Percentages Explained Step by Step

🎯 By the end of this lesson...

You should be able to:

Explain what a percentage represents.
Convert fractions and decimals to percentages.
Convert percentages to fractions and decimals.
Find a percentage of a quantity.
Express one quantity as a percentage of another.
Calculate percentage increase and decrease.
Calculate percentage change.
Use percentage multipliers.
Calculate repeated percentage change.
Solve reverse-percentage problems.
Calculate simple and compound interest.
Apply percentages to practical situations.

🤔 Think About This...

🛍️

A sale at a clothing shop

A pair of shoes normally costs US$75.

The shop advertises:

20% OFF

Does this mean that you pay US$20?

Does it mean that the shoes now cost 20 cents?

What does 20% off actually mean?

A percentage tells us how many parts are being considered out of every 100 equal parts.

1️⃣ What Does Percentage Mean?

The word percent means out of every one hundred.

0%

None of the whole

25%

25 out of 100

50%

Half of the whole

100%

The complete whole

1% = 1/100

25% = 25/100

80% = 80/100

A percentage is a fraction out of 100, even when the denominator is not shown.

🔲 Visualising a Percentage

Imagine a square divided into 100 equal smaller squares.

If 35 of the squares are shaded, then 35 out of the 100 parts are shaded.

35

out of

100

35/100 = 35%

35% means 35 parts out of every 100 equal parts.

💡 Did You Know?

Percentages appear almost everywhere in everyday life.

🔋 Phone battery

📚 School marks

⚽ Ball possession

🏦 Bank interest

🛒 Shop discounts

🌦️ Weather forecasts

Percentages allow us to compare quantities using the same scale of 100.

2️⃣ Converting a Fraction to a Percentage

There are two useful methods.

Method 1: Multiply by 100%

Fraction × 100% = Percentage

Example 1: Convert 3/5 to a percentage

3/5 × 100%

= 60%

3/5 = 60%

Method 2: Create a fraction out of 100

Example 2: Convert 3/4 to a percentage

Multiply the denominator by 25 to obtain 100.

3/4 = 75/100

3/4 = 75%

🧠 StudyNest Method: Fraction to Percentage

Step 1: Look at the denominator.

Step 2: Ask whether you can easily create an equivalent fraction out of 100.

Step 3: If that is not convenient, multiply the fraction by 100%.

Step 4: Simplify or divide carefully.

Step 5: Include the percentage sign.

3️⃣ Converting a Decimal to a Percentage

To convert a decimal to a percentage, multiply it by 100%.

Decimal × 100% = Percentage

Example 3: Convert 0.72 to a percentage

0.72 × 100%

= 72%

0.72 = 72%

Example 4: Convert 1.35 to a percentage

1.35 × 100%

= 135%

1.35 = 135%

A percentage can be greater than 100%.

135% represents 1.35 times the original whole.

4️⃣ Converting a Percentage to a Fraction

Example 5: Convert 40% to a fraction

Write 40 over 100.

40% = 40/100

Simplify using the HCF.

40/100 = 2/5

40% = 2/5

Example 6: Convert 12.5% to a fraction

12.5% = 12.5/100

Multiply the numerator and denominator by 10 to remove the decimal.

125/1000

Simplify.

12.5% = 1/8

5️⃣ Converting a Percentage to a Decimal

To convert a percentage to a decimal, divide it by 100.

Percentage ÷ 100 = Decimal

Example 7: Convert 68% to a decimal

68 ÷ 100 = 0.68

68% = 0.68

Example 8: Convert 7.5% to a decimal

7.5 ÷ 100 = 0.075

7.5% = 0.075

🔁 Fraction, Decimal and Percentage

Fraction

3/4

→

Decimal

0.75

→

Percentage

75%

Fraction

Decimal

Percentage

1/2

0.5

50%

1/4

0.25

25%

3/4

0.75

75%

1/5

0.2

20%

1/10

0.1

10%

6️⃣ Finding a Percentage of a Quantity

In percentage calculations, the word of normally means multiply.

Percentage of a quantity

= Percentage/100 × Quantity

Example 9: Find 30% of 240

30/100 × 240

= 72

30% of 240 is 72.

