makes this true:

n \times \left(\frac{a}{b} \div n\right) = \frac{a}{b}

Reading division as “how much is in one share” is the sharing meaning of division. Its older name is the partitive meaning, from the word “part”. The other way to read a division, “how many of these fit”, is the next lesson, Dividing fractions: how many fit.

Intuition

Part 1: sharing a unit fraction

Draw the bed as a long rectangle. Cut it into 3 equal sections, side by side, and shade one. That is the bare \displaystyle \frac{1}{3}.

To share the third among 4 children, cut it into 4 equal rows. Now carry those row lines across the whole bed, so every section is cut the same way. The bed is now 3 sections by 4 rows, which is 3 \times 4 = 12 pieces, all the same size. Each child gets one row of the third, and that row is one of the 12 pieces.

Figure 1The bare third is cut into 4 equal shares, and the lines run across the whole bed. Each share is 1 of 12 equal pieces, so \displaystyle \frac{1}{3} \div 4 = \frac{1}{12}.

So each child gets one twelfth of the bed:

\frac{1}{3} \div 4 = \frac{1}{12}

Carrying the lines across is the key move. Without it you would only know that each child has “one fourth of the third”. That is true, but it does not say how much of the bed it is. The lines across the whole bed cut it into equal pieces, so the share gets a name as a fraction of the bed.

The share is smaller than the third, as a share should be. The third is split among more than one child, so each child gets less than all of it.

Part 2: any unit fraction, and checking by multiplying

Nothing in Part 1 depended on the numbers 3 and 4. Take \displaystyle \frac{1}{b} of the bed and share it among n children. Cut the bed into b sections, then cut every section into n rows. That makes b \times n equal pieces, and one child’s share is one of them:

\frac{1}{b} \div n = \frac{1}{b \times n}

More children means more rows, and more rows means smaller pieces. Sharing \displaystyle \frac{1}{3} among 4 gives twelfths; sharing it among 6 would give eighteenths, which are smaller.

Checking a share. Multiplication and division undo each other (Multiplication and division undo each other). You know 12 \div 4 = 3 because 4 \times 3 = 12. The same is true with fractions. If each of 4 children has \displaystyle \frac{1}{12} of the bed, then together they have \displaystyle 4 \times \frac{1}{12} = \frac{4}{12} of it (A fraction times a whole number). Dividing the top and the bottom by 4 gives an equal fraction:

4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3}

Four shares put back together make the third the children started with, so the share is right. The two facts \displaystyle \frac{1}{3} \div 4 = \frac{1}{12} and \displaystyle 4 \times \frac{1}{12} = \frac{1}{3} say the same thing in two ways. This also works backward: if you know the share and the number of children, you can multiply to find the amount that was shared.

Part 3: stories, tape diagrams and number lines

Here is a garden story. Hemant has \displaystyle \frac{1}{5} of a bag of soil left, and he spreads it equally over 2 beds. How much of the bag goes on each bed? The amount is \displaystyle \frac{1}{5} of a bag, the number of equal shares is 2, and the question is the size of one share. That is \displaystyle \frac{1}{5} \div 2.

A tape diagram draws the whole bag as a long strip. Cut the strip into 5 equal parts and shade one: that is the soil that is left. Split the shaded fifth into 2 equal pieces. Under the tape, a number line from 0 to 1 measures the same bag. If every fifth were split into 2, the line would be marked in tenths, so the first piece ends at \displaystyle \frac{1}{10}.

Figure 2Half of the leftover fifth reaches only to \displaystyle \frac{1}{10} on the number line, so \displaystyle \frac{1}{5} \div 2 = \frac{1}{10} of the bag.

\frac{1}{5} \div 2 = \frac{1}{10}

Each bed gets \displaystyle \frac{1}{10} of a bag. Check: \displaystyle 2 \times \frac{1}{10} = \frac{2}{10} = \frac{1}{5}.

The tape is the bed picture drawn thin, with only one direction to cut in. The number line adds something: it shows the share as a point, a distance from 0. You can see at once that \displaystyle \frac{1}{10} is halfway from 0 to \displaystyle \frac{1}{5}, which is what sharing between 2 should give.

