Common fractions (Grade 8) — Mathematics Grades 8–11

Adding, multiplying and dividing fractions, and calculating with percentages.

Gr 8 Term 3 — Common fractionsTerm 3

Practise Common fractions in the app →

What gets asked

Add or subtract fractions with different denominators.
Multiply or divide fractions and mixed numbers.
Calculate a percentage of an amount, or a percentage change.
Solve a word problem that shares or groups amounts using fractions.

You must be able to

Find a common denominator before adding or subtracting fractions.
Multiply fractions straight across, without a common denominator.
Divide by a fraction by multiplying by its reciprocal.
Work out a percentage increase or decrease using the original amount.

Traps that cost marks

Adding the numerators and the denominators separately, e.g. $\frac{2}{5} + \frac{1}{3} = \frac{3}{8}$ . Change both fractions to the same denominator first, then add the numerators only.
Finding a common denominator before multiplying two fractions. Multiplying is different from adding. Just multiply the numerators, then multiply the denominators.
Turning the first fraction upside down instead of the one you are dividing by. Keep the first fraction as it is. Flip the second fraction, then multiply.
Comparing the change to the new amount instead of the amount you started with. Percentage change always compares to the original amount, not the amount you end up with.

Worked example

Calculate:

\frac{2}{3} - \frac{1}{4}

Common denominator is 12: $\frac{2}{3} = \frac{8}{12}$ and $\frac{1}{4} = \frac{3}{12}$ .
Subtract the numerators: $\frac{8}{12} - \frac{3}{12} = \frac{5}{12}$ .
The answer is $\frac{5}{12}$ , already in simplest form.

Equivalent fractions

Fractions that look different but are worth the same, and the three ways to rewrite one.

numerator: The top number of a fraction. It counts how many parts you have.
denominator: The bottom number. It says how many equal parts the whole was cut into.
equivalent fractions: Different fractions that are worth the same, like $\frac{1}{2}$ and $\frac{3}{6}$ .
mixed number: A whole number and a fraction written together, such as $2 \frac{1}{3}$ .
improper fraction: A fraction whose numerator is at least as big as its denominator, like $\frac{7}{3}$ or $\frac{5}{5}$ .
reciprocal: The fraction turned upside down. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$ .

Every fraction has a numerator on top and a denominator below. The denominator tells you how many equal parts the whole was cut into. The numerator tells you how many of those parts you have.

Equivalent fractions look different but are worth the same. Multiply the top and the bottom by the same number: $\frac{1}{2} = \frac{3}{6}$ . Adding the same number to both instead would change the value.

To simplify, divide the top and the bottom by the same number. Keep going until nothing divides into both. $\frac{18}{24} = \frac{3}{4}$ , dividing each by $6$ . That is its simplest form.

A mixed number can be rewritten as an improper fraction. Multiply the whole number by the denominator, then add the numerator. So $2 \frac{1}{3} = \frac{7}{3}$ , because $2 \times 3 + 1 = 7$ . You may write the answer back as a mixed number at the end.

The reciprocal of a fraction is that fraction turned over. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$ , and the reciprocal of $5$ is $\frac{1}{5}$ . For a mixed number, make it improper first.

Rules to remember

$\frac{a}{b} = \frac{a \times k}{b \times k}$
the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$

Examples

Write

\frac{3}{5}

with a denominator of

20

Worked answer

Ask what the $5$ was multiplied by: $5 \times 4 = 20$ .
Do exactly the same on top: $3 \times 4 = 12$ .
So $\frac{3}{5} = \frac{12}{20}$ .

Answer: $\frac{12}{20}$

Multiplying only the bottom would make the fraction smaller than it was.

A class has

30

learners and

18

of them walk to school. Write

\frac{18}{30}

in its simplest form.

Worked answer

Look for the biggest number that divides into both.
$6$ works: $18 \div 6 = 3$ and $30 \div 6 = 5$ .
So $\frac{18}{30} = \frac{3}{5}$ .

