Percentages and Percentage Change
Welcome, students 👋 In this lesson, you will learn how percentages help us compare quantities, describe growth and decrease, and model real situations such as sales, tax, population change, and investment returns. Percentages are everywhere in daily life and in IB Mathematics: Applications and Interpretation HL, especially when we work with financial models, numerical data, and algebraic reasoning.
Learning objectives
- Explain the main ideas and terminology behind percentages and percentage change.
- Apply IB Mathematics: Applications and Interpretation HL reasoning and procedures related to percentages and percentage change.
- Connect percentages and percentage change to the broader topic of Number and Algebra.
- Summarize how percentages and percentage change fit within Number and Algebra.
- Use evidence and examples related to percentages and percentage change in IB Mathematics: Applications and Interpretation HL.
Understanding percentages
A percentage means “out of 100.” The symbol $\%$ tells us that a value is being compared to a whole on a base of $100$. For example, $25\%$ means $25$ out of every $100$, which is the same as the fraction $\frac{25}{100}$ and the decimal $0.25$.
This idea is important because percentages make comparisons easier. A price discount of $20\%$ is easier to interpret than saying the price is reduced by $0.20$ of its original value. Percentages also help when comparing data sets of different sizes. For example, if one school has $50$ students out of $200$ who take part in a club, and another has $30$ out of $100$, the percentages are both $25\%$, so the participation rate is the same.
To convert between forms:
- Percentage to decimal: divide by $100$.
- Decimal to percentage: multiply by $100$.
- Percentage to fraction: write it over $100$ and simplify if possible.
For example, $7\% = \frac{7}{100} = 0.07$.
A common mistake is to think that a percentage is always a “part” without considering the whole. In reality, a percentage only makes sense when there is a reference amount, called the base value or original value. If a shirt is discounted by $30\%$, the discount is $30\%$ of the original price, not $30\%$ of the sale price.
Percentage change: increase and decrease
Percentage change compares how much a quantity changes relative to its starting value. The general formula is
$$\text{Percentage change} = \frac{\text{change in value}}{\text{original value}} \times 100\%.$$
If the result is positive, the quantity has increased. If it is negative, the quantity has decreased.
For an increase,
$$\text{Percentage increase} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\%.$$
For a decrease,
$$\text{Percentage decrease} = \frac{\text{original value} - \text{new value}}{\text{original value}} \times 100\%.$$
Here is a real-world example. Suppose a phone costs $\$800$ and later costs $\$920$. The increase is
$$920 - 800 = 120.$$
The percentage increase is
$$\frac{120}{800} \times 100\% = 15\%.$$
So the price increased by $15\%$.
Now consider a price drop from $\$60$ to $\$45$. The decrease is
$$60 - 45 = 15.$$
The percentage decrease is
$$\frac{15}{60} \times 100\% = 25\%.$$
So the price decreased by $25\%$.
Notice that percentage change is not symmetric. An increase of $20\%$ followed by a decrease of $20\%$ does not return to the original value. For example, if a value starts at $100$, then after a $20\%$ increase it becomes
$$100 \times 1.2 = 120.$$
A $20\%$ decrease from $120$ gives
$$120 \times 0.8 = 96.$$
This shows why percentages are linked to multiplicative reasoning, not just simple subtraction.
Percentage multipliers and repeated change
In IB Mathematics, percentage change is often written using a multiplier. A multiplier turns a percentage into a factor.
- A $p\%$ increase uses the multiplier $1 + \frac{p}{100}$.
- A $p\%$ decrease uses the multiplier $1 - \frac{p}{100}$.
For example, a $12\%$ increase has multiplier
$$1 + \frac{12}{100} = 1.12.$$
A $7\%$ decrease has multiplier
$$1 - \frac{7}{100} = 0.93.$$
This is powerful because repeated percentage change becomes a sequence or exponential model. Suppose a city population grows by $4\%$ each year. If the initial population is $P_0$, then after one year it becomes
$$P_1 = 1.04P_0.$$
After two years,
$$P_2 = 1.04P_1 = 1.04^2P_0.$$
After $n$ years,
$$P_n = P_0(1.04)^n.$$
This formula is an example of how percentages connect to sequences and financial models in the Number and Algebra topic. The same structure appears in savings accounts, inflation, depreciation, and investment growth.