Every sharing story has the same three parts: an amount that is a fraction of some whole, a number of equal shares, and the question “how much is in one share?” The answer is a fraction of the same whole: the same bag, the same bed, the same gallon. To write a story for \displaystyle \frac{1}{7} \div 3, pick a whole, take \displaystyle \frac{1}{7} of it, and share that among 3 people, beds or rows.

Part 4: sharing a fraction that is not a unit fraction

Now the amount has more than 1 on top. Ayasha’s group has \displaystyle \frac{3}{4} of a bed for 3 children. Three fourths is 3 pieces, each \displaystyle \frac{1}{4} of the bed. The pieces are already there, so hand one to each child:

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

When the number of children divides the top number evenly, the pieces can be handed out as they are, and only the top changes: \displaystyle \frac{a}{b} \div n = \frac{a \div n}{b}.

What if it does not divide evenly? Taeyang’s group has \displaystyle \frac{2}{3} of a bed for 4 children. Two thirds will not hand out to 4 children in whole thirds. So cut finer, as in Part 1: cut every third into 4 rows, right across the bed. Now the bed has 3 \times 4 = 12 pieces, and the two thirds is 2 \times 4 = 8 of them. Eight pieces hand out to 4 children as 2 pieces each.

Figure 3One child’s share of \displaystyle \frac{2}{3} among 4 children is one row of the shaded part, 2 pieces of 12. That row is also \displaystyle \frac{1}{4} of the two thirds.

\frac{2}{3} \div 4 = \frac{2}{12} = \frac{1}{6}

Now look at the same picture another way. One child’s share is one row of the shaded part, 1 row out of 4. So it is \displaystyle \frac{1}{4} of the two thirds, and “a fraction of” an amount means multiplying (A fraction times a fraction):

\frac{2}{3} \div 4 = \frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}

Sharing among 4 is the same as taking \displaystyle \frac{1}{4} of the amount. In general, sharing among n is taking \displaystyle \frac{1}{n}:

\frac{a}{b} \div n = \frac{1}{n} \times \frac{a}{b} = \frac{a}{b \times n}

The rule from Part 2 is the case where a = 1.

Results

A unit fraction shared. For counting numbers b and n, one share of \displaystyle \frac{1}{b} among n is

\frac{1}{b} \div n = \frac{1}{b \times n}

Any fraction shared. Sharing among n is taking \displaystyle \frac{1}{n} of the amount:

\frac{a}{b} \div n = \frac{1}{n} \times \frac{a}{b} = \frac{a}{b \times n}

When n divides the top evenly, the same share can be written by handing out the pieces: \displaystyle \frac{a}{b} \div n = \frac{a \div n}{b}. It is an equal fraction to \displaystyle \frac{a}{b \times n}.

Checking by multiplying back. The n shares together are the amount:

n \times \frac{a}{b \times n} = \frac{a}{b}

Why it is true

Start with \displaystyle \frac{a}{b} of a whole. It is a pieces, each \displaystyle \frac{1}{b} of the whole. Cut every one of the b parts of the whole into n equal rows, with the lines running all the way across. The whole now has b \times n equal pieces, so each small piece is \displaystyle \frac{1}{b \times n} of the whole.

Each old piece of size \displaystyle \frac{1}{b} became n small pieces, one in each row. Give each of the n sharers one row of the amount. A row holds one small piece from each of the a old pieces, so every sharer gets a small pieces, all the same. That share is \displaystyle \frac{a}{b \times n}.

Does it fit the definition? The n shares hold n \times a small pieces, which is every small piece in the amount:

n \times \frac{a}{b \times n} = \frac{n \times a}{b \times n} = \frac{a}{b}

The last step divides the top and the bottom by n, which gives an equal fraction. No other share works: a bigger share, taken n times, would be more than \displaystyle \frac{a}{b}, and a smaller one would be less.

The multiplying form agrees. By the rule for multiplying fractions, \displaystyle \frac{1}{n} \times \frac{a}{b} = \frac{1 \times a}{n \times b}, and n \times b = b \times n by the commutative property, so it is \displaystyle \frac{a}{b \times n}.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.

