Percentages and Percentage Change

Welcome, students! 🌟 In everyday life, percentages appear everywhere: shop discounts, exam scores, interest rates, population growth, and even the statistics in sports and social media. In IB Mathematics: Applications and Interpretation SL, percentages are more than just a way to talk about “out of 100.” They are a powerful tool for comparing amounts, describing change, and building numerical models. By the end of this lesson, you should be able to explain what percentages mean, calculate percentage change, and use these ideas in realistic mathematical situations.

Learning goals for this lesson:

Understand the meaning of percentages and percentage change
Use correct terminology such as percentage increase, percentage decrease, and multiplier
Solve problems involving discounts, tax, growth, and shrinkage
Connect percentages to algebraic reasoning and financial modelling
Recognize how technology can help with percentage calculations and interpretation

What is a Percentage? 📊

A percentage is a fraction out of $100$. The word “percent” literally means “per hundred.” So, $45\%$ means $45$ out of every $100$, which can also be written as $\frac{45}{100}$ or $0.45$.

This is useful because percentages let us compare quantities fairly, even when the totals are different. For example, if one class has $18$ students out of $30$ passing a test and another has $24$ students out of $40$ passing, the percentages are both $60\%$. That makes comparison easier than using raw numbers alone.

To convert between forms:

Percentage to decimal: divide by $100$
Decimal to percentage: multiply by $100$
Fraction to percentage: convert to a decimal, then multiply by $100$

For example:

$75\% = \frac{75}{100} = 0.75$
$0.08 = 8\%$
$\frac{3}{4} = 0.75 = 75\%$

A common real-world example is a test score. If students scores $42$ marks out of $50$, the percentage score is

$$\frac{42}{50} \times 100 = 84\%$$

This means students answered $84$ out of every $100$ marks correctly. 🎯

Percentage of a Quantity

A percentage can also be used to find part of a whole. The formula is:

$$\text{percentage of a quantity} = \frac{p}{100} \times \text{quantity}$$

If a jacket costs $80$ and there is a $15\%$ discount, the discount amount is

$$\frac{15}{100} \times 80 = 12$$

So the price reduction is $12$, and the new price is

$$80 - 12 = 68$$

This is the kind of calculation used constantly in shopping, taxes, and budgeting. For example, if sales tax is $7\%$, the final price of an item with price $50$ is

$$50 + \frac{7}{100} \times 50 = 50 + 3.5 = 53.5$$

So the final cost is $53.50$.

In IB Mathematics: Applications and Interpretation SL, you are expected to move smoothly between arithmetic and interpretation. That means not only calculating the answer, but also explaining what it means in context. If a phone plan gives a $20\%$ discount, it does not mean the customer pays $20\%$ of the original price. It means the customer saves $20\%$ and pays the remaining $80\%$.

Percentage Change: Increase and Decrease

Percentage change compares how much a quantity has changed relative to its original value. The formula is:

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

Here, the change is:

$$\text{change} = \text{new value} - \text{original value}$$

If the result is positive, it is a percentage increase. If the result is negative, it is a percentage decrease.

Example: Increase

A population rises from $2000$ to $2300$.

The change is

$$2300 - 2000 = 300$$

The percentage increase is

$$\frac{300}{2000} \times 100 = 15\%$$

So the population increased by $15\%$.

Example: Decrease

A game console drops in price from $400$ to $340$.

The change is

$$340 - 400 = -60$$

The percentage change is

$$\frac{-60}{400} \times 100 = -15\%$$

This means the price decreased by $15\%$. The negative sign is important because it shows direction.

A frequent mistake is using the new value as the base instead of the original value. In percentage change, the original value is always the reference point unless the question says otherwise. That is why “original” is such an important word in the terminology.

Multipliers and Repeated Percentage Change

One of the most useful ideas in this topic is the multiplier. Instead of finding the increase or decrease step by step each time, we can write a new value directly in terms of the original value.

If a quantity increases by $r\%$, the multiplier is

$$1 + \frac{r}{100}$$

If a quantity decreases by $r\%$, the multiplier is

$$1 - \frac{r}{100}$$

Example: Growth

A salary increases by $6\%$. The multiplier is

$$1 + \frac{6}{100} = 1.06$$

If the original salary is $3000$, the new salary is

$$3000 \times 1.06 = 3180$$

Example: Reduction

A laptop is reduced by $25\%$. The multiplier is

$$1 - \frac{25}{100} = 0.75$$

If the original price is $1200$, the sale price is

$$1200 \times 0.75 = 900$$

Multipliers are especially important in financial models, because they make repeated change easy to calculate. For example, if an investment grows by $4\%$ each year, then after one year it is multiplied by $1.04$, after two years by $1.04^2$, and after $n$ years by $1.04^n$.