🧠 Mental Percentage Shortcuts

10%

Divide by 10.

5%

Find 10%, then halve it.

1%

Divide by 100.

50%

Find one half.

25%

Find one quarter.

75%

Find three quarters.

Example 10: Find 15% of 360 mentally

10% of 360 = 36

5% of 360 = 18

15% = 10% + 5%

36 + 18 = 54

15% of 360 is 54.

🌍 Real-Life Example: School-Fee Deposit

Example 11

The fees for a course are US$400.

A parent must pay a deposit of 25%.

25/100 × 400

= 100

The deposit is US$100.

7️⃣ Expressing One Quantity as a Percentage of Another

Part/Whole × 100%

Example 12: A learner scores 36 marks out of 45

36/45 × 100%

= 80%

The learner scored 80%.

Place the part on top and the original whole underneath.

Reversing the fraction gives the wrong percentage.

8️⃣ Increasing a Quantity by a Percentage

Example 13: Increase US$250 by 12%

First find the increase.

12/100 × 250 = 30

Add it to the original amount.

250 + 30 = 280

US$250 increased by 12% is US$280.

The percentage amount is not the new total. Add the increase to the original amount.

9️⃣ Decreasing a Quantity by a Percentage

Example 14: Reduce US$75 by 20%

First find the reduction.

20/100 × 75 = 15

Subtract it from the original amount.

75 − 15 = 60

After the 20% discount, the shoes cost US$60.

🔢 Percentage Multipliers

A multiplier lets us calculate the new quantity in one step.

Increase by r%

Multiplier = 1 + r/100

Decrease by r%

Multiplier = 1 − r/100

Example 15: Increase 250 by 12%

Multiplier = 1 + 0.12 = 1.12

250 × 1.12 = 280

Example 16: Decrease 75 by 20%

Multiplier = 1 − 0.20 = 0.80

75 × 0.80 = 60

A multiplier greater than 1 produces an increase.

A multiplier between 0 and 1 produces a decrease.

🔟 Percentage Change

Percentage change compares the amount of change with the original value.

Percentage change

= Change/Original value × 100%

Example 17: A price rises from US$80 to US$92

Find the increase.

Change = 92 − 80 = 12

12/80 × 100%

= 15%

The price increased by 15%.

In the denominator, use the original value—not the new value.

🧠 StudyNest Method: Percentage Change

Step 1: Identify the original value.

Step 2: Find the amount of increase or decrease.

Step 3: Divide the change by the original value.

Step 4: Multiply by 100%.

Step 5: State whether it is an increase or decrease.

1️⃣1️⃣ Repeated Percentage Change

When a percentage change happens repeatedly, each new percentage is calculated from the latest value.

Example 18: A population of 2000 grows by 5% each year for two years

The multiplier is:

1 + 0.05 = 1.05

Apply it twice.

2000 × 1.05²

= 2205

After two years, the population is 2205.

Two increases of 5% do not produce exactly a 10% increase.

The second increase is calculated from the new, larger amount.

1️⃣2️⃣ Reverse Percentages

A reverse-percentage question gives the final value and asks us to find the original value.

Original price 100%

→

15% discount 85% remains

→

Sale price US$68

Example 19: A shirt costs US$68 after a 15% discount

After the discount, 85% of the original price remains.

85% = 0.85

Original price × 0.85 = 68

Original price = 68 ÷ 0.85

= 80

The original price was US$80.

🧠 StudyNest Method: Reverse Percentages

Step 1: Decide what percentage of the original value the final amount represents.

After an increase:

100% + the increase

After a decrease:

100% − the decrease

Step 2: Convert that percentage to a decimal multiplier.

Step 3: Divide the final amount by the multiplier.

Step 4: Check by applying the percentage change to your answer.

Do not simply add the percentage reduction to the final price. The percentage was calculated from the unknown original price.

1️⃣3️⃣ Simple Interest

Simple interest is calculated using the original amount every year.

Simple interest

I = P × r/100 × t

where:

I is the interest earned or charged;
P is the original amount;
r is the interest rate per year;
t is the number of years.