Example 1

Half a bed among three children

Hemant’s class plants beans in \displaystyle \frac{1}{2} of a raised bed. Three children share the bean half equally. What fraction of the bed does each child plant?

Solution

The amount is \displaystyle \frac{1}{2} of the bed and there are 3 equal shares, so the question is \displaystyle \frac{1}{2} \div 3.

Cut the bed into 2 equal sections and shade one for beans. Cut the bean half into 3 equal rows, and carry the lines across the whole bed so every piece is the same size. The bed now has 2 \times 3 = 6 pieces, and each child plants one of them:

\frac{1}{2} \div 3 = \frac{1}{6}

Each child plants \displaystyle \frac{1}{6} of the bed. The bean half holds 3 of the 6 pieces, one for each child, so nothing is left over.

Example 2

Three rows in a sixth

Taeyang has \displaystyle \frac{1}{6} of a bed. He plants it in 3 equal rows: carrots, lettuce and radishes. What fraction of the bed is radishes?

Solution

Taeyang’s sixth is shared into 3 equal rows, so the radish row is \displaystyle \frac{1}{6} \div 3.

Cut the bed into 6 sections, then cut every section into 3 rows. The bed has 6 \times 3 = 18 equal pieces, and the radish row is one of them:

\frac{1}{6} \div 3 = \frac{1}{18}

Radishes cover \displaystyle \frac{1}{18} of the bed. The radish row is a third of Taeyang’s part but only an eighteenth of the bed. The question asked about the bed, so the answer is \displaystyle \frac{1}{18}.

Example 3

Divide, then multiply back

Find \displaystyle \frac{1}{8} \div 4, and check the answer by multiplying.

Solution

This is a unit fraction shared among 4, so

\frac{1}{8} \div 4 = \frac{1}{8 \times 4} = \frac{1}{32}

In a picture: cut a whole into 8 sections and every section into 4 rows, which makes 32 equal pieces; one share of an eighth is one of them.

Check: four shares should make back the eighth. \displaystyle 4 \times \frac{1}{32} = \frac{4}{32}, and dividing the top and the bottom by 4 gives \displaystyle \frac{1}{8}. The share is right.

Example 4

Working backward from the share

Ayasha’s group of 6 children shared one section of a bed equally. Each child got \displaystyle \frac{1}{24} of the bed. What fraction of the bed was the section?

Solution

Call the section s. The story says \displaystyle s \div 6 = \frac{1}{24}. Division and multiplication undo each other, so the 6 shares put back together make the section:

s = 6 \times \frac{1}{24} = \frac{6}{24}

Dividing the top and the bottom by 6 gives \displaystyle \frac{6}{24} = \frac{1}{4}. The section was \displaystyle \frac{1}{4} of the bed.

Check with the rule: \displaystyle \frac{1}{4} \div 6 = \frac{1}{4 \times 6} = \frac{1}{24}, which is the share the story gave.

Example 5

Plant food on a number line

Hemant has \displaystyle \frac{1}{3} of a gallon of plant food. He shares it equally among 5 beds. How much does each bed get? Show it on a number line.

Solution

The amount is \displaystyle \frac{1}{3} of a gallon and there are 5 shares, so the question is \displaystyle \frac{1}{3} \div 5.

Draw a number line from 0 to 1 gallon and mark the thirds. The plant food reaches from 0 to \displaystyle \frac{1}{3}. Split that stretch into 5 equal jumps. If every third were split the same way, the line would be marked in fifteenths, because 3 \times 5 = 15. So the first jump ends at \displaystyle \frac{1}{15}:

\frac{1}{3} \div 5 = \frac{1}{15}

Each bed gets \displaystyle \frac{1}{15} of a gallon. Check: \displaystyle 5 \times \frac{1}{15} = \frac{5}{15} = \frac{1}{3}, dividing the top and the bottom by 5.

Example 6

Two ways to make twelfths

Ayasha shares \displaystyle \frac{1}{4} of a bed among 3 children. Taeyang shares \displaystyle \frac{1}{3} of a bed of the same size among 4 children. Whose children get more of a bed?

Solution

Find each share with the rule.