Answer: $\frac{3}{5}$

Dividing the top and the bottom by different numbers would leave a fraction that is not equal.

Write

3 \frac{2}{5}

as an improper fraction, then write down its reciprocal.

Worked answer

Whole times denominator: $3 \times 5 = 15$ .
Add the numerator: $15 + 2 = 17$ .
So $3 \frac{2}{5} = \frac{17}{5}$ .
Turn that over to get $\frac{5}{17}$ .

Answer: $\frac{17}{5}$ , and the reciprocal $\frac{5}{17}$

The reciprocal comes from the improper form, never from the fraction part on its own.

Traps

Multiplying only the denominator when building an equivalent fraction. Whatever you do to the bottom, do to the top. $\frac{3}{5} = \frac{12}{20}$ .
Dividing the top and the bottom by different numbers when simplifying. Use the same number on both, or the new fraction is not worth the same.
Adding the whole number straight on to the numerator instead of multiplying first. Whole times denominator, then add the top: $2 \times 3 + 1 = 7$ , giving $\frac{7}{3}$ .
Turning over only the fraction part of a mixed number to find its reciprocal. Make it improper first. The reciprocal of $3 \frac{2}{5}$ is $\frac{5}{17}$ .

Learn and practise “Equivalent fractions” in the app →

Calculate with fractions

Adding, subtracting, multiplying and dividing fractions, and why each one is done differently.

common denominator: A denominator that both fractions can be rewritten with. For halves and thirds it is $6$ .
simplest form: A fraction where nothing except $1$ divides into the top and the bottom.
square root: The number that multiplies by itself to give it. $\frac{9}{16} = \frac{3}{4}$ .
cube root: The number used as a factor three times to give it, on the top and the bottom.

You cannot add halves to thirds as they stand. Rewrite both over a common denominator, then add only the numerators. For $\frac{2}{3} - \frac{1}{4}$ the common denominator is $12$ , and the answer is $\frac{5}{12}$ , already in its simplest form.

Multiplying needs no common denominator at all. Multiply the tops, then multiply the bottoms: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ . The word of also means multiply, so $\frac{2}{5}$ of $250$ is $100$ .

Dividing by a fraction means multiplying by its reciprocal. Only the second fraction turns over. The answer may come out bigger than you started with: $10 \div \frac{1}{5} = 50$ , because fifty fifths fit into ten.

Change every mixed number into an improper fraction before you multiply or divide. A power works on the top and the bottom together: $(\frac{2}{3})^{2} = \frac{4}{9}$ . A square root does the same, and so does a cube root.

In a word problem, read carefully what the fraction is of. A second fraction is usually of what is left, not of the amount you started with. Work out what is left first, then take the fraction of that.

Rules to remember

$\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$
$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Examples

Calculate:

2 \frac{1}{2} - \frac{3}{4}

Worked answer

Change the mixed number: $2 \frac{1}{2} = \frac{5}{2}$ .
Use $4$ as the common denominator: $\frac{5}{2} = \frac{10}{4}$ .
Subtract the tops: $\frac{10}{4} - \frac{3}{4} = \frac{7}{4}$ .
Write it back as $1 \frac{3}{4}$ .

Answer: $\frac{7}{4}$ , or $1 \frac{3}{4}$

Subtracting the fraction parts alone would need $\frac{1}{2} - \frac{3}{4}$ , which drops below zero.

A feeding scheme cooks

250

portions.

\frac{2}{5}

of them go to Grade 8. How many is that?

Worked answer

The word of tells you to multiply.
$\frac{2}{5} \times 250 = 100$ .
So Grade 8 gets $100$ portions.

Answer: $100$ portions

Dividing by $\frac{2}{5}$ instead would give $625$ , more than the whole batch.

A tuck shop has

10

litres of juice. Each cup holds

\frac{1}{5}

of a litre. How many cups?