This connects percentages to sequences and algebra. A repeated percentage change creates a geometric sequence, where each term is multiplied by the same constant factor. In IB AI SL, this link is very important because many real-world situations can be modelled this way.

Applications in Finance and Real Life 💰

Percentages are central to financial literacy. Interest rates, inflation, depreciation, commission, and tax all rely on percentage change.

Compound Growth

If money is invested at interest rate $r\%$ per period, compound growth can be modelled by

$$A = P\left(1 + \frac{r}{100}\right)^n$$

where $A$ is the final amount, $P$ is the initial amount, and $n$ is the number of periods.

For example, if $500$ is invested at $3\%$ per year for $4$ years:

$$A = 500(1.03)^4$$

Using technology or a calculator gives approximately

$$A \approx 562.75$$

So the investment grows to about $\$562.75.

Depreciation

Some assets lose value over time, such as cars and electronics. If a car loses $12\%$ of its value each year, the multiplier is

$$1 - \frac{12}{100} = 0.88$$

If the starting value is $18,000$, then after $3$ years the value is

$$18000(0.88)^3$$

This gives about $\$12,283.97.

These models are important because they show how percentages can be used to predict future values. They are also examples of numerical modelling, one of the core ideas in Number and Algebra.

Technology and Interpretation

Technology is very useful for percentage problems, especially when the numbers are awkward or when repeated changes happen many times. A spreadsheet can quickly calculate discounts, tax, growth over several years, or a table of values for a model.

For example, if a quantity changes by $5\%$ each year, a calculator can evaluate

$$P(1.05)^n$$

for many values of $n$. This helps students see the pattern and interpret the results. Technology also reduces arithmetic errors, so more time can be spent thinking about what the numbers mean.

In IB Mathematics: Applications and Interpretation SL, interpretation matters as much as calculation. If a graph shows that a quantity is decreasing at a steady percentage rate, students should be able to explain that the model is multiplicative, not additive. A decrease of $10\%$ each year does not mean subtracting the same amount every year; it means multiplying by $0.9$ each year.

Common Misconceptions and How to Avoid Them

A few mistakes happen often in percentage questions:

Using the wrong base value

Percentage change always compares change to the original value.

Confusing percentage points with percentages

If a rate increases from $20\%$ to $25\%$, it rises by $5$ percentage points, not $5\%$.

Adding percentages directly when changes are repeated

Two increases of $10\%$ do not make one increase of $20\%$ overall.
Instead, the multiplier is

$$1.1 \times 1.1 = 1.21$$

so the overall increase is $21\%$.

Forgetting that decrease multipliers are less than $1$

A $30\%$ decrease uses the multiplier $0.7$.

Checking your answer in context is a good habit. Does the result make sense? If the price of an item after a discount is higher than the original, something has gone wrong.

Conclusion

Percentages and percentage change are essential tools in Number and Algebra because they connect number sense, algebraic manipulation, and real-world modelling. They help describe relative size, growth, loss, and comparison in a way that is easy to understand and widely used. In IB Mathematics: Applications and Interpretation SL, students should be able to calculate percentages, interpret percentage change, use multipliers, and connect these ideas to financial and numerical models. Mastering this topic builds a strong foundation for sequences, exponential models, and data interpretation.

Study Notes

A percentage means “per hundred,” so $p\% = \frac{p}{100} = \frac{p}{100}$.
To find a percentage of a quantity, use $\frac{p}{100} \times \text{quantity}$.
Percentage change is calculated by $\frac{\text{new} - \text{original}}{\text{original}} \times 100$.
A positive percentage change is an increase; a negative percentage change is a decrease.
The multiplier for an increase of $r\%$ is $1 + \frac{r}{100}$.
The multiplier for a decrease of $r\%$ is $1 - \frac{r}{100}$.
Repeated percentage change creates a geometric sequence.
Compound growth can be modelled by $A = P\left(1 + \frac{r}{100}\right)^n$.
Depreciation can be modelled by $A = P\left(1 - \frac{r}{100}\right)^n$.
Always use the original value as the reference point unless the question says otherwise.
Technology such as calculators and spreadsheets helps with repeated percentage calculations and interpretation.
Percentages are widely used in finance, shopping, tax, interest, inflation, and depreciation.