\frac{1}{4} \div 3 = \frac{1}{4 \times 3} = \frac{1}{12}

\frac{1}{3} \div 4 = \frac{1}{3 \times 4} = \frac{1}{12}

The shares are the same, \displaystyle \frac{1}{12} of a bed each. The reason is 4 \times 3 = 3 \times 4, the commutative property. In pictures, 4 sections of 3 rows and 3 sections of 4 rows both cut the bed into 12 equal pieces. Ayasha starts with a smaller part, but she shares it among fewer children, and the two balance exactly.

Example 7

Handing out pieces that are already there

Hemant’s group has \displaystyle \frac{4}{5} of a bed for tomatoes, shared equally by 2 children. How much of the bed does each child get?

Solution

The question is \displaystyle \frac{4}{5} \div 2. Four fifths is 4 pieces, each \displaystyle \frac{1}{5} of the bed. Two children can take 2 pieces each:

\frac{4}{5} \div 2 = \frac{2}{5}

Sharing between 2 is taking \displaystyle \frac{1}{2}, so the multiplying form must agree, and it does:

\frac{1}{2} \times \frac{4}{5} = \frac{4}{10} = \frac{2}{5}

Each child gets \displaystyle \frac{2}{5} of the bed. Check: \displaystyle 2 \times \frac{2}{5} = \frac{4}{5}.

Example 8

Five sixths among four children

Taeyang’s class has \displaystyle \frac{5}{6} of a bed for flowers, and 4 children to plant it. How much of the bed does each child get?

Solution

The question is \displaystyle \frac{5}{6} \div 4. Five sixths cannot be handed out to 4 children in whole sixths, so cut finer. Cut every sixth into 4 rows. The bed has 6 \times 4 = 24 pieces, and the flower part is 5 \times 4 = 20 of them. Shared among 4 children, that is 20 \div 4 = 5 pieces each:

\frac{5}{6} \div 4 = \frac{5}{24}

Taking \displaystyle \frac{1}{4} of the amount gives the same: \displaystyle \frac{1}{4} \times \frac{5}{6} = \frac{5}{24}.

Check: \displaystyle 4 \times \frac{5}{24} = \frac{20}{24}, and dividing the top and the bottom by 4 gives \displaystyle \frac{5}{6}.

Is the size sensible? The flower part is a bit less than the whole bed, so a fourth of it should be a bit less than a fourth of the bed. It is: \displaystyle \frac{5}{24} is a bit less than \displaystyle \frac{6}{24}, which is \displaystyle \frac{1}{4}.

Boundaries

The number of shares is a counting number, 1 or more. You cannot share among 0 children, and \displaystyle \frac{a}{b} \div 0 names no amount: no share, put together 0 times, gives back an amount that is not 0. The bottom b can never be 0 either, since a fraction with 0 on the bottom names no amount.

Sharing among 1 is allowed and changes nothing: one share is the whole amount, so \displaystyle \frac{a}{b} \div 1 = \frac{a}{b}.

The shares must be equal. If the children split the third in any way they like, there is no single size of a share, and the division has no answer.

The answer is a fraction of the original whole, not of the part being shared. In Example 2 the radish row is \displaystyle \frac{1}{3} of Taeyang’s part and \displaystyle \frac{1}{18} of the bed. Both are true, of different wholes. The division \displaystyle \frac{1}{6} \div 3 gives the second.

Here the number you divide by is a whole number. Dividing by a fraction, such as \displaystyle 4 \div \frac{1}{3}, is not a sharing among people, since nobody shares among a third of a child. The plain way to read it is “how many fit”, which is Dividing fractions: how many fit.

Where this goes next

The next lesson, Dividing fractions: how many fit, reads division the other way: how many copies of a unit fraction fit inside a whole number. Sharing and fitting are the two meanings of dividing with fractions, and every fraction division story is one or the other.

The fact found in Part 4, that dividing by n is the same as multiplying by \displaystyle \frac{1}{n}, grows later into dividing by any fraction (Dividing by multiplying by the reciprocal). Deciding which meaning a story asks for is Word problems that divide fractions.

Exercises

Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 18 with Part 3, 19 to 23 with Part 4, and the reasoning group draws on all four parts.