Worked answer

Dividing by a fraction means multiplying by its reciprocal.
The reciprocal of $\frac{1}{5}$ is $5$ .
$10 \times 5 = 50$ .

Answer: $50$ cups

The answer is bigger than $10$ , because many small cups fit into ten litres.

Traps

Adding the numerators and the denominators separately, e.g. $\frac{2}{5} + \frac{1}{3} = \frac{3}{8}$ . Change both fractions to the same denominator first, then add the numerators only.
Finding a common denominator before multiplying two fractions. Multiplying is different from adding. Just multiply the numerators, then multiply the denominators.
Turning the first fraction upside down instead of the one you are dividing by. Keep the first fraction as it is. Flip the second fraction, then multiply.
Squaring only the numerator, so $(\frac{2}{3})^{2}$ becomes $\frac{4}{3}$ . The power works on both parts: $(\frac{2}{3})^{2} = \frac{4}{9}$ .

Learn and practise “Calculate with fractions” in the app →

Percentages of amounts

Three ways to write the same number, and how to work out a percentage of an amount.

percentage: A fraction out of $100$ . So $25%$ means $25$ out of every $100$ .
decimal fraction: A fraction written with a comma instead of a line, such as $0, 25$ .
the whole: The full amount that a part is being compared with.

A percentage is a fraction out of $100$ . So $25%$ means $\frac{25}{100}$ , and $\frac{25}{100} = \frac{1}{4}$ . A quarter, $0, 25$ and $25%$ are three ways of writing one number.

To turn a common fraction into a decimal fraction, divide the top by the bottom: $1 \div 4 = 0, 25$ . To turn a decimal fraction into a percentage, multiply by $100$ : $0, 25 \times 100 = 25$ .

To find a percentage of an amount, write the percentage over $100$ and multiply. So $15%$ of R $240$ is $\frac{15}{100} \times 240 = 36$ , which is R $36$ .

To write one quantity as a percentage of another, put the part on top and the whole underneath. Then multiply by $100$ . If $30$ of $40$ learners are present, that is $\frac{30}{40} \times 100 = 75$ , so $75%$ .

Both quantities must be in the same unit before you compare them. You cannot put $80$ cents over R $4$ as it stands. Change the rands to cents first, so R $4$ becomes $400$ cents.

Rules to remember

A percentage of $p$ means the fraction $\frac{p}{100}$ .
To write a part as a percentage, divide it by the whole and multiply by $100$ .

Examples

Write

\frac{3}{5}

as a decimal fraction and as a percentage.

Worked answer

Divide the top by the bottom: $3 \div 5 = 0, 6$ .
Now multiply by $100$ : $0, 6 \times 100 = 60$ .
So $\frac{3}{5}$ is $0, 6$ , which is $60%$ .

Answer: $0, 6$ and $60%$

Reading the digits straight off would give $0, 35$ , which is not a conversion at all.

A shop takes

20%

off a school bag marked R

350

. How much comes off?

Worked answer

Write the percentage as a fraction: $\frac{20}{100}$ .
Multiply it by the marked price: $\frac{20}{100} \times 350 = 70$ .
So R $70$ comes off the price.

Answer: R $70$

Dividing R $350$ by $20$ would give R $17, 50$ , which answers a different question.

A learner has R

4

and spends

80

cents on sweets at the spaza. What percentage is that?

Worked answer

Put both into the same unit: R $4$ is $400$ cents.
The part goes on top: $\frac{80}{400}$ .
Multiply by $100$ : $\frac{80}{400} \times 100 = 20$ .
So she spent $20%$ of her money.

Answer: $20%$

Comparing $80$ with $4$ without changing the unit gives a nonsense answer.