A useful real-world example is depreciation. If a car loses $15\%$ of its value each year and starts at $\$30{,}000, then after one year its value is
$$30{,}000 \times 0.85 = 25{,}500.$$
After two years,
$$30{,}000 \times 0.85^2.$$
The exponential model shows that each year’s decrease is based on the current value, not the original value. This is why financial models often use multipliers.
Solving percentage problems with algebra
Many percentage questions in IB AI HL require algebraic setup. Let the original value be $x$.
If a quantity increases by $18\%$, the new value is
$$1.18x.$$
If a quantity decreases by $35\%$, the new value is
$$0.65x.$$
This makes it easy to solve unknowns.
Example: A jacket is on sale for $\$72$ after a $20\% discount. What was the original price?
Let the original price be $x$. Since there is a $20\%$ decrease,
$$0.8x = 72.$$
Solving gives
$$x = \frac{72}{0.8} = 90.$$
So the original price was $\$90.
Example: A value increases from $40$ to $x$ after a $25\%$ increase.
Use the multiplier:
$$x = 1.25 \times 40 = 50.$$
So the new value is $50$.
Sometimes a problem asks for the original value after a percent increase. Example: After a $10\%$ increase, a quantity becomes $88$. Find the original value.
Set up the equation
$$1.1x = 88.$$
Then
$$x = 80.$$
These problems show a major idea in Number and Algebra: translate verbal information into algebraic expressions, then solve. Technology can help check calculations, but the mathematical reasoning must still be clear.
Compound change, inflation, and financial contexts
Percentage change often happens over time. In finance, this is called compound growth or compound decrease. If an amount changes by the same percentage each period, the model uses repeated multiplication.
For a growth rate of $r\%$ per period, the model is
$$A = P\left(1 + \frac{r}{100}\right)^n,$$
where $P$ is the initial amount, $A$ is the final amount, and $n$ is the number of periods.
For a decrease of $r\%$ per period, the model is
$$A = P\left(1 - \frac{r}{100}\right)^n.$$
Example: If a savings account has $\$5000$ and grows by $3\%$ each year for $4 years, then
$$A = 5000(1.03)^4.$$
This gives the final amount after compound growth.
Inflation works similarly. If prices rise by $2.5\%$ each year, then the multiplier is $1.025$. This helps explain why the same money may buy fewer goods over time. If a basket of goods costs $\$200$ now, then after one year of $2.5\% inflation the estimated cost is
$$200(1.025) = 205.$$
Financial literacy is part of mathematical modelling. Percentages help students interpret news reports, loan offers, savings plans, and economic trends. In IB AI HL, this also supports the use of technology to analyze data, make predictions, and evaluate models.
Common mistakes and how to avoid them
A key mistake is using the wrong base value. If a population grows from $2000$ to $2400$, the percentage increase is based on $2000$, not $2400$.
Another mistake is adding percentages directly across different stages. For example, a $10\%$ increase and then another $10\%$ increase does not make a $20\%$ increase overall. The correct factor is
$$1.1 \times 1.1 = 1.21,$$
so the total increase is $21\%$.
Also, remember that percentage points are different from percentages. If a rate changes from $8\%$ to $11\%$, the increase is $3$ percentage points, but the relative percentage increase is
$$\frac{11 - 8}{8} \times 100\% = 37.5\%.$$
This distinction matters in statistics, economics, and data interpretation.
Conclusion
Percentages and percentage change are essential tools for describing comparison, growth, and decrease in a clear and practical way 📊 They connect directly to algebra through multipliers, equations, and exponential models. They also support financial modelling, data analysis, and interpretation of real-world information. In Number and Algebra, percentage reasoning helps students move from arithmetic thinking to more powerful algebraic and modelling strategies. For IB Mathematics: Applications and Interpretation HL, mastering percentages means being able to read contexts carefully, choose the correct base value, and use formulas accurately to solve problems.
Study Notes
- A percentage means “out of $100$.”
- Convert between forms using $\frac{p}{100}$ and decimal multiplication.
- Percentage change uses
$$\frac{\text{change}}{\text{original}} \times 100\%.$$
- A $p\%$ increase has multiplier $1 + \frac{p}{100}$.
- A $p\%$ decrease has multiplier $1 - \frac{p}{100}$.
- Repeated percentage change leads to exponential models such as
$$A = P\left(1 \pm \frac{r}{100}\right)^n.$$
- The original value is the base for all percentage calculations.
- Percentage increases and decreases are not reversible with the same percentage.
- Percentage points are different from percentage change.
- Percentages connect Number and Algebra to financial models, data interpretation, and real-world decision-making.