Year 9 Maths Advanced Working with Numbers

Operations with Percentages

Q: How do you calculate a percentage increase?

Multiply by 1 plus the percentage as a decimal. To increase by 6 per cent, multiply by 1.06. To find the percentage increase between two amounts, divide the increase by the original amount and multiply by 100.

Q: Why doesn't a 30% increase followed by a 30% decrease cancel out?

The decrease is taken from the larger, increased amount. Multiplying gives 1.3 times 0.7, which is 0.91, so the overall result is a 9 per cent decrease.

Q: How do you find a percentage of a percentage?

Write both as decimals and multiply. 20 per cent of 35 per cent is 0.2 times 0.35, which is 0.07, or 7 per cent.

20 practice questions 1 video lesson Theory + worked examples

Percentages compare amounts out of $100$ and describe increases and decreases. This NSW Year 9 Advanced maths subtopic revises the Stage 4 skills (NESA outcome MA4‑FRC‑C‑01) and moves on to repeated and reverse percentage changes.

You will write one quantity as a percentage of another, increase or decrease amounts, find an original amount after a change, and find percentage increases such as a school's enrolment growth. Harder questions compare percentage points with percentage change, apply two changes in a row, find the change in a rectangle's area, strengthen a drink and prove that a rise then a fall of $p\%$ always gives a loss. It is part of Year 9 Advanced Working with Numbers.

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Theory

Every percentage increase or decrease is a multiplication: up $6\%$ is $\times 1.06$, down $35\%$ is $\times 0.65$. This Year 9 Advanced maths page revises the skills from earlier years and extends them to percentage change, finding the original amount, successive changes, percents of percents, mixtures and percentage points.

A percentage is a number out of $100$: $r\%$ means $\dfrac{r}{100}$. To write one quantity as a percentage of another, put both in the same units, divide the part by the whole and multiply by $100\%$.

A multiplier turns a change into one multiplication. An increase of $r\%$ leaves $100\% + r\%$ of the original; a decrease of $r\%$ leaves $100\% - r\%$. So a $17\%$ decrease is $\times 0.83$.

Percentage change compares the change with the original amount. When the new amount is known and the original is not, divide by the multiplier. This is a reverse percentage.

Changes made one after another multiply, because each one acts on the amount left by the one before. That is why a rise and a fall of the same percentage do not cancel.

Every percentage change is a multiplication. Increase by $r\%$: multiply by $1 + \dfrac{r}{100}$. Decrease by $r\%$: multiply by $1 - \dfrac{r}{100}$.

Successive changes multiply: $1.3 \times 0.7 = 0.91$. Up $30\%$ then down $30\%$ is an overall $9\%$ decrease, not zero.

Percentage of a quantity, and one quantity as a percentage of another:

\[r\% \text{ of } A = \dfrac{r}{100} \times A \qquad \dfrac{\text{part}}{\text{whole}} \times 100\%\]

\frac{r}{100} \times A

Increase or decrease by $r\%$, and percentage change:

\[\text{new} = \text{original} \times \left(1 \pm \dfrac{r}{100}\right) \qquad \text{percentage change} = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%\]

new = original \times (1 \pm \frac{r}{100})

Reverse a change by dividing by the multiplier: $\text{original} = \dfrac{\text{new}}{\text{multiplier}}$.

Percent of a percent. \(20\%\) of \(35\%\) is \(0.2 \times 0.35 = 0.07 = 7\%\) of the whole. Multiply the decimals; never add or average the percentages.

Area and other products. If a length rises \(20\%\) and a width rises \(25\%\), the area is multiplied by \(1.2 \times 1.25 = 1.5\): a \(50\%\) increase, not \(45\%\).

Percentage points. A score going from \(40\%\) to \(50\%\) rises by \(10\) percentage points, which is a \(25\%\) increase on the original \(40\%\).

How to solve a percentage problem

Identify the original (the $100\%$) amount. In reverse problems it is the unknown.
Write each change as a multiplier: $1.06$ for up $6\%$, $0.65$ for down $35\%$.
Multiply for successive changes or percents of percents; divide to go back to the original.
For a change, divide the difference by the original, not the new amount.
Check by applying your answer forwards. Round money to the nearest cent.

Example 1 — Multipliers and percentage change

(i) Decrease $\$380$ by $17\%$.
(ii) A price drops from $\$840$ to $\$714$. Find the percentage change.
(iii) Write $45$ minutes as a percentage of $2.5$ hours.

Solution

(i) A $17\%$ decrease leaves $83\%$, so multiply by $0.83$.

$380 \times 0.83$

$=$

$\mathbf{\$315.40}$

380 \times 0.83 = 315.40

(ii) Divide the change by the original amount.

$\text{change}$	$=$	$714 - 840 = -126$
$\dfrac{-126}{840} \times 100\%$	$=$	$-15\%$

\frac{- 126}{840} \times 100 % = - 15 %

A $15\%$ decrease.

(iii) Use the same units: $2.5$ hours $= 150$ minutes.

$\dfrac{45}{150} \times 100\%$

$=$

$\mathbf{30\%}$

\frac{45}{150} \times 100 % = 30 %

Example 2 — Finding the original amount

(i) After a $35\%$ discount, a tent costs $\$247$. Find the original price.
(ii) After a $6\%$ rise, weekly rent is $\$583$. Find the rent before the rise.

Solution

The amount given is after the change, so it is not $100\%$. Write it as a multiple of the original, then divide.