Part 1: sharing a unit fraction

\displaystyle \frac{1}{2} \div 4
\displaystyle \frac{1}{5} \div 3
\displaystyle \frac{1}{4} \div 5
Two children share \displaystyle \frac{1}{3} of a bed equally. What fraction of the bed does each child get?
Describe an area model for \displaystyle \frac{1}{6} \div 2: how many sections, how many rows, how many pieces in all, and how many pieces in one share. Use it to find the share.
Half of a bed is shared equally by 6 children. What fraction of the bed does each child get?

Part 2: the rule, and multiplying back

Find \displaystyle \frac{1}{7} \div 2 and check it by multiplying.
\displaystyle \frac{1}{9} \div 3
\displaystyle \frac{1}{10} \div 5
A unit fraction of a bed is shared equally among 4 children, and each child gets \displaystyle \frac{1}{20} of the bed. What fraction of the bed was shared?
A section that is \displaystyle \frac{1}{6} of a bed is shared equally, and each child gets \displaystyle \frac{1}{30} of the bed. How many children shared it?
Hemant says \displaystyle \frac{1}{4} \div 2 = \frac{1}{2}. Test his answer by multiplying back, and find the right share.

Part 3: stories, tape diagrams and number lines

Ayasha has \displaystyle \frac{1}{8} of a gallon of plant food and shares it equally between 2 beds. How much does each bed get?
Use a number line from 0 to 1 to find \displaystyle \frac{1}{3} \div 3. Say where the tick marks go.
Taeyang has \displaystyle \frac{1}{4} of a bag of seeds and shares it equally among 4 children. Describe a tape diagram for the story, and find each child’s share of the bag.
Write a garden story for \displaystyle \frac{1}{5} \div 4, and answer it.
Ayasha cuts \displaystyle \frac{1}{3} meter of twine into 7 equal pieces to tie up bean plants. How long is each piece?
Which is larger, \displaystyle \frac{1}{5} \div 3 or \displaystyle \frac{1}{4} \div 4? By how much?

Part 4: sharing any fraction

\displaystyle \frac{2}{3} \div 2
\displaystyle \frac{6}{7} \div 3
\displaystyle \frac{2}{5} \div 3
\displaystyle \frac{3}{8} \div 4
Hemant’s class has \displaystyle \frac{3}{4} of a bed shared equally by 5 children. Write the share as a multiplication, and find it.

Reasoning

Taeyang finds \displaystyle \frac{4}{9} \div 2 by halving the top and gets \displaystyle \frac{2}{9}. Ayasha doubles the bottom and gets \displaystyle \frac{4}{18}. Show that both are right, and explain what each one did to the bed.
Explain why sharing an amount \displaystyle \frac{a}{b} that is not 0 among n children, with n more than 1, always gives each child less than \displaystyle \frac{a}{b}, and why multiplying the share by n always gives back \displaystyle \frac{a}{b}.