Example 20: US$600 invested at 8% simple interest for 3 years

I = 600 × 8/100 × 3

I = 144

Total amount = 600 + 144

Interest earned = US$144

Total amount = US$744

1️⃣4️⃣ Compound Interest

Compound interest is calculated using the growing balance.

Interest added in one period becomes part of the amount used in the following period.

Amount

A = P(1 + r/100)ᵗ

Example 21: US$600 invested at 8% compound interest for 3 years

A = 600(1 + 8/100)³

A = 600(1.08)³

A = 755.8272

Round money to the nearest cent.

Total amount = US$755.83

Subtract the original investment to find the interest earned.

755.83 − 600 = 155.83

Compound interest earned = US$155.83

⚖️ Simple Interest and Compound Interest

Simple Interest

Calculated from the original amount each year.

The amount of interest added each year remains the same.

Compound Interest

Calculated from the latest balance.

Interest is earned on earlier interest as well as the original amount.

🌍 More Real-Life Examples

🛒 Shopping Example

A school bag costs US$48 and is discounted by 15%.

Discount = 15/100 × 48

= 7.20

Sale price = 48 − 7.20

The customer pays US$40.80.

📚 School Attendance Example

A class has 40 learners, and 34 attend school.

34/40 × 100%

= 85%

Attendance is 85%.

⚽ Football Example

A team wins 18 of the 30 matches it plays.

18/30 × 100%

= 60%

The team won 60% of its matches.

🧠 The StudyNest Percentage Strategy

Step 1: Identify the original whole.

Step 2: Decide what the question is asking:

a conversion, a percentage of a quantity, percentage change, a new value or an original value.

Step 3: Choose the correct formula or method.

Step 4: Calculate one clear stage at a time.

Step 5: Include the percentage sign or correct units.

Step 6: Check whether the answer is reasonable.

🎯 Exam Tips

In percentage-change questions, divide by the original value.

Distinguish between the percentage amount and the final amount.

Use percentage multipliers for repeated change and compound interest.

In reverse-percentage questions, divide the final value by the correct multiplier.

Keep the full calculator value during your working and round only at the end.

⚠ Common Mistakes

Mistake 1:

Forgetting that percent means out of 100.

Mistake 2:

Multiplying by 100 when converting a percentage to a decimal.

Percentage to decimal requires division by 100.

Mistake 3:

Finding the percentage increase but forgetting to add it to the original quantity.

Mistake 4:

Finding the percentage decrease but forgetting to subtract it from the original quantity.

Mistake 5:

Dividing percentage change by the new value instead of the original value.

Mistake 6:

Assuming a 20% increase followed by a 20% decrease returns a value to its starting amount.

Mistake 7:

Adding a discount percentage to a sale price when finding the original price.

Mistake 8:

Confusing the interest earned with the total account balance.

🎮 Your Turn!

🟢 Foundation Questions

Convert 3/10 to a percentage.
Convert 7/20 to a percentage.
Convert 0.46 to a percentage.
Convert 1.08 to a percentage.
Convert 65% to a decimal.
Convert 12% to a fraction in its simplest form.
Find 35% of 260.
Find 7.5% of 480.

🔵 Developing Questions

A learner scores 54 marks out of 72. Express the mark as a percentage.
Express 24 as a percentage of 80.
Increase 450 by 18%.
Decrease 320 by 25%.
A jacket costing US$90 is reduced by 30%. Find its sale price.
The number of learners in a club increases from 40 to 52. Find the percentage increase.
A phone loses value from US$300 to US$225. Find the percentage decrease.

🔴 Challenge Questions

A quantity of 800 increases by 4% each year for two years. Find its value after two years.
A television costs US$680 after a 15% discount. Find its original price.
After a salary increase of 8%, a worker earns US$756 per month. Find the original salary.
Find the simple interest on US$900 at 6% per year for four years.
US$1200 is invested at 5% compound interest per year for three years. Find the total amount.

✅ Show Answers

30%
35%
46%
108%
0.65
3/25
91
36
75%
30%
531
240
US$63
Increase = 12

12/40 × 100% = 30%
Decrease = 75

75/300 × 100% = 25%
800 × 1.04² = 865.28
680 ÷ 0.85 = US$800
756 ÷ 1.08 = US$700
I = 900 × 0.06 × 4

= US$216
1200 × 1.05³

= US$1389.15

🧠 Remember This

Percent means out of 100.

