Calculating Percentages, Percentage Change, and Discounts

Good to see you again. Last lesson established that percentages, fractions, and decimals represent the same proportion: for example, $25\%=0.25=\frac{1}{4}$. This lesson puts that connection to work.

You will learn to calculate a percentage of an amount, work out a percentage increase or decrease, and deal with discounts. The aim is not to depend on one memorised layout. Instead, you will identify what the percentage refers to, choose a sensible calculation, and check whether your answer fits the situation.

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Percentage means a part of the original amount

When a question asks for a percentage of an amount, it is asking for a part of that amount.

For example, “Find $18\%$ of $250$” means find $18$ parts out of every $100$ parts of $250$.

Since:

$$18\%=\frac{18}{100}=0.18$$

we can calculate:

$$18\%\text{ of }250=0.18\times 250=45$$

The word of is a strong clue that multiplication is needed.

$$\text{percentage amount}=\frac{\text{percentage}}{100}\times \text{original amount}$$

For any question, first identify:

• the whole or original amount;
• the percentage;
• whether you need the percentage amount itself, or a new total after an increase or reduction.

For instance, $18\%$ of $250$ is $45$. It is not $205$, $232$, or $268$, because those would involve subtracting from or adding to $250$. The wording has only asked for the percentage part.

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Choose a method that makes sense

There is more than one correct way to calculate a percentage. A good exam strategy is to choose the method that makes the numbers easiest to handle.

Method 1: Convert the percentage to a decimal

This method works for every percentage, including awkward ones such as $13\%$, $7.5\%$, or $68\%$.

To find $13\%$ of $460$:

$$13\%=0.13$$ $$0.13\times 460=59.8$$

So:

$$13\%\text{ of }460=59.8$$

If this were money, write it as $\$59.80$.

Method 2: Use familiar fraction equivalents

Some percentages have useful fraction forms:

Percentage	Fraction equivalent	Fast meaning
$50\%$	$\frac{1}{2}$	halve
$25\%$	$\frac{1}{4}$	divide by $4$
$75\%$	$\frac{3}{4}$	find a quarter, then multiply by $3$
$10\%$	$\frac{1}{10}$	divide by $10$
$20\%$	$\frac{1}{5}$	divide by $5$

For example:

$$75\%\text{ of }24=\frac{3}{4}\times 24$$ $$24÷4=6$$ $$6\times 3=18$$

Therefore:

$$75\%\text{ of }24=18$$

Method 3: Build the percentage from $10\%$, $5\%$, $1\%$, or $50\%$

This is particularly useful without a calculator.

To find $35\%$ of $240$:

$$10\%\text{ of }240=24$$ $$30\%\text{ of }240=72$$ $$5\%\text{ of }240=12$$

Now combine $30\%$ and $5\%$:

$$72+12=84$$

So:

$$35\%\text{ of }240=84$$

A quick mental check helps: $35\%$ is a little more than one third, and one third of $240$ is $80$, so $84$ is plausible.

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Percentage change: compare the change with the starting amount

Percentage change tells you how large a change is relative to where you started.

Suppose a price rises from $\$80$ to $\$100$. The actual increase is:

$$100-80=20$$

But an increase of $\$20$ means different things depending on the original price. Compared with the original $\$80$, it is:

$$\frac{20}{80}=0.25$$ $$0.25\times 100=25$$

So the price has increased by:

$$25\%$$

The general calculation is:

$$\text{percentage change}=\frac{\text{change in amount}}{\text{initial amount}}\times 100$$

The most important word is initial. The starting amount is the base used for comparison.

A decrease example

A laptop is reduced from $\$840$ to $\$714$.

First, find the size of the reduction:

$$840-714=126$$

Then compare that reduction with the original price:

$$\frac{126}{840}=0.15$$ $$0.15\times 100=15$$

Therefore, the laptop’s price has decreased by:

$$15\%$$

Do not divide by the final price of $\$714$. The question asks how much the price changed from its original value, so $840$ must be the denominator.

In the worksheet, compare Questions 1, 2, and 6. The contexts differ—savings, a sale price, and rainfall—but the comparison is always made with the first value given. A negative difference can show a decrease during working, but the final response should normally state the direction clearly: for example, a $15\%$ decrease.

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An increase or decrease gives a new total

Some questions ask for the percentage change. Others give you a percentage change and ask for the new amount. These are different tasks.

Consider a jacket costing $\$120$ that increases by $30\%$.