Traps

Reading the two numbers off as digits, so $\frac{1}{4}$ is written as $0, 14$ . Divide the top by the bottom instead: $1 \div 4 = 0, 25$ .
Moving the comma the wrong way between a decimal fraction and a percentage. Going to a percentage multiplies by $100$ : $0, 6 \times 100 = 60$ .
Dividing by the percentage instead of multiplying by the fraction over $100$ . Write the percentage over $100$ first, then multiply it by the amount.
Putting the whole on top and the part underneath. The part goes on top and the whole underneath, then multiply by $100$ .

Learn and practise “Percentages of amounts” in the app →

Percentage increase and decrease

Prices that go up and prices that come down, and how to say how big the change was.

original amount: The amount you started with, before any increase or decrease.
new amount: The amount you are left with once the increase or decrease has been applied.
percentage change: How much an amount grew or shrank, written as a percentage of what it was.

Every change is worked out on the original amount. If airtime costs R $50$ and goes up by $20%$ , the increase is $\frac{20}{100} \times 50 = 10$ , so R $10$ . The new amount is $50 + 10 = 60$ , which is R $60$ .

Read the last line of the question with care. Sometimes it wants the increase on its own, and sometimes it wants the new amount. Those are two different numbers, and only one of them scores.

There is a quicker way for a decrease. Taking $15%$ off leaves $85%$ behind, so multiply by $\frac{85}{100}$ just once. For R $200$ that gives $\frac{85}{100} \times 200 = 170$ , so R $170$ .

To find a percentage change, work out the change first. Then divide it by the original amount and multiply by $100$ . A price going from R $300$ to R $345$ changes by $45$ , and $\frac{45}{300} \times 100 = 15$ , so $15%$ .

When two changes follow each other, the second one works on the amount after the first. A R $200$ item rising $10%$ becomes R $220$ . A second rise of $10%$ is then taken of R $220$ , not of R $200$ .

Rules to remember

A percentage change is the change divided by the original amount, times $100$ .
Taking $15%$ off is the same as multiplying by $\frac{85}{100}$ .

Examples

A taxi fare of R

25

goes up by

12%

. What is the new fare?

Worked answer

The increase is $\frac{12}{100} \times 25 = 3$ , so R $3$ .
That $12%$ is taken of the old fare, R $25$ .
New fare: $25 + 3 = 28$ , so R $28$ .

Answer: R $28$

The question asked for the new fare, not for the R $3$ that was added on.

A jacket marked R

400

is reduced by

15%

. What does it cost now?

Worked answer

Taking $15%$ off leaves $85%$ of the price.
Multiply just once: $\frac{85}{100} \times 400 = 340$ .
So the jacket now costs R $340$ .

Answer: R $340$

Using $\frac{15}{100}$ here would give the amount taken off, not the price paid.

Prepaid electricity cost R

120

last month and R

150

this month. What is the percentage increase?

Worked answer

Find the change: $150 - 120 = 30$ .
Divide by the original amount: $\frac{30}{120}$ .
Multiply by $100$ : $\frac{30}{120} \times 100 = 25$ .
So the increase is $25%$ .

Answer: $25%$

Dividing by R $150$ instead would give $20%$ , which compares with the wrong amount.

Traps

Giving the amount of the increase as the final answer instead of the new amount. Read the last line again. If it asks for the new price, add the increase on first.
Using $\frac{15}{100}$ for the short method of a $15%$ decrease. A $15%$ decrease leaves $85%$ , so the short method multiplies by $\frac{85}{100}$ .
Comparing the change to the new amount instead of the amount you started with. Percentage change always compares to the original amount, not the amount you end up with.
Taking the second percentage of the original amount in a two-stage question. Work out the amount after the first change, then take the second percentage of that.

Learn and practise “Percentage increase and decrease” in the app →

Grade 8 Common fractions

What gets asked

You must be able to

Traps that cost marks

Worked example

Equivalent fractions

Rules to remember

Examples

Traps

Calculate with fractions

Rules to remember

Examples

Traps

Percentages of amounts

Rules to remember

Examples

Traps

Percentage increase and decrease

Rules to remember

Examples

Traps

About this material