Year 5AC9M5N03

Equivalent Fractions

Compare and order fractions with the same and related denominators, including mixed numerals, applying knowledge of factors and multiples.

Read this first

Different fractions can name the same amount. Cutting a cake into more slices does not give you more cake — it just gives you more, smaller pieces.

Look at where the shading stops. Every bar stops in exactly the same place. That is what makes these fractions equivalent — they are different names for one amount.

Words you need

Numerator: The top number. It counts how many pieces you have.
Denominator: The bottom number. It tells you how many equal pieces the whole was cut into.
Equivalent: Equal in value. Two fractions are equivalent when they name the same amount, even though they look different.
Simplest form: A fraction written with the smallest numbers possible. $\tfrac{4}{8}$ in simplest form is $\tfrac{1}{2}$ .

Worked example

e.g. Let's find a fraction equivalent to

\tfrac{3}{4}

with a denominator of 12.

Start with $\tfrac{3}{4}$ . The whole is cut into 4 equal pieces and we have 3 of them.
We want 12 pieces instead of 4. Ask: what do I multiply 4 by to get 12? The answer is 3, because $4 \times 3 = 12$ .
Here is the rule that matters: whatever you do to the bottom, you must do to the top. If you cut every piece into 3 smaller pieces, you also end up holding 3 times as many pieces.
So multiply both numbers by 3: $\tfrac{3}{4} = \tfrac{3 \times 3}{4 \times 3} = \tfrac{9}{12}$ .

Check the picture. The shading stops in the same place both times, so $\tfrac{3}{4}$ and $\tfrac{9}{12}$ really are the same amount. Nothing was added — the pieces were just cut smaller.

Practice

The pictures fade as you go. By the last few you are on your own.

1Look at the bars. What fraction with a denominator of 6 is equivalent to

\tfrac{1}{3}

?with a picture

Answer

\tfrac{2}{6}

. Each third splits into 2 sixths, so 1 third becomes 2 sixths. Both numbers are multiplied by 2.

2The number line is cut into eighths. Which two labels sit at the same point?with a picture

Answer

\tfrac{1}{2}

and

\tfrac{4}{8}

are at the same point, so they are equivalent. On a number line, equivalent fractions always land on exactly the same spot.

3Find the missing number:

\tfrac{2}{5} = \tfrac{?}{15}

with a hint

Need a hint?

What do you multiply 5 by to get 15? Do the same thing to the 2.

Answer

\tfrac{6}{15}

. Since

5 \times 3 = 15

, multiply the top by 3 as well:

2 \times 3 = 6

4Write

\tfrac{6}{10}

in its simplest form.with a hint

Need a hint?

This time you are going the other way — divide instead of multiply. What number goes into both 6 and 10?

Answer

\tfrac{3}{5}

. Both 6 and 10 divide by 2, giving

\tfrac{6 \div 2}{10 \div 2} = \tfrac{3}{5}

. Nothing divides into both 3 and 5, so that is as simple as it goes.

5Find the missing number:

\tfrac{3}{7} = \tfrac{12}{?}

on your own

Answer28. The top was multiplied by 4 (

3 \times 4 = 12

), so the bottom must be too:

7 \times 4 = 28

6Write

\tfrac{9}{12}

in its simplest form.on your own

Answer

\tfrac{3}{4}

. Both divide by 3:

\tfrac{9 \div 3}{12 \div 3} = \tfrac{3}{4}

7Ella ate

\tfrac{2}{6}

of a pizza. Sam ate

\tfrac{1}{3}

of an identical pizza. Who ate more?on your own

AnswerNeither — they ate the same.

\tfrac{1}{3}

is equivalent to

\tfrac{2}{6}

, because

1 \times 2 = 2

and

3 \times 2 = 6

Think about it

Ravi says

\tfrac{1}{4}

must be bigger than

\tfrac{1}{2}

because 4 is bigger than 2. What would you say to him?

One good answerThe bigger the denominator, the smaller each piece is, because the whole is being shared between more pieces. Cutting a cake into 4 gives smaller slices than cutting it into 2. A picture of both bars settles it immediately.

Why does multiplying the top and bottom by the same number not change the value of a fraction?

One good answerBecause you are cutting every existing piece into the same number of smaller pieces. You end up with more pieces, but each one is proportionally smaller, so the total amount is unchanged. Multiplying top and bottom by the same number is really multiplying by 1.

Can you find three different fractions equivalent to

\tfrac{1}{2}

? How many do you think exist?

One good answerFor example

\tfrac{2}{4}

\tfrac{3}{6}

\tfrac{50}{100}

. There are infinitely many — pick any whole number and multiply both the top and the bottom by it.

Mia simplified

\tfrac{8}{12}

\tfrac{4}{6}

. Is she right? Is she finished?

One good answerShe is right but not finished.

\tfrac{4}{6}

is equivalent to

\tfrac{8}{12}

, but both 4 and 6 still divide by 2, giving

\tfrac{2}{3}

. Simplest form means no number other than 1 divides into both.

Common mistakes

Adding the same number to the top and bottom instead of multiplying — writing

\tfrac{1}{2} = \tfrac{2}{3}

Adding changes the value. Check with a picture:

\tfrac{1}{2}

is half a bar,

\tfrac{2}{3}

is more than half. Equivalent fractions come from multiplying or dividing both numbers, never from adding.

Thinking a bigger denominator always means a bigger fraction.

The denominator counts how many pieces the whole was cut into, so a bigger denominator means smaller pieces.

\tfrac{1}{8}

is smaller than

\tfrac{1}{3}

Stopping too early when simplifying — leaving

\tfrac{4}{6}

instead of

\tfrac{2}{3}

Always check afterwards whether any number still divides into both. If one does, keep going.

Forgetting the pieces must be equal.

A bar cut into 4 uneven pieces does not show quarters. Fractions only work when every piece is exactly the same size.

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