First find the increase:

$$30\%\text{ of }120=0.30\times 120=36$$

Then add the increase to the original price:

$$120+36=156$$

The new price is:

$$\$156$$

A percentage increase has two parts:

• the original amount, which is $100\%$;
• the extra percentage.

So a $30\%$ increase gives:

$$100\%+30\%=130\%$$

As a decimal multiplier:

$$130\%=1.30$$

This gives a shorter calculation:

$$120\times 1.30=156$$

The multiplier $1.30$ is not a fact to memorise separately. It comes from keeping all of the original $100\%$ and adding another $30\%$.

For an increase of $p\%$:

$$\text{new amount}=\text{original amount}\times \left(1+\frac{p}{100}\right)$$

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Discounts are percentage decreases

A discount is money taken off an original price. It is therefore a percentage decrease.

Suppose headphones have an original price of $\$142$ and are reduced by $12\%$.

Method 1: Find the discount, then subtract

Find $12\%$ of $\$142$:

$$0.12\times 142=17.04$$

This is the amount saved:

$$\$17.04$$

Subtract it from the original price:

$$142-17.04=124.96$$

The sale price is:

$$\$124.96$$

Method 2: Find the percentage that remains

A $12\%$ discount means the customer pays:

$$100\%-12\%=88\%$$

Convert $88\%$ to a decimal:

$$88\%=0.88$$

Then calculate:

$$142\times 0.88=124.96$$

Both methods must give the same answer. Method 1 makes the saving visible; Method 2 is often quicker when you only need the final price.

For a discount of $p\%$:

$$\text{sale price}=\text{original price}\times \left(1-\frac{p}{100}\right)$$

The key idea is that, after a discount, you pay the remaining percentage, not the discount percentage itself.

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Finding an original price from a discounted price

A common exam trap is to reverse a discount incorrectly.

If an item costs $\$40$ after a $20\%$ discount, it is tempting to add $20\%$ of $\$40$. That would be wrong because $\$40$ is not the original $100\%$; it is the remaining $80\%$.

Let the original price be $x$.

$$100\%-20\%=80\%$$ $$80\%=0.80$$

So:

$$0.80x=40$$ $$x=40÷0.80$$ $$x=50$$

The original price was:

$$\$50$$

A useful check confirms this:

$$20\%\text{ of }50=10$$ $$50-10=40$$

The original value is $\$50$, not $\$48$. Adding $20\%$ to the sale price does not undo a $20\%$ discount, because the two percentages would be calculated from different starting amounts.

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Exam checks that prevent common mistakes

Before finalising a percentage answer, use these quick checks.

1. Check the decimal conversion

$$3\%=0.03$$ $$30\%=0.30$$

Confusing these makes an answer ten times too large.

2. Check whether you need the part or the final total

If a $\$200$ item has a $15\%$ discount:

$$15\%\text{ of }200=30$$

The discount is $\$30$, but the sale price is:

$$200-30=170$$

3. Use the initial amount for percentage change

For a change from $80$ to $100$:

$$\frac{20}{80}\times 100=25\%$$

The denominator is the starting value, $80$.

4. Check the direction and size

• After an increase, the final amount should be greater than the original.
• After a discount, the final price should be lower than the original.
• A $50\%$ discount should halve the price.
• A $1\%$ change should be small, so an answer that changes the amount greatly is suspicious.

5. Round money only at the end

Keep full calculator values during working when possible. Round the final monetary answer to two decimal places:

$$\$124.956\approx \$124.96$$

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A flexible routine for percentage questions

When you see a percentage problem, use this sequence:

1. Identify the original whole. What amount does $100\%$ represent?
2. Decide what the question asks for. Is it a percentage amount, a new total, a percentage change, a discount, or an original value?
3. Translate the percentage. Use a decimal, fraction, or mental building blocks.
4. Calculate from the correct base amount. For percentage change, this is the initial value.
5. Add or subtract only when the context requires it.
6. Check the answer against the situation and include the correct unit, such as $\$$, kg, or $\%$.

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The central ideas are these:

$$p\%\text{ of an amount}=\frac{p}{100}\times \text{amount}$$ $$\text{percentage change}=\frac{\text{change}}{\text{initial amount}}\times 100$$

For an increase, retain the original $100\%$ and add the extra percentage. For a discount, retain the percentage left after the reduction. Once you identify what represents $100\%$, the calculation becomes much more reliable.

Next, you will extend this proportional thinking to ratios, rates, and everyday comparison problems.