How to Calculate Percentage Increase and Decrease

Grade: 7-8 | Topic: Arithmetic

What You Will Learn

After working through this page you will be able to calculate percentage increase and percentage decrease using a single reliable formula. You will also learn how to find the new value after a given percentage change, reverse the calculation to find the original value, and handle successive percentage changes correctly.

Theory

The percentage change formula

Percentage change measures how much a value has grown or shrunk relative to where it started. The universal formula is:

$\text{Percentage change} = \frac{\text{New value} - \text{Original value}}{\text{Original value}} \times 100\%$

• If the result is positive, the value increased.
• If the result is negative, the value decreased.

Key rule: Always divide by the original value -- the one you started with before the change happened.

Finding the new value after a percentage change

Sometimes you know the original value and the percentage, and you need the result. Use the multiplier approach:

For an increase of $p\%$:

$\text{New value} = \text{Original} \times \left(1 + \frac{p}{100}\right)$

For a decrease of $p\%$:

$\text{New value} = \text{Original} \times \left(1 - \frac{p}{100}\right)$

The expression in parentheses is called the multiplier. A 20% increase has a multiplier of 1.20. A 20% decrease has a multiplier of 0.80.

Percentage change	Multiplier
5% increase	1.05
12% increase	1.12
25% decrease	0.75
40% decrease	0.60

Reverse percentage: finding the original value

When you know the value after a percentage change and need to work backwards:

$\text{Original value} = \frac{\text{New value}}{\text{Multiplier}}$

For example, if a shirt costs $63 after a 10% discount, the sale price represents 90% of the original:

$\text{Original} = \frac{63}{0.90} = 70$

The original price was $70.

Successive percentage changes

When two or more percentage changes happen one after another, you cannot simply add or subtract the percentages. Each change applies to the result of the previous one. Multiply the multipliers together:

$\text{Final value} = \text{Original} \times \text{Multiplier}_1 \times \text{Multiplier}_2$

For example, a 20% increase followed by a 10% decrease:

$\text{Final} = \text{Original} \times 1.20 \times 0.90 = \text{Original} \times 1.08$

The overall effect is an 8% increase -- not 10%, as you might expect from $20\% - 10\%$.

Worked Examples

Example 1: Calculating percentage increase (easy)

Problem: A plant was 40 cm tall last month. Now it is 52 cm tall. What is the percentage increase in height?

Step 1: Find the amount of change. $52 - 40 = 12 \text{ cm}$

Step 2: Divide by the original value. $\frac{12}{40} = 0.30$

Step 3: Multiply by 100. $0.30 \times 100 = 30\%$

Answer: The plant's height increased by 30%.

Example 2: Calculating percentage decrease (easy)

Problem: A shop reduced the price of a book from $25 to $20. What is the percentage decrease?

Step 1: Find the amount of change. $25 - 20 = 5$

Step 2: Divide by the original price. $\frac{5}{25} = 0.20$

Step 3: Multiply by 100. $0.20 \times 100 = 20\%$

Answer: The price decreased by 20%.

Example 3: Finding the new value after a percentage increase (medium)

Problem: A town has a population of 12,000. The population grows by 8% this year. What is the new population?

Step 1: Calculate the multiplier. $1 + \frac{8}{100} = 1.08$

Step 2: Multiply the original population by the multiplier. $12{,}000 \times 1.08 = 12{,}960$

Answer: The new population is 12,960.

Example 4: Reverse percentage -- finding the original price (medium)

Problem: After a 15% markup, a retailer sells a product for $92. What was the cost price?

Step 1: The selling price represents 115% of the cost price. The multiplier is 1.15.

Step 2: Divide the selling price by the multiplier. $\frac{92}{1.15} = 80$

Step 3: Verify -- $80 \times 1.15 = 92$. Correct.

Answer: The cost price was $80.

Example 5: Successive percentage changes (challenging)

Problem: A stock's value increases by 25% in the first year and then decreases by 20% in the second year. If the stock started at $160, what is its value after two years? What is the overall percentage change?

Step 1: Apply the first change (25% increase). $160 \times 1.25 = 200$

Step 2: Apply the second change (20% decrease) to the new value. $200 \times 0.80 = 160$

Step 3: Calculate the overall percentage change. $\frac{160 - 160}{160} \times 100 = 0\%$

Step 4: Alternatively, combine the multipliers: $1.25 \times 0.80 = 1.00$. This confirms zero overall change.

Answer: The stock returns to $160, a net change of 0%. Even though the percentages were "25% up and 20% down," the overall effect is no change because each percentage applies to a different base.

Common Mistakes

Mistake 1: Dividing by the new value instead of the original

❌ A price goes from $50 to $65. Student calculates: $\frac{15}{65} \times 100 = 23.1\%$.

✅ $\frac{65 - 50}{50} \times 100 = \frac{15}{50} \times 100 = 30\%$

Why this matters: Percentage change is always relative to the starting point. Using the new value as the denominator gives an incorrect, smaller answer for increases and a larger answer for decreases.

Mistake 2: Adding successive percentages

❌ A 30% increase followed by a 30% decrease means no change.

✅ Start with 100. After 30% increase: $100 \times 1.30 = 130$. After 30% decrease: $130 \times 0.70 = 91$. The result is 91, a 9% overall decrease.

Why this matters: Each percentage change uses the previous result as its base. A 30% decrease of 130 removes more than the 30% increase on 100 added, because 30% of 130 is larger than 30% of 100.

Mistake 3: Confusing percentage change with percentage points

❌ "Interest rate went from 3% to 5%, so it increased by 2%."

✅ The rate increased by 2 percentage points. The percentage increase is $\frac{5 - 3}{3} \times 100 = 66.7\%$.

Why this matters: In everyday language, people sometimes say "2% increase" when they mean 2 percentage points. In math class, be precise. Percentage change measures the relative size of the change compared to the original.

Practice Problems

Try these on your own before checking the answers:

1. A shirt's price goes from $40 to $50. What is the percentage increase?
2. A tank held 80 litres. After a leak, it holds 68 litres. What is the percentage decrease?
3. An investment of $5,000 earns 6% interest in one year. What is the new total?
4. After a 20% discount, a pair of shoes costs $56. What was the original price?
5. A city's population increases by 10% one year, then decreases by 10% the next. If it started at 200,000, what is the population after two years?

Click to see answers

1. Change = $50 - 40 = 10$. $\frac{10}{40} \times 100 = 25\%$
2. Change = $80 - 68 = 12$. $\frac{12}{80} \times 100 = 15\%$
3. $5000 \times 1.06 = 5{,}300$, so the new total is $5,300
4. Multiplier = $1 - 0.20 = 0.80$. Original = \frac{56}{0.80} = \70$
5. After year 1: $200{,}000 \times 1.10 = 220{,}000$. After year 2: $220{,}000 \times 0.90 = 198{,}000$

Summary

• Percentage change = $\frac{\text{New} - \text{Original}}{\text{Original}} \times 100$. Always divide by the original value.
• Use multipliers for quick calculations: multiply by $(1 + p/100)$ for an increase, $(1 - p/100)$ for a decrease.
• To find the original value, divide the new value by the multiplier (reverse percentage).
• Successive percentage changes must be calculated step by step -- never simply add or subtract the percentages.

Related Topics

• Percentages — Complete Guide
• How to Calculate Percentage of a Number
• Percentage Word Problems with Solutions

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