Answers

\displaystyle \frac{1}{8}. Two sections, each cut into 4 rows, make 2 \times 4 = 8 pieces; one share is one.
\displaystyle \frac{1}{15}. \displaystyle \frac{1}{5 \times 3} = \frac{1}{15}.
\displaystyle \frac{1}{20}. \displaystyle \frac{1}{4 \times 5} = \frac{1}{20}.
\displaystyle \frac{1}{6} of the bed. Three sections, each cut into 2 rows, make 6 pieces; each child gets one.
\displaystyle \frac{1}{12}. Cut the bed into 6 sections and shade one. Cut every section into 2 rows. That is 6 \times 2 = 12 pieces in all, and one share is 1 piece.
\displaystyle \frac{1}{12} of the bed. \displaystyle \frac{1}{2} \div 6 = \frac{1}{2 \times 6} = \frac{1}{12}.
\displaystyle \frac{1}{14}. Check: \displaystyle 2 \times \frac{1}{14} = \frac{2}{14} = \frac{1}{7}.
\displaystyle \frac{1}{27}. \displaystyle \frac{1}{9 \times 3} = \frac{1}{27}.
\displaystyle \frac{1}{50}. \displaystyle \frac{1}{10 \times 5} = \frac{1}{50}.
\displaystyle \frac{1}{5} of the bed. Four shares put back make the amount: \displaystyle 4 \times \frac{1}{20} = \frac{4}{20} = \frac{1}{5}.
5 children. One share is \displaystyle \frac{1}{6 \times n}, and 6 \times 5 = 30. Check: \displaystyle 5 \times \frac{1}{30} = \frac{5}{30} = \frac{1}{6}.
He is wrong: \displaystyle 2 \times \frac{1}{2} = 1, a whole bed, not \displaystyle \frac{1}{4}. The right share is \displaystyle \frac{1}{8}, since \displaystyle 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4}.
\displaystyle \frac{1}{16} of a gallon. \displaystyle \frac{1}{8} \div 2 = \frac{1}{8 \times 2} = \frac{1}{16}.
\displaystyle \frac{1}{9}. Put tall ticks at the thirds, then split each third into 3 with short ticks, so the line is marked in ninths. The share runs from 0 to the first short tick, \displaystyle \frac{1}{9}.
\displaystyle \frac{1}{16} of the bag. Draw the bag as a tape cut into 4 equal parts and shade one. Split the shaded fourth into 4 equal pieces; if every fourth were split that way the tape would have 16 pieces, and one child’s share is one of them.
One story: Hemant has \displaystyle \frac{1}{5} of a bed left and 4 children share it equally; how much of the bed does each child get? Each gets \displaystyle \frac{1}{20} of the bed, since \displaystyle \frac{1}{5 \times 4} = \frac{1}{20}. Any story works if it shares \displaystyle \frac{1}{5} of a whole into 4 equal shares and asks for one share.
\displaystyle \frac{1}{21} meter. \displaystyle \frac{1}{3} \div 7 = \frac{1}{3 \times 7} = \frac{1}{21}.
\displaystyle \frac{1}{5} \div 3 is larger, by \displaystyle \frac{1}{240}. The shares are \displaystyle \frac{1}{15} and \displaystyle \frac{1}{16}; cutting into 15 pieces makes bigger pieces than cutting into 16, and \displaystyle \frac{1}{15} - \frac{1}{16} = \frac{16}{240} - \frac{15}{240} = \frac{1}{240}.
\displaystyle \frac{1}{3}. Two thirds is 2 pieces; each of 2 shares gets one.
\displaystyle \frac{2}{7}. Six sevenths is 6 pieces; 6 \div 3 = 2 pieces each.
\displaystyle \frac{2}{15}. Cut each fifth into 3: \displaystyle \frac{2}{5} = \frac{6}{15}, and 6 \div 3 = 2 pieces each. Or \displaystyle \frac{1}{3} \times \frac{2}{5} = \frac{2}{15}.
\displaystyle \frac{3}{32}. \displaystyle \frac{1}{4} \times \frac{3}{8} = \frac{3}{32}. Check: \displaystyle 4 \times \frac{3}{32} = \frac{12}{32} = \frac{3}{8}.
\displaystyle \frac{1}{5} \times \frac{3}{4} = \frac{3}{20} of the bed. Check: \displaystyle 5 \times \frac{3}{20} = \frac{15}{20} = \frac{3}{4}.
Both name the same amount: \displaystyle \frac{4}{18} = \frac{2}{9}, dividing the top and the bottom by 2. Taeyang handed out pieces that were already there: four ninths is 4 pieces, and 2 children take 2 each. Ayasha cut finer: she cut every ninth into 2, so the bed has 18 pieces and each child gets one from each of the 4 ninths, 4 eighteenths. Both pass the check: \displaystyle 2 \times \frac{2}{9} = \frac{4}{9}.
Less: the n shares are equal and together make \displaystyle \frac{a}{b}. If one share were as big as \displaystyle \frac{a}{b}, then two or more of them would already be more than \displaystyle \frac{a}{b}, so each share must be smaller. Multiplying back: the share is defined as the size of one of n equal shares of \displaystyle \frac{a}{b}, so n of them put together are the whole amount \displaystyle \frac{a}{b}. In numbers, \displaystyle n \times \frac{a}{b \times n} = \frac{n \times a}{b \times n}, which is \displaystyle \frac{a}{b} after dividing the top and the bottom by n.