$0.65 \times \text{original}$	$=$	$247$
$\text{original}$	$=$	$247 \div 0.65 = \mathbf{\$380}$

original = 247 \div 0.65 = 380

$1.06 \times \text{old rent}$	$=$	$583$
$\text{old rent}$	$=$	$583 \div 1.06 = \mathbf{\$550}$

old rent = 583 \div 1.06 = 550

Check: $380 \times 0.65 = 247$ and $550 \times 1.06 = 583$. Taking $6\%$ off $\$583$ would give $\$548.02$, which is wrong.

Example 3 — Changes one after another

(i) A share price rises $30\%$, then falls $30\%$. Find the overall percentage change.
(ii) A price rises by $60\%$. By what percentage must the new price fall to return to the original?

Solution

(i) Multiply the multipliers.

$1.3 \times 0.7$	$=$	$0.91$
$0.91 - 1$	$=$	$-0.09$

1.3 \times 0.7 = 0.91

An overall $9\%$ decrease. The fall is $30\%$ of a larger amount, so the two changes do not cancel.

(ii) Let the original be $P$. The new price is $1.6P$, and it must fall by $0.6P$. Divide by the new price.

$\dfrac{0.6P}{1.6P} \times 100\%$

$=$

$\mathbf{37.5\%}$

\frac{0.6 P}{1.6 P} \times 100 % = 37.5 %

Check with $\$100$: up to $\$160$, and $37.5\%$ of $\$160$ is $\$60$, back to $\$100$.

Example 4 — Parts of a whole and mixtures

(i) In a club, $70\%$ of members are adults. $40\%$ of the adults and $90\%$ of the children can swim. What percentage of the club can swim?
(ii) $800$ mL of a drink is $12\%$ cordial. How much water must be added to make it $10\%$ cordial?

Solution

(i) Find each group as a fraction of the whole club, then add.

$\text{adults}$	$=$	$0.7 \times 0.4 = 0.28$
$\text{children}$	$=$	$0.3 \times 0.9 = 0.27$
$\text{total}$	$=$	$0.55 = \mathbf{55\%}$

0.28 + 0.27 = 0.55

(ii) The cordial stays at $0.12 \times 800 = 96$ mL. Adding $x$ mL of water changes only the total.

$\dfrac{96}{800 + x}$	$=$	$0.1$
$96$	$=$	$80 + 0.1x$
$x$	$=$	$\mathbf{160}\ \text{mL}$

\frac{96}{800 + x} = 0.1, x = 160

Check: $\dfrac{96}{960} = 0.1 = 10\%$.

Common pitfalls

Divide by the original. From $\$840$ to $\$714$ is a $15\%$ decrease ($\dfrac{126}{840}$), not $17.6\%$ ($\dfrac{126}{714}$).

Watch out for reverse problems. If $\$247$ is the price after $35\%$ off, adding $35\%$ of $\$247$ does not give the original. Divide by $0.65$.

Changes multiply, they do not add. Down $20\%$ then down another $20\%$ is $0.8 \times 0.8 = 0.64$: a $36\%$ decrease, not $40\%$.

Same units first. $45$ minutes out of $2.5$ hours is $\dfrac{45}{150}$, not $\dfrac{45}{2.5}$.

Frequently asked questions

How do you calculate a percentage increase?

Multiply by $1$ plus the percentage as a decimal. To increase by $6\%$, multiply by $1.06$. To find the percentage increase between two amounts, divide the increase by the original amount and multiply by $100\%$.

How do you find the original price before a discount?

Work out what percentage of the original is left, write it as a decimal, and divide the sale price by it. After $35\%$ off, $65\%$ is left, so divide by $0.65$.

Why doesn't a 30% increase followed by a 30% decrease cancel out?

The decrease is taken from the larger, increased amount. $1.3 \times 0.7 = 0.91$, so the overall result is a $9\%$ decrease.

What is the difference between percent and percentage points?

Percentage points measure the gap between two percentages. A rise from $40\%$ to $50\%$ is $10$ percentage points, but it is a $25\%$ increase, because $10$ is $25\%$ of $40$.

How do you find a percentage of a percentage?

Write both as decimals and multiply. $20\%$ of $35\%$ is $0.2 \times 0.35 = 0.07 = 7\%$.

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Percentages and Money

\(\text{change}\)	\(=\)	\(714 - 840 = -126\)
\(\dfrac{-126}{840} \times 100\%\)	\(=\)	\(-15\%\)

\(0.65 \times \text{original}\)	\(=\)	\(247\)
\(\text{original}\)	\(=\)	\(247 \div 0.65 = \mathbf{\$380}\)

\(1.06 \times \text{old rent}\)	\(=\)	\(583\)
\(\text{old rent}\)	\(=\)	\(583 \div 1.06 = \mathbf{\$550}\)

\(1.3 \times 0.7\)	\(=\)	\(0.91\)
\(0.91 - 1\)	\(=\)	\(-0.09\)

\(\text{adults}\)	\(=\)	\(0.7 \times 0.4 = 0.28\)
\(\text{children}\)	\(=\)	\(0.3 \times 0.9 = 0.27\)
\(\text{total}\)	\(=\)	\(0.55 = \mathbf{55\%}\)

\(\dfrac{96}{800 + x}\)	\(=\)	\(0.1\)
\(96\)	\(=\)	\(80 + 0.1x\)
\(x\)	\(=\)	\(\mathbf{160}\ \text{mL}\)

Operations with Percentages

📖 Theory

How to solve a percentage problem

Common pitfalls

Frequently asked questions

Theory