Fraction or decimal to percentage → multiply by 100%.

Percentage to decimal → divide by 100.

Percentage of a quantity → percentage/100 × quantity.

Percentage change → change/original value × 100%.

Increase → use a multiplier greater than 1.

Decrease → use a multiplier between 0 and 1.

Reverse percentage → divide the final value by the multiplier.

🎯 Before Moving On...

✅ Can I explain what percentage means?
✅ Can I convert between fractions, decimals and percentages?
✅ Can I find a percentage of a quantity?
✅ Can I express one quantity as a percentage of another?
✅ Can I calculate percentage increase and decrease?
✅ Can I calculate percentage change?
✅ Can I use a multiplier for repeated changes?
✅ Can I solve a reverse-percentage problem?
✅ Can I distinguish simple and compound interest?

🎉 Well Done!

You can now use percentages in calculations and practical situations.

Remember:

Identify the original whole → Decide what is being asked → Choose the correct method → Check your answer

📈 Your Progress

Lesson 8 of 17

Real Numbers Progress 47%

Percentages

🎬 Connect percentages to one hundred

🎯 By the end of this lesson...

🤔 Think About This...

A sale at a clothing shop

1️⃣ What Does Percentage Mean?

🔲 Visualising a Percentage

💡 Did You Know?

2️⃣ Converting a Fraction to a Percentage

Method 1: Multiply by 100%

Example 1: Convert 3/5 to a percentage

Method 2: Create a fraction out of 100

Example 2: Convert 3/4 to a percentage

🧠 StudyNest Method: Fraction to Percentage

3️⃣ Converting a Decimal to a Percentage

Example 3: Convert 0.72 to a percentage

Example 4: Convert 1.35 to a percentage

4️⃣ Converting a Percentage to a Fraction

Example 5: Convert 40% to a fraction

Example 6: Convert 12.5% to a fraction

5️⃣ Converting a Percentage to a Decimal

Example 7: Convert 68% to a decimal

Example 8: Convert 7.5% to a decimal

🔁 Fraction, Decimal and Percentage

Fraction

Decimal

Percentage

6️⃣ Finding a Percentage of a Quantity

Example 9: Find 30% of 240

🧠 Mental Percentage Shortcuts

10%

5%

1%

50%

25%

75%

Example 10: Find 15% of 360 mentally

🌍 Real-Life Example: School-Fee Deposit

Example 11

7️⃣ Expressing One Quantity as a Percentage of Another

Example 12: A learner scores 36 marks out of 45

8️⃣ Increasing a Quantity by a Percentage

Example 13: Increase US$250 by 12%

9️⃣ Decreasing a Quantity by a Percentage

Example 14: Reduce US$75 by 20%

🔢 Percentage Multipliers

Increase by r%

Decrease by r%

Example 15: Increase 250 by 12%

Example 16: Decrease 75 by 20%

🔟 Percentage Change

Example 17: A price rises from US$80 to US$92

🧠 StudyNest Method: Percentage Change

1️⃣1️⃣ Repeated Percentage Change

Example 18: A population of 2000 grows by 5% each year for two years

1️⃣2️⃣ Reverse Percentages

Example 19: A shirt costs US$68 after a 15% discount

🧠 StudyNest Method: Reverse Percentages

1️⃣3️⃣ Simple Interest

Example 20: US$600 invested at 8% simple interest for 3 years

1️⃣4️⃣ Compound Interest

Example 21: US$600 invested at 8% compound interest for 3 years

⚖️ Simple Interest and Compound Interest

Simple Interest

Compound Interest

🌍 More Real-Life Examples

🛒 Shopping Example

📚 School Attendance Example

⚽ Football Example

🧠 The StudyNest Percentage Strategy

🎯 Exam Tips

⚠ Common Mistakes

🎮 Your Turn!

🟢 Foundation Questions

🔵 Developing Questions

🔴 Challenge Questions

🧠 Remember This

🎯 Before Moving On...

🎉 Well Done!

📈 Your Progress