Introduction to Percentage
You see percentages everywhere. A shop says "30% off on all shoes." Your test result shows "85%." The news says "60% of the city received rain today." But what does this number really mean?
The word percent comes from the Latin per centum, which means "per hundred" or "out of 100". So when you score 85%, it means you got 85 out of every 100 marks. When a shop offers 30% off, it means for every Rs. 100 you would pay, you save Rs. 30.
Percentages make it easy to compare different quantities. Suppose Rina scored 18 out of 25 in English and 40 out of 50 in Maths. Which subject did she do better in? It is hard to compare directly because the totals are different. But if you convert both to percentages (72% and 80%), the comparison is instant.
In this chapter, you will learn what percentage means, how to convert fractions and decimals into percentages and back, how to find the percentage of a number, and how to solve everyday problems using percentages.
What is Introduction to Percentage?
Definition: A percentage is a number expressed as a fraction of 100. The symbol for percentage is %.
If you say 45%, you mean:
45% = 45 out of 100 = 45/100
Key Terms:
| Term | Meaning | Example |
|---|---|---|
| Percentage (%) | A number out of 100 | 25% means 25 out of 100 |
| Fraction form | Percentage written as a fraction with denominator 100 | 25% = 25/100 = 1/4 |
| Decimal form | Percentage written as a decimal | 25% = 0.25 |
| Whole (Base) | The total quantity that is being considered | If 25% of 200 students passed, then 200 is the whole |
| Part | The portion of the whole | 25% of 200 = 50 students (the part) |
Important:
- 100% means the whole thing (all of it).
- 50% means half.
- 25% means one-quarter.
- 0% means nothing at all.
- Percentages can be more than 100%. For example, 150% means 1.5 times the original.
Introduction to Percentage Formula
Converting Fraction to Percentage:
Percentage = (Fraction) x 100 %
Where:
- Multiply the fraction by 100
- Add the % symbol
Converting Percentage to Fraction:
Fraction = Percentage / 100
Where:
- Write the percentage number over 100
- Simplify the fraction to its lowest terms
Converting Percentage to Decimal:
Decimal = Percentage / 100
Where:
- Move the decimal point two places to the left
- Example: 75% = 0.75
Converting Decimal to Percentage:
Percentage = Decimal x 100 %
Where:
- Move the decimal point two places to the right
- Example: 0.45 = 45%
Finding Percentage of a Number:
Percentage of a number = (Percentage / 100) x Number
Finding what percentage one number is of another:
Percentage = (Part / Whole) x 100 %
Types and Properties
Percentages are used in several types of problems:
Type 1: Converting Fractions to Percentages
Multiply the fraction by 100 and add %. Example: 3/5 = (3/5) x 100 = 60%.
Type 2: Converting Percentages to Fractions
Write the percentage over 100 and simplify. Example: 40% = 40/100 = 2/5.
Type 3: Converting Decimals to Percentages
Multiply the decimal by 100 and add %. Example: 0.35 = 0.35 x 100 = 35%.
Type 4: Converting Percentages to Decimals
Divide by 100 (move decimal point two places left). Example: 62% = 62/100 = 0.62.
Type 5: Finding Percentage of a Given Number
Use the formula: (Percentage/100) x Number. Example: 20% of 350 = (20/100) x 350 = 70.
Type 6: Finding What Percentage One Number is of Another
Use the formula: (Part/Whole) x 100. Example: What percentage is 45 of 180? Answer: (45/180) x 100 = 25%.
Useful Fraction-Percentage Equivalents:
| Fraction | Percentage |
|---|---|
| 1/2 | 50% |
| 1/4 | 25% |
| 3/4 | 75% |
| 1/5 | 20% |
| 2/5 | 40% |
| 3/5 | 60% |
| 1/8 | 12.5% |
| 1/3 | 33.33% (approx.) |
| 2/3 | 66.67% (approx.) |
| 1/10 | 10% |
Solved Examples
Example 1: Convert a Fraction to Percentage
Problem: Convert 3/4 to a percentage.
Solution:
Given:
- Fraction = 3/4
Using the formula:
- Percentage = (3/4) x 100
- = 300/4
- = 75%
Answer: 3/4 = 75%
Example 2: Convert a Percentage to Fraction
Problem: Convert 45% to a fraction in simplest form.
Solution:
Given:
- Percentage = 45%
Steps:
- Write as fraction: 45/100
- Find HCF of 45 and 100 = 5
- Divide both by 5: 45/100 = 9/20
Answer: 45% = 9/20
Example 3: Convert a Decimal to Percentage
Problem: Convert 0.72 to a percentage.
Solution:
Given:
- Decimal = 0.72
Using the formula:
- Percentage = 0.72 x 100
- = 72%
Answer: 0.72 = 72%
Example 4: Convert a Percentage to Decimal
Problem: Convert 35% to a decimal.
Solution:
Given:
- Percentage = 35%
Steps:
- Decimal = 35 / 100
- = 0.35
Answer: 35% = 0.35
Example 5: Find Percentage of a Number
Problem: Find 20% of 450.
Solution:
Given:
- Percentage = 20%
- Number = 450
Using the formula:
- 20% of 450 = (20/100) x 450
- = (1/5) x 450
- = 450/5
- = 90
Answer: 20% of 450 = 90
Example 6: What Percentage is One Number of Another
Problem: Ravi scored 36 out of 40 in a test. What is his percentage?
Solution:
Given:
- Part (marks obtained) = 36
- Whole (total marks) = 40
Using the formula:
- Percentage = (36/40) x 100
- = (9/10) x 100
- = 90%
Answer: Ravi scored 90%
Example 7: Shopping Discount Problem
Problem: A shirt costs Rs. 800. A shop offers 15% discount. How much do you save? What is the selling price?
