How to Use the Calculator?
To use the calculator, simply enter the initial value in the first box and the final value in the second box; the calculator will instantly show you whether the value has increased or decreased, and by what percentage. It is useful for analyzing changes in prices, scores, savings, population, or any other quantity that changes over time.
What Is Percentage Change?
Percentage change is one of the most useful ideas in mathematics because it helps us compare how much something has increased or decreased relative to its original value.
You see it in exam marks, prices of products, population growth, discounts in shops, sports statistics, and even weather reports. If the price of a notebook rises from 50 to 60, it is not enough to simply say that the price went up by 10. We often want to know how big that increase is compared to the original price. That is exactly what percentage change tells us.
Let us begin with the meaning of the word “change.” In mathematics, change simply means the difference between the new value and the old value. If something becomes larger, the change is called an increase. If something becomes smaller, the change is called a decrease.
The word “percentage” means “out of 100,” so percentage change tells us how large the change is out of every 100 parts of the original amount.
The Formula for Percentage Change
The most important thing to remember is that percentage change is always compared with the original value, not the new one. Many people make mistakes because they divide by the wrong number. The base or reference value must be the starting value, which is the original amount before the increase or decrease happened.
Here is the formula for percentage change:
$$\text{Percentage Change} = \frac{\text{Change}}{\text{Original Value}} \times 100\%$$
And since the change itself is found by subtracting the original value from the new value, we can also write:
$$\text{Change} = \text{New Value} – \text{Original Value}$$
So the full formula is:
$$\text{Percentage Change} = \frac{\text{New Value} – \text{Original Value}}{\text{Original Value}} \times 100\%$$
This formula works for both increase and decrease. If the answer comes out positive, it means there was a percentage increase. If the answer comes out negative, it means there was a percentage decrease.
Percentage Increase
Let us now understand percentage increase.
Suppose the number of people in a club rises from 20 to 25. First, we find the increase. The number has gone up by $25 – 20 = 5$. Now we compare that increase with the original number, which is 20. So we divide by 20 and multiply by 100.
$$\begin{aligned} \text{Percentage Increase} &= \frac{25 – 20}{20} \times 100\% \\[15pt] &= \frac{5}{20} \times 100\% \\[15pt] &= 25\% \end{aligned}$$
So the club membership increased by 25%. Notice carefully that we divided by 20, not 25, because 20 was the original number.
Percentage Decrease
Now let us look at percentage decrease.
Suppose the price of a flash drive falls from $\$40$ to $\$30$ . The amount of decrease is $40 – 30 = 10$. Since the original price was 40, we divide by 40 and multiply by 100.
$$\begin{aligned} \text{Percentage Decrease} &= \frac{40 – 30}{40} \times 100\% \\[15pt] &= \frac{10}{40} \times 100\% \\[15pt] &= 25\% \end{aligned}$$
So the price decreased by 25%. Again, the denominator is the original value, which is 40.
More Examples
Let us try several examples.
Example 1
A student scored 60 marks in one test and 75 marks in the next test. Find the percentage increase in marks.
The increase is $75 – 60 = 15$. The original mark is 60.
$$\begin{aligned} \text{Percentage Increase} &= \frac{15}{60} \times 100\% \\[15pt] &= 25\% \end{aligned}$$
So the student’s marks increased by 25%. This tells us the improvement was one-fourth of the original mark.
Example 2
The attendance in a class dropped from 50 students to 45 students. Find the percentage decrease in attendance.
The decrease is $50 – 45 = 5$. The original attendance is 50.
$$\begin{aligned} \text{Percentage Decrease} &= \frac{5}{50} \times 100\% \\[15pt] &= 10\% \end{aligned}$$
So attendance decreased by 10%.
Example 3
Suppose the population of a village was 2,400 last year and 2,760 this year. Find the percentage increase.
First, find the increase:
$$2760 – 2400 = 360$$
Now compare with the original population:
$$\begin{aligned} \text{Percentage Increase} &= \frac{360}{2400} \times 100\% \\[15pt] &= 15\% \end{aligned}$$
So the population increased by 15%.
Example 4
Now consider a water tank that contained 900 liters of water but now contains 720 liters. Find the percentage decrease.
First, find the decrease:
$$900 – 720 = 180$$
Now divide by the original amount:
$$\begin{aligned} \text{Percentage Decrease} &= \frac{180}{900} \times 100\% \\[15pt] &= 20\% \end{aligned}$$
So the amount of water decreased by 20%.
