Year 11 General Consumer Arithmetic: Loans And Investments

Percentages And Applications

Q: How do you convert a percentage to a decimal?

Divide by 100. For example, 8% is 0.08, and 25% is 0.25.

Q: How do you increase a number by a percentage?

Multiply by 1 plus the percentage as a decimal. A 15% increase means multiply by 1.15. A 7.5% increase means multiply by 1.075.

Q: How do you decrease a number by a percentage?

Multiply by 1 minus the percentage as a decimal. A 20% decrease means multiply by 0.80. A 35% decrease means multiply by 0.65.

Q: How do you remove GST from a price?

GST is 10%, so the multiplier is 1.10. To find the price before GST, divide the GST-inclusive price by 1.10.

Q: Why doesn't increasing then decreasing by the same percent return the original?

Because the percentages are applied to different bases. 100 times 1.10 times 0.90 is 99, not 100. The 10% decrease is taken from the larger 110, not the original 100.

13 practice questions 2 video lessons Theory + worked examples

Theory

A percentage is a fraction out of 100. The fastest way to apply one is the multiplier method: a 15% increase means $\times 1.15$; a 20% decrease means $\times 0.80$; adding GST means $\times 1.10$. Reverse problems, successive discounts, and depreciation all follow the same pattern.

A percentage is a fraction out of $100$. To convert: divide the percent by $100$ to get a decimal. So $8\% = 0.08$ and $25\% = 0.25$.

An increase of $r\%$ means the new amount is the original plus $r\%$ of itself — multiply by $\left(1 + \dfrac{r}{100}\right)$. A decrease of $r\%$ means multiply by $\left(1 - \dfrac{r}{100}\right)$. These multipliers are the workhorse of percentage problems.

GST (Goods and Services Tax) is $10\%$ in Australia. To add GST, multiply by $1.10$; to remove GST, divide by $1.10$. Depreciation is repeated decrease — an asset losing $r\%$ per year for $n$ years uses the multiplier $\left(1 - \dfrac{r}{100}\right)^n$.

Successive changes multiply. A \(20\%\) discount followed by a further \(10\%\) off gives \(\times 0.80 \times 0.90 = \times 0.72\) — a \(28\%\) discount overall, not \(30\%\).

Convert any percent change to a single multiplier

Divide by the multiplier to undo the change

Convert a percentage to a decimal:

\[ r\% = \dfrac{r}{100} \]

r % = \frac{r}{100}

The three core operations:

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

\[ \text{increase by } r\%: \quad A \times \left(1 + \dfrac{r}{100}\right) \]

\[ \text{decrease by } r\%: \quad A \times \left(1 - \dfrac{r}{100}\right) \]

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

Percentage change:

\[ \text{\% change} = \dfrac{\text{change}}{\text{original}} \times 100\% \]

% change = \frac{change}{original} \times 100

Reverse problem (find original from new):

\[ \text{original} = \dfrac{\text{new amount}}{\text{multiplier}} \]

Depreciation (lose $r\%$ per year for $n$ years):

\[ \text{value} = \text{original} \times \left(1 - \dfrac{r}{100}\right)^n \]

value = P {(1 - \frac{r}{100})}^{n}

Quick multipliers: \(+8\% \to \times 1.08\); \(-15\% \to \times 0.85\); \(+\text{GST} \to \times 1.10\); \(-20\% \to \times 0.80\).

How to apply a percentage change

Convert the percent to a multiplier: $1 + r/100$ for an increase, $1 - r/100$ for a decrease.
Multiply the original amount by that multiplier to get the new amount.
To reverse, divide the new amount by the multiplier to recover the original.

For successive changes, multiply all the multipliers together: a \(25\%\) discount then a further \(10\%\) off is \(\times 0.75 \times 0.90 = \times 0.675\).

Example 1 — Increase

A jacket priced at $\$160$ is marked up by $15\%$. Find the new price.

Solution

A $15\%$ increase means multiply by $1.15$.

$\text{new}$	$=$	$160 \times 1.15$
$\text{new}$	$=$	$\$184$

new = 184

The new price is $\textbf{\$184}$.

