Percentages — How to Calculate, Convert, and Solve Percentage Problems

Grade: 7-8 | Topic: Arithmetic

What You Will Learn

By the end of this guide you will be able to convert between fractions, decimals, and percentages, calculate the percentage of any number, and determine percentage increase or decrease. You will also gain confidence solving real-world percentage word problems involving discounts, taxes, and tips.

Theory

What is a percentage?

A percentage is a way of expressing a number as a fraction of 100. The word itself comes from the Latin per centum, meaning "out of one hundred." When you see the symbol $\%$, think "divided by 100":

$45\% = \frac{45}{100} = 0.45$

Percentages are everywhere in daily life — discounts at the store, interest rates, test scores, and nutrition labels all use them. Understanding percentages means you can interpret and compare these numbers confidently.

Converting between fractions, decimals, and percentages

These three forms all represent the same value. Knowing how to move between them is an essential skill.

Fraction to percentage: Divide the numerator by the denominator, then multiply by 100:

$\text{Percentage} = \frac{a}{b} \times 100\%$

For example, $\frac{3}{5} = 3 \div 5 = 0.6$, and $0.6 \times 100 = 60\%$.

Decimal to percentage: Multiply by 100 (or shift the decimal point two places to the right):

$0.375 \times 100 = 37.5\%$

Percentage to decimal: Divide by 100 (or shift the decimal point two places to the left):

$62\% = 62 \div 100 = 0.62$

Percentage to fraction: Write the percentage over 100 and simplify:

$75\% = \frac{75}{100} = \frac{3}{4}$

Finding a percentage of a number

To find $p\%$ of a number $n$, use the formula:

$p\% \text{ of } n = \frac{p}{100} \times n$

For example, 20% of 150:

$\frac{20}{100} \times 150 = 0.20 \times 150 = 30$

Finding what percentage one number is of another

When you know the part and the whole, you can find the percentage:

$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%$

For example, if you scored 36 out of 45 on a test:

$\frac{36}{45} \times 100 = 80\%$

Percentage increase and decrease

Percentage change tells you how much a value has grown or shrunk relative to its original size:

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

A positive result indicates an increase; a negative result indicates a decrease. You can also compute the new value directly:

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

For an increase, $p$ is positive. For a decrease, $p$ is negative (or you can write $1 - \frac{p}{100}$).

Reverse percentage problems

Sometimes you know the value after a percentage change and need to find the original. In that case, rearrange the formula:

$\text{Original value} = \frac{\text{New value}}{1 + \frac{p}{100}}$

For example, a jacket costs $68 after a 15% discount. The sale price represents $100\% - 15\% = 85\%$ of the original price:

$\text{Original price} = \frac{68}{0.85} = 80$

So the original price was $80.

Worked Examples

Example 1: Finding a percentage of a number (easy)

Problem: What is 35% of 240?

Step 1: Convert the percentage to a decimal. $35\% = \frac{35}{100} = 0.35$

Step 2: Multiply by the number. $0.35 \times 240 = 84$

Answer: 84

Example 2: Finding what percentage a number is of another (medium)

Problem: A class has 32 students. Today, 24 students are present. What percentage of the class is present?

Step 1: Identify the part and the whole. Part = 24 (students present), Whole = 32 (total students).

Step 2: Apply the percentage formula. $\frac{24}{32} \times 100 = \frac{2400}{32} = 75\%$

Step 3: Verify — $75\%$ of 32 is $0.75 \times 32 = 24$. Correct.

Answer: 75% of the class is present.

Example 3: Percentage increase (medium)

Problem: A bicycle was priced at $120 last year. This year the price rose to $138. What is the percentage increase?

Step 1: Calculate the amount of change. $138 - 120 = 18$

Step 2: Divide the change by the original value. $\frac{18}{120} = 0.15$

Step 3: Multiply by 100 to express as a percentage. $0.15 \times 100 = 15\%$

Answer: The price increased by 15%.

Example 4: Percentage decrease and discount (medium)

Problem: A store offers a 20% discount on a $65 item. What is the sale price?

Step 1: Calculate the discount amount. $\frac{20}{100} \times 65 = 0.20 \times 65 = 13$

Step 2: Subtract the discount from the original price. $65 - 13 = 52$

Answer: The sale price is $52.

Example 5: Reverse percentage — finding the original value (challenging)

Problem: After a 12% tax is added, a meal costs $33.60. What was the price of the meal before tax?

Step 1: Recognise that $33.60 represents 112% of the original price. $100\% + 12\% = 112\% = 1.12$

Step 2: Divide the final amount by 1.12 to find the original price. $\frac{33.60}{1.12} = 30$

Step 3: Verify — $12\%$ of $30 is $0.12 \times 30 = 3.60$, and $30 + 3.60 = 33.60$. Correct.

Answer: The price before tax was $30.00.

Common Mistakes

Mistake 1: Using the wrong base when calculating percentage change

❌ A price goes from $80 to $100. Student calculates: $\frac{20}{100} \times 100 = 20\%$.

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

Why this matters: Percentage change must always be divided by the original value, not the new value. Using the new value as the denominator underestimates an increase and overestimates a decrease.

Mistake 2: Forgetting to convert the percentage to a decimal before multiplying

❌ $25\% \times 60 = 25 \times 60 = 1500$

✅ $25\% \times 60 = 0.25 \times 60 = 15$

Why this matters: The percent symbol means "divided by 100." If you multiply by 25 instead of 0.25, your answer will be 100 times too large. Always convert first.

Mistake 3: Adding successive percentages directly

❌ A 10% increase followed by a 10% decrease returns to the original. Not true!

✅ Start with 100. After a 10% increase: $100 \times 1.10 = 110$. After a 10% decrease: $110 \times 0.90 = 99$. The result is 99, not 100.

Why this matters: Each percentage applies to a different base. The 10% decrease is applied to the already-increased amount, so it removes more than the original increase added. Successive percentage changes must be calculated step by step.

Practice Problems

Try these on your own before checking the answers:

1. What is 18% of 250?
2. A student scored 42 out of 60 on a quiz. What percentage is that?
3. A town's population grew from 4,000 to 4,600. What is the percentage increase?
4. A laptop originally costs $800. It is on sale for 15% off. What is the sale price?
5. After a 25% markup, a product sells for $75. What was the original cost?

Click to see answers

1. $0.18 \times 250 = 45$
2. $\frac{42}{60} \times 100 = 70\%$
3. Change = $4600 - 4000 = 600$. $\frac{600}{4000} \times 100 = 15\%$
4. Discount = $0.15 \times 800 = 120$. Sale price = 800 - 120 = \680$
5. 75 \div 1.25 = \60$

Summary

• A percentage expresses a value as a fraction of 100 — convert by multiplying or dividing by 100.
• To find $p\%$ of a number, compute $\frac{p}{100} \times n$.
• To find what percentage one number is of another, use $\frac{\text{Part}}{\text{Whole}} \times 100$.
• Percentage change is always calculated relative to the original value: $\frac{\text{change}}{\text{original}} \times 100$.
• Successive percentage changes must be computed step by step because each applies to a different base.

Related Topics

• How to Calculate Percentage of a Number
• How to Calculate Percentage Increase and Decrease
• Percentage Word Problems with Solutions
• Converting Between Fractions, Decimals, and Percentages
• Fractions — Complete Guide

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