Example 5
Imagine two shops both increase the price of an item by $\$20$ . In the first shop, the price goes from 100 to 120. In the second shop, the price goes from 200 to 220. The increase in both cases is $\$20$, but the percentage increase is not the same because the starting prices are different. In the first case:
$$\frac{20}{100} \times 100\% = 20\%$$
In the second case:
$$\frac{20}{200} \times 100\% = 10\%$$
This shows why percentage change gives more useful information than just the amount of change. It tells us how significant the change is compared with where we started.
Finding the New Value After a Percentage Increase
Sometimes you need to find the new value after a percentage increase or decrease. This is closely related to percentage change, so it is important to know.
Suppose a cell phone costs $\$800$ and the price increases by 15%. First, find 15% of 800.
$$\begin{aligned} 15\% \text{ of } 800 &= \frac{15}{100} \times 800 \\[15pt] &= 120 \end{aligned}$$
Then add this increase to the original price:
$$800 + 120 = 920$$
So the new price is $\$920$.
You can also do this with a multiplier. If there is an increase of 15%, then the new value becomes $100% + 15% = 115%$ of the original. Since $115% = 1.15$, we can write:
$$\begin{aligned} \text{New Value} &= 800 \times 1.15 \\[15pt] &= 920 \end{aligned}$$
This method is often faster once you understand it.
Finding the New Value After a Percentage Decrease
For a decrease, suppose a laptop costs $\$500$ and gets a discount of 20%. First, find 20% of 500.
$$\begin{aligned} 20\% \text{ of } 500 &= \frac{20}{100} \times 500 \\[15pt] &= 100 \end{aligned}$$
Now subtract the discount from the original price:
$$500 – 100 = 400$$
So the new price is $400.
Using the multiplier method, a decrease of 20% means the new value is $100% – 20% = 80%$ of the original. Since $80% = 0.8$, we get:
$$\begin{aligned} \text{New Value} &= 500 \times 0.8 \\[15pt] &= 400 \end{aligned}$$
This idea is very useful in business and everyday life because discounts, tax changes, and population changes are often given in percentages.
Increase and Decrease by the Same Percentage
There is also an interesting idea that confuses many people: an increase and then a decrease by the same percentage do not cancel each other.
For example, imagine a party speaker price is $\$1,000$. First it increases by 10%. The new price becomes:
$$1000 \times 1.10 = 1100$$
Then it decreases by 10%. Now the decrease is based on 1,100, not on 1,000.
$$1100 \times 0.90 = 990$$
So the final price is $\$990$, not $\$1,000$. Even though there was a 10% increase and then a 10% decrease, the final result is a net decrease of 10, which is a 1% decrease from the original price. This happens because the second percentage is calculated on a different base.
Let us verify that net change:
$$\text{Net Change} = 990 – 1000 = -10$$
$$\text{Percentage Change} = \frac{-10}{1000} \times 100\% = -1\%$$
So the overall change is a 1% decrease.
Repeated Percentage Change and Compounding
This idea becomes even more important in compound situations, such as repeated yearly growth or repeated discounts.
Suppose a value increases by 5% every year for two years. It is not correct to just add $5% + 5% = 10%$ and assume the final value is exactly 10% more. Instead, each year’s increase is based on the updated value. If the original amount is 200, then after one year:
$$200 \times 1.05 = 210$$
After the second year:
$$210 \times 1.05 = 220.5$$
So the total increase is:
$$220.5 – 200 = 20.5$$
And the percentage increase is:
$$\frac{20.5}{200} \times 100\% = 10.25\%$$
So the total increase is 10.25%, not exactly 10%. This happens because of compounding.
Reverse Percentage Change
Let us also understand how to reverse a percentage change, because that can be tricky.
Suppose after a 20% increase, a value becomes 240. What was the original value? Many people wrongly subtract 20% of 240, but that is not correct because 240 is the new value, not the original.
Let the original value be $x$. After a 20% increase, the new value is $1.2x$.
$$\begin{aligned} 1.2x &= 240 \\[15pt] x &= \frac{240}{1.2} \\[15pt] x &= 200 \end{aligned}$$
So the original value was 200.
Similarly, if a price after a 25% discount becomes 600, then the new price is 75% of the original. Let the original price be $x$.
$$\begin{aligned} 0.75x &= 600 \\[15pt] x &= \frac{600}{0.75} \\[15pt] x &= 800 \end{aligned}$$
So the original price was 800.
Final Summary
So, percentage change tells us how much a quantity has changed compared with its original value.
The first step is always to find the difference between the new and original values. The second step is to divide by the original value. The third step is to multiply by 100 to turn the answer into a percentage. If the quantity went up, the result is a percentage increase. If the quantity went down, the result is a percentage decrease.
When finding a new value after a known percentage change, you can either find the amount of increase or decrease first, or use a decimal multiplier such as $1.15$ for a 15% increase or $0.8$ for a 20% decrease.