Example 2 — Reverse: original price

After a $20\%$ discount, a coat sells for $\$192$. Find its original price.

Solution

The multiplier is $1 - 0.20 = 0.80$. Divide the sale price by it.

$\text{original}$	$=$	$\dfrac{192}{0.80}$
$\text{original}$	$=$	$\$240$

original = 240

The original price was $\textbf{\$240}$.

Example 3 — Successive changes

A $\$240$ item is reduced by $25\%$, then a further $10\%$ is taken off the discounted price. Find the final price.

Solution

Multiply by both discount multipliers in turn.

$\text{final}$	$=$	$240 \times 0.75 \times 0.90$
$\text{final}$	$=$	$240 \times 0.675$
$\text{final}$	$=$	$\$162$

final = 162

The final price is $\textbf{\$162}$ — a $32.5\%$ total discount.

Example 4 — Depreciation

A car is bought for $\$30{,}000$ and loses $20\%$ of its value each year. Find its value after $3$ years.

Solution

Each year multiplies the value by $0.80$; over $3$ years that's $0.80^3$.

$\text{value}$	$=$	$30{,}000 \times 0.80^3$
$\text{value}$	$=$	$30{,}000 \times 0.512$
$\text{value}$	$=$	$\$15{,}360$

value = 15360

After $3$ years the car is worth $\textbf{\$15{,}360}$.

Common pitfalls

Up then down doesn't return to start. Increase $\$100$ by $10\%$ then decrease by $10\%$ gives $\$99$, not $\$100$. The second percentage is taken from the larger amount.

Adding discount percentages. A $25\%$ discount followed by a further $10\%$ discount is not $35\%$ off. Multiply the multipliers: $0.75 \times 0.90 = 0.675$, so the total discount is $32.5\%$.

Removing GST by subtracting 10%. If the price is $\$110$ including GST, the pre-GST price is $110 \div 1.10 = \$100$, not $\$110 - 11 = \$99$. Always divide.

Forgetting to convert percentage to decimal. "$8\%$ of $200$" is $200 \times 0.08 = 16$, not $200 \times 8 = 1{,}600$.

Frequently asked questions

How do you convert a percentage to a decimal?

Divide the percent by $100$. So $8\% = 0.08$ and $25\% = 0.25$. The decimal is what you multiply by.

How do you increase a number by a percentage?

Multiply by $1 + r/100$. A $15\%$ increase means multiply by $1.15$. A $7.5\%$ increase means multiply by $1.075$.

How do you decrease a number by a percentage?

Multiply by $1 - r/100$. A $20\%$ discount means multiply by $0.80$. A $35\%$ discount means multiply by $0.65$.

How do you find the original price before a discount?

Divide the discounted price by the discount multiplier. After $20\%$ off, the multiplier is $0.80$. If the sale price is $\$192$, the original was $192 \div 0.80 = \$240$.

How do you remove GST from a price?

GST is $10\%$, so the multiplier is $1.10$. Divide the GST-inclusive price by $1.10$ to get the price before GST.

Why doesn't increasing then decreasing by the same percent return the original?

Because the percentages are applied to different bases. $100 \times 1.10 \times 0.90 = 99$, not $100$. The decrease is taken from the larger $110$, not the starting $100$.

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Simple Interest

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\(\text{new}\)	\(=\)	\(160 \times 1.15\)
\(\text{new}\)	\(=\)	\(\$184\)

\(\text{original}\)	\(=\)	\(\dfrac{192}{0.80}\)
\(\text{original}\)	\(=\)	\(\$240\)

\(\text{final}\)	\(=\)	\(240 \times 0.75 \times 0.90\)
\(\text{final}\)	\(=\)	\(240 \times 0.675\)
\(\text{final}\)	\(=\)	\(\$162\)

\(\text{value}\)	\(=\)	\(30{,}000 \times 0.80^3\)
\(\text{value}\)	\(=\)	\(30{,}000 \times 0.512\)
\(\text{value}\)	\(=\)	\(\$15{,}360\)

Percentages And Applications

📖 Theory

How to apply a percentage change

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

Video Lessons

Practice Questions