Introduction to Percentages

Hello, students! Today, we’re going to learn about percentages. A percentage is a way to express a number as a part of 100. For example, if you score 75 out of 100 in a test, you can say you scored 75%.

What is a Percentage?

• Definition: A percentage shows how much out of 100 something is.
• Symbol: We use the symbol “%” to represent percentages.
• Example: If you have 25 apples and you give away 5, you have given away $$\frac{5}{25} \times 100 = 20%$$ of your apples.

Key Rules of Percentages

1. Finding a Percentage of a Number:
   • To find a percentage of a number, you can convert the percentage to a fraction and then multiply.
   • Example: To find 20% of 50, you can calculate $$0.20 \times 50 = 10$$.
2. Increasing or Decreasing by a Percentage:
   • To increase a number by a percentage, find the percentage and add it to the original number.
   • To decrease a number by a percentage, find the percentage and subtract it from the original number.
   • Example: To increase 100 by 10%, calculate $$100 + (0.10 \times 100) = 110$$.
3. Converting Fractions and Decimals to Percentages:
   • To convert a fraction to a percentage, divide the top number by the bottom number, and then multiply by 100.
   • Example: For the fraction $$\frac{1}{4}$$, you calculate $$\frac{1}{4} = 0.25$$ and then $$0.25 \times 100 = 25%$$.

Tips and Tricks

• Remember: “Per cent” means “per hundred.” So, if you see 50%, think of it as 50 out of 100.
• Practice with real-life examples: Look for percentages in shops, like discounts.
• Use a calculator: When in doubt, a calculator can help you find percentages quickly.

Practice Questions

Easy Level Questions

1. What is 50% of 20?
2. What is 25% of 80?
3. If you have 60 marbles and give away 30, what percentage have you given away?
4. Convert 0.5 to a percentage.
5. What is 10% of 200?
6. If a shirt costs £40 and is on sale for 20% off, how much do you save?
7. What is 75% of 40?
8. How much is 30% of 90?
9. If you score 90 out of 100, what is your percentage?
10. Convert 3/5 to a percentage.

Medium Level Questions

1. Increase 150 by 20%.
2. Decrease 250 by 10%.
3. What is 15% of 200?
4. If a student scores 45 out of 60, what percentage is that?
5. A bike costs £300 and is reduced by 15%. What is the new price?
6. Convert 0.75 to a percentage.
7. What percentage is 5 out of 20?
8. Find 40% of 250.
9. If you have £80 and spend 25%, how much do you have left?
10. If a cake is cut into 8 pieces and 3 pieces are eaten, what percentage has been eaten?

Hard Level Questions

1. If a jacket costs £120 and is 25% off, how much do you pay?
2. Convert 7/8 to a percentage.
3. If you run 5 km and that is 25% of your daily goal, what is your goal?
4. A class has 30 students, and 18 are boys. What percentage are boys?
5. If you received a score of 88% on a test with 50 questions, how many did you get right?
6. Find 65% of £80.
7. A car costs £10,000 and depreciates by 10% in a year. What is its value after one year?
8. If you have a pizza cut into 12 slices and eat 4, what percentage is left?
9. Increase 200 by 15% and then decrease that result by 10%. What is the final amount?
10. If a student improves their score from 70% to 85%, what is the increase in percentage points?

Answers and Explanations

Easy Level Answers

1. Answer: 10. Explanation: 50% of 20 is calculated as $$0.50 \times 20 = 10$$.
2. Answer: 20. Explanation: 25% of 80 is found by $$0.25 \times 80 = 20$$.
3. Answer: 50%. Explanation: If you give away 30 out of 60 marbles, it’s $$\frac{30}{60} \times 100 = 50%$$.
4. Answer: 50%. Explanation: To convert 0.5 to a percentage, multiply by 100: $$0.5 \times 100 = 50%$$.
5. Answer: 20. Explanation: 10% of 200 is calculated as $$0.10 \times 200 = 20$$.
6. Answer: £8. Explanation: The savings from 20% off £40 is calculated by $$0.20 \times 40 = 8$$.
7. Answer: 30. Explanation: 75% of 40 is calculated as $$0.75 \times 40 = 30$$.
8. Answer: 27. Explanation: 30% of 90 is found by $$0.30 \times 90 = 27$$.
9. Answer: 90%. Explanation: If you score 90 out of 100, you have $$\frac{90}{100} \times 100 = 90%$$.
10. Answer: 60%. Explanation: To convert $$\frac{3}{5}$$ to a percentage, calculate $$\frac{3}{5} \times 100 = 60%$$.

Medium Level Answers

1. Answer: 180. Explanation: To increase 150 by 20%, calculate $$150 + (0.20 \times 150) = 180$$.
2. Answer: 225. Explanation: To decrease 250 by 10%, calculate $$250 – (0.10 \times 250) = 225$$.
3. Answer: 30. Explanation: 15% of 200 is calculated as $$0.15 \times 200 = 30$$.
4. Answer: 75%. Explanation: Scoring 45 out of 60 gives $$\frac{45}{60} \times 100 = 75%$$.
5. Answer: £255. Explanation: The sale price of £300 at 15% off is $$300 – (0.15 \times 300) = 255$$.
6. Answer: 75%. Explanation: To convert 0.75 to a percentage, multiply by 100: $$0.75 \times 100 = 75%$$.
7. Answer: 25%. Explanation: The percentage of 5 out of 20 is calculated as $$\frac{5}{20} \times 100 = 25%$$.
8. Answer: 100. Explanation: 40% of 250 is calculated as $$0.40 \times 250 = 100$$.
9. Answer: £60. Explanation: Spending 25% of £80 leaves you with $$80 – (0.25 \times 80) = 60$$.
10. Answer: 50%. Explanation: Eating 3 out of 8 pieces means $$\frac{3}{8} \times 100 = 37.5%$$ eaten.

Hard Level Answers

1. Answer: £90. Explanation: If a £120 jacket is 25% off, you save £30, so you pay $$120 – 30 = 90$$.
2. Answer: 87.5%. Explanation: To convert $$\frac{7}{8}$$ to a percentage, calculate $$\frac{7}{8} \times 100 = 87.5%$$.
3. Answer: 20 km. Explanation: If 5 km is 25% of your goal, the full goal is $$\frac{5}{0.25} = 20$$ km.
4. Answer: 60%. Explanation: If there are 30 students and 18 are boys, then the boys make up $$\frac{18}{30} \times 100 = 60%$$.
5. Answer: 44. Explanation: If you score 88% on a 50-question test, you got $$0.88 \times 50 = 44$$ questions right.
6. Answer: £52. Explanation: 65% of £80 is calculated as $$0.65 \times 80 = 52$$.
7. Answer: £9,000. Explanation: A £10,000 car that depreciates by 10% loses £1,000, making it worth $$10,000 – 1,000 = 9,000$$.
8. Answer: 66.67%. Explanation: Eating 4 out of 12 slices means $$\frac{4}{12} \times 100 = 33.33%$$ left.
9. Answer: 207. Explanation: Increasing 200 by 15% gives you 230, and decreasing that by 10% results in $$230 – (0.10 \times 230) = 207$$.
10. Answer: 15 percentage points. Explanation: The improvement from 70% to 85% is an increase of $$85 – 70 = 15$$ percentage points.

Feel free to ask if you have any questions or need further explanations! Happy learning!