Grade 9 Common fractions

Equivalent fractions

The same amount can be written many ways. Here is how to move between them without changing it.

numerator

The number on top. It counts how many parts you have.

denominator

The number underneath. It says how many equal parts one whole was cut into.

equivalent fractions

Fractions that look different but have the same value, such as $\frac{1}{2}$ and $\frac{3}{6}$.

mixed number

A whole number written next to a fraction, such as $2\frac{3}{5}$.

improper fraction

A fraction whose numerator is at least as big as its denominator, such as $\frac{13}{5}$ or $\frac{4}{4}$.

simplest form

A fraction whose numerator and denominator share no factor except $1$.

Multiply the numerator and the denominator by the same number and the value does not change. That is how equivalent fractions are made. $\frac{3}{8}$ and $\frac{15}{40}$ are one amount written two ways.

To simplify, divide the top and the bottom by a factor they share. Carry on until the only factor left in common is $1$. Divide, never subtract: taking $1$ off the top and the bottom gives a different number.

For a mixed number, multiply the whole number by the denominator, add the numerator, and keep the denominator. You may write the answer back the other way: $\frac{13}{5}$ is $2\frac{3}{5}$, and a marker often wants the mixed number.

The fraction line means divide, top by bottom. So $\frac{3}{4}=3÷4=0,75$. Some of these divisions stop and some run on forever. $\frac{1}{3}$ gives $0,333\dots$, written $0,\dot{3}$ so that nobody thinks it stops.

Rules to remember

• $\frac{a}{b}=\frac{a\times k}{b\times k}$ for any $k$ except $0$
• $2\frac{3}{5}=\frac{13}{5}$
• $\frac{3}{4}=0,75$

Examples

Traps

• Multiplying only the denominator when changing a fraction. The numerator must be multiplied by the same number, or the value changes.
• Adding the whole number straight onto the numerator, so $2\frac{3}{5}$ becomes $\frac{5}{5}$. Multiply by the denominator first: $2\times 5+3=13$, which gives $\frac{13}{5}$.
• Taking the same number off the top and the bottom to simplify. Only dividing keeps the value. $\frac{18}{24}$ becomes $\frac{3}{4}$, never $\frac{17}{23}$.
• Dividing the denominator by the numerator. The line means top divided by bottom: $\frac{3}{4}=3÷4=0,75$.

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Calculate with fractions

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

reciprocal

The fraction turned upside down. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.

Adding needs equal-sized parts. You cannot add halves to thirds any more than you can add taxis to loaves. Rewrite both over the same denominator first, then add only the numerators.

Multiplying does not need a common denominator. Multiply the tops, multiply the bottoms. $\frac{2}{3}\times \frac{4}{5}=\frac{8}{15}$.

Dividing by a fraction means multiplying by its reciprocal. Turn the second fraction over and multiply. Only the second one turns over.

Change every mixed number to an improper fraction before multiplying or dividing. $2\frac{1}{3}$ becomes $\frac{7}{3}$, because $2\times 3+1=7$. You may change the answer back at the end: $\frac{7}{5}$ is $1\frac{2}{5}$.

A power applies to the whole fraction, top and bottom. ${\left(\frac{2}{3}\right)}^2=\frac{4}{9}$, not $\frac{4}{3}$.

Rules to remember

• $\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$
• $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$

Examples

Traps

• Adding numerators and denominators separately, so $\frac{1}{2}+\frac{1}{3}=\frac{2}{5}$. Make the denominators the same first, then add only the tops: $\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$.
• Turning the first fraction over instead of the one you are dividing by. Keep the first, flip the second: $\frac{3}{4}÷\frac{2}{5}=\frac{3}{4}\times \frac{5}{2}$.
• Looking for a common denominator before multiplying. Multiplying needs no common denominator. Multiply the tops and the bottoms as they are.
• Squaring only the numerator, so ${\left(\frac{2}{3}\right)}^2$ becomes $\frac{4}{3}$. The power applies to both: ${\left(\frac{2}{3}\right)}^2=\frac{4}{9}$.

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Percentages and problems

Working out percentages of money and amounts, and knowing what to compare them to.

percentage

A fraction out of $100$. $25\%$ means $\frac{25}{100}$, which is a quarter.

original amount

The amount you started with, before any increase or decrease.

To find a percentage of an amount, write the percentage over $100$ and multiply. $15\%$ of R240 is $\frac{15}{100}\times 240$, which is R36.

An increase or decrease is worked out on the ORIGINAL amount, always. If airtime costs R50 and goes up by $20\%$, the increase is $\frac{20}{100}\times 50=10$, so the new price is R60.

Read the question carefully. Sometimes it wants the increase itself, and sometimes it wants the new total. These are different numbers and only one of them scores.

In a two-stage problem, the second fraction is usually of what is LEFT, not of what you started with. Work out what is left first, then take the fraction of that.

Rules to remember

• percentage of an amount $=\frac{p}{100}\times \text{amount}$

Examples

Traps

• Giving the increase when the question asked for the new amount. Read the last line again. If it asks for the new price, add the increase on before you answer.
• Working the percentage change out on the new value instead of the original. The base is always what you started with. Divide the change by the original amount.
• Taking the second fraction of the whole amount instead of the remainder. Work out what is left after the first step, then take the fraction of that number.

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