Solution:
Given:
- Price = Rs. 800
- Discount = 15%
Finding the discount amount:
- Discount = 15% of 800
- = (15/100) x 800
- = 15 x 8
- = Rs. 120
Selling price:
- Selling price = 800 - 120 = Rs. 680
Answer: You save Rs. 120. The selling price is Rs. 680.
Example 8: Attendance Problem
Problem: A class has 50 students. On Monday, 8 students were absent. What percentage of students were present?
Solution:
Given:
- Total students = 50
- Absent = 8
- Present = 50 - 8 = 42
Finding the percentage:
- Percentage present = (42/50) x 100
- = 4200/50
- = 84%
Answer: 84% of students were present.
Example 9: Mixed Number Fraction to Percentage
Problem: Convert 2 1/4 to a percentage.
Solution:
Given:
- Mixed number = 2 1/4
Steps:
- Convert to improper fraction: 2 1/4 = 9/4
- Percentage = (9/4) x 100
- = 900/4
- = 225%
Answer: 2 1/4 = 225%
Example 10: Water Tank Problem
Problem: A tank holds 500 litres of water. It is 65% full. How many litres of water are in the tank? How many more litres are needed to fill it completely?
Solution:
Given:
- Capacity = 500 litres
- Filled = 65%
Water in tank:
- 65% of 500 = (65/100) x 500
- = 32500/100
- = 325 litres
Water needed:
- Empty = 100% - 65% = 35%
- 35% of 500 = (35/100) x 500 = 175 litres
Answer: The tank has 325 litres. It needs 175 more litres to be full.
Real-World Applications
Percentages are used in daily life all the time:
Shopping and Discounts: Shops use percentages to show discounts. "Flat 50% off" means you pay only half the original price.
Exam Results: Marks are converted to percentages so you can compare performance across subjects with different total marks.
Bank Interest: Banks give interest on savings at a certain percentage per year. If the rate is 6% per annum, you earn Rs. 6 for every Rs. 100 deposited for one year.
Taxes: GST (Goods and Services Tax) is calculated as a percentage. If GST is 18% on a product worth Rs. 1000, the tax is Rs. 180.
Nutrition Labels: Food packets show "Daily Value %" to tell you how much of your daily requirement a serving provides.
Sports: A cricketer's strike rate, a basketball player's shooting percentage, and a football team's win percentage all use this concept.
Weather: "There is a 70% chance of rain" means that out of 100 similar weather conditions, rain occurs in about 70 of them.
Key Points to Remember
- Percent means "per hundred" or "out of 100."
- The symbol for percentage is %.
- To convert a fraction to percentage: multiply by 100 and add %.
- To convert a percentage to fraction: divide by 100 and simplify.
- To convert a decimal to percentage: multiply by 100 and add %.
- To convert a percentage to decimal: divide by 100 (shift decimal two places left).
- To find percentage of a number: (Percentage/100) x Number.
- To find what percentage one number is of another: (Part/Whole) x 100.
- 100% = the whole, 50% = half, 25% = one-quarter, 10% = one-tenth.
- Percentages can be greater than 100%. Example: 200% means twice the original.
- Percentages make it easy to compare quantities with different totals.
Practice Problems
- Convert 7/20 to a percentage.
- Convert 85% to a fraction in simplest form.
- Convert 0.08 to a percentage.
- Convert 125% to a decimal.
- Find 35% of 600.
- Meera scored 72 out of 80 in Science. What is her percentage?
- A school has 1200 students. 45% are girls. How many girls and how many boys are in the school?
- A shopkeeper gives 12% discount on a bag priced at Rs. 1500. Find the discount and selling price.
Frequently Asked Questions
Q1. What does percentage mean?
Percentage means 'per hundred' or 'out of 100'. When we say 40%, we mean 40 out of every 100. The symbol % represents percentage.
Q2. How do you convert a fraction to a percentage?
Multiply the fraction by 100 and add the % symbol. For example, 3/5 = (3/5) x 100 = 60%.
Q3. How do you convert a percentage to a fraction?
Write the percentage as a fraction with denominator 100, then simplify. For example, 60% = 60/100 = 3/5.
Q4. How do you find 25% of a number?
25% of a number means one-quarter of it. Divide the number by 4. For example, 25% of 80 = 80/4 = 20. Or use the formula: (25/100) x 80 = 20.
Q5. Can a percentage be more than 100%?
Yes. A percentage greater than 100% means the value is more than the whole. For example, if a population grows by 150%, it means it has become 2.5 times the original (100% original + 150% growth = 250% of original).
Q6. What is the difference between percentage and percentile?
Percentage is the marks obtained out of 100 (or converted to out of 100). Percentile tells you what percentage of other students scored below you. For example, 90th percentile means you scored better than 90% of students.
Q7. How do you convert a decimal to a percentage?
Multiply the decimal by 100 and add %. Move the decimal point two places to the right. For example, 0.65 = 0.65 x 100 = 65%.
Q8. How do you find what percentage one number is of another?
Use the formula: Percentage = (Part / Whole) x 100. For example, to find what percentage 15 is of 60: (15/60) x 100 = 25%.
Related Topics
- Percentage Increase and Decrease
- Profit and Loss
- Converting Fractions to Decimals
- Introduction to Ratio
- Discount Calculation
- Simple Interest
- Compound Interest
- Applications of Compound Interest
- Sales Tax and VAT
- Growth and Decay
- Finding Percentage of a Number
- Converting Between %, Fraction and Decimal
- Word Problems on Comparing Quantities
- Word Problems on Profit and Loss
