Fractions Explained Step by Step

🎯 By the end of this lesson...

You should be able to:

Identify the numerator and denominator of a fraction.
Recognise proper, improper and mixed fractions.
Find equivalent fractions.
Simplify fractions fully.
Convert between mixed numbers and improper fractions.
Compare and order fractions.
Add and subtract fractions.
Multiply and divide fractions.
Solve real-life problems involving fractions.

🤔 Think About This...

🍕

Who ate the most?

Three friends share identical pizzas.

Tariro eats ½ of a pizza.
Rudo eats ¼ of a pizza.
Tino eats ⅜ of a pizza.

Who ate the greatest amount?

Looking only at the top or bottom number is not enough.

To compare fractions correctly, we must understand what the numerator and denominator represent.

1️⃣ What Is a Fraction?

A fraction represents a part of a whole or a part of a group.

3 5

Numerator: 3

The number of parts being considered.

Denominator: 5

The total number of equal parts.

The denominator tells us how many equal pieces make one whole. The numerator tells us how many of those pieces we have.

2️⃣ Types of Fractions

Proper Fraction

3/5

The numerator is smaller than the denominator.

Its value is less than 1.

Improper Fraction

7/4

The numerator is greater than or equal to the denominator.

Its value is at least 1.

Mixed Number

1 ¾

A whole number and a proper fraction written together.

Unit Fraction

1/8

A fraction with numerator 1.

3️⃣ Equivalent Fractions

Equivalent fractions have the same value, even though they are written using different numbers.

½ = 2/4 = 3/6 = 4/8

We create an equivalent fraction by multiplying or dividing the numerator and denominator by the same non-zero number.

Example 1: Find an equivalent fraction for 3/5

Multiply both the numerator and denominator by 2.

3 × 2
─────
5 × 2

3/5 = 6/10

The value has not changed because both parts were multiplied by the same number.

4️⃣ Simplifying Fractions

Simplifying a fraction means writing it using the smallest possible whole-number numerator and denominator.

We use the Highest Common Factor from Lesson 3.

Example 2: Simplify 18/24

Find the HCF of 18 and 24.

HCF of 18 and 24 = 6

Divide both the numerator and denominator by 6.

18 ÷ 6
──────
24 ÷ 6

18/24 = 3/4

18/24 simplifies to 3/4.

🧠 StudyNest Method: Simplifying Fractions

Step 1: Find the HCF of the numerator and denominator.

Step 2: Divide the numerator by the HCF.

Step 3: Divide the denominator by the same HCF.

Step 4: Check that the new numerator and denominator have no common factor other than 1.

Whatever you do to the numerator, you must also do to the denominator.

5️⃣ Improper Fractions and Mixed Numbers

Improper Fraction to Mixed Number

Example 3: Convert 11/4 to a mixed number

Divide the numerator by the denominator.

11 ÷ 4 = 2 remainder 3

The whole-number answer becomes the whole part.

The remainder becomes the new numerator.

The denominator remains 4.

11/4 = 2 ¾

Mixed Number to Improper Fraction

Example 4: Convert 3 2/5 to an improper fraction

Multiply the whole number by the denominator.

3 × 5 = 15

Add the numerator.

15 + 2 = 17

Keep the same denominator.

3 2/5 = 17/5

🧠 StudyNest Conversion Methods

Improper → Mixed

Divide numerator by denominator.
Write the quotient as the whole number.
Use the remainder as the numerator.
Keep the original denominator.

Mixed → Improper

Multiply whole number by denominator.
Add the numerator.
Place the answer over the original denominator.

6️⃣ Comparing Fractions

The best comparison method depends on how the fractions are written.

Same Denominator

Example 5: Compare 3/8 and 5/8

The denominators are equal, so compare the numerators.

5 > 3

5/8 > 3/8

Different Denominators

Example 6: Compare 2/3 and 3/5

Find a common denominator.

LCM of 3 and 5 = 15

2/3 = 10/15

3/5 = 9/15

Since 10/15 > 9/15, therefore 2/3 > 3/5.

🧠 StudyNest Method: Comparing Fractions

Step 1: Check whether the denominators are already the same.

Step 2: If they are the same, compare the numerators.

Step 3: If the denominators differ, find a common denominator.

Step 4: Rewrite the fractions as equivalent fractions.

Step 5: Compare the new numerators.

7️⃣ Adding Fractions

Fractions with the Same Denominator

Example 7

2/7 + 3/7

Add the numerators and keep the denominator.

2/7 + 3/7 = 5/7

Fractions with Different Denominators

Example 8

1/4 + 2/3

Find the LCM of 4 and 3.

LCM = 12

Rewrite both fractions.

1/4 = 3/12

2/3 = 8/12

Add the numerators.

3/12 + 8/12 = 11/12

1/4 + 2/3 = 11/12

8️⃣ Subtracting Fractions

Example 9

5/6 − 1/4

Find the LCM of 6 and 4.

LCM = 12

5/6 = 10/12

1/4 = 3/12

10/12 − 3/12 = 7/12

5/6 − 1/4 = 7/12

Never add or subtract the denominators.

The denominator names the size of the pieces and must first be made common.

🧠 StudyNest Method: Adding or Subtracting

Step 1: Check the denominators.

Step 2: If they differ, find the LCM.

Step 3: Rewrite each fraction using the common denominator.

Step 4: Add or subtract the numerators.

Step 5: Keep the common denominator.

Step 6: Simplify the answer fully.

9️⃣ Multiplying Fractions

To multiply fractions:

multiply the numerators;
multiply the denominators;
simplify the answer.

Example 10

2/3 × 5/8

2 × 5
─────
3 × 8

10/24

Simplify by dividing by 2.

2/3 × 5/8 = 5/12

✂️ Cancelling Before Multiplying

We can simplify common factors before multiplying.

Example 11

3/4 × 8/9

Cancel common factors diagonally.

3 and 9 divide by 3

8 and 4 divide by 4

1/1 × 2/3

3/4 × 8/9 = 2/3

Cancelling before multiplying keeps the numbers smaller and reduces calculation errors.

🔟 Dividing Fractions

Dividing by a fraction is the same as multiplying by its reciprocal.

The reciprocal is found by exchanging the numerator and denominator.

Reciprocal of 3/5 = 5/3

Example 12

2/3 ÷ 4/5

Keep the first fraction, change division to multiplication, and use the reciprocal of the second fraction.

2/3 × 5/4

10/12

Simplify.

2/3 ÷ 4/5 = 5/6

🧠 StudyNest Method: Dividing Fractions

Step 1: Keep the first fraction unchanged.

Step 2: Change the division sign to multiplication.

Step 3: Replace the second fraction with its reciprocal.

Step 4: Cancel common factors if possible.

Step 5: Multiply the fractions.

Step 6: Simplify the final answer.

Only the second fraction is replaced by its reciprocal.

1️⃣1️⃣ Calculations with Mixed Numbers

Before multiplying or dividing mixed numbers, convert them to improper fractions.

Example 13

1 ½ × 2 ⅓

Convert both mixed numbers.

1 ½ = 3/2

2 ⅓ = 7/3

3/2 × 7/3

Cancel the common factor 3.

1/2 × 7/1 = 7/2

Convert to a mixed number.

1 ½ × 2 ⅓ = 3 ½

🌍 Real-Life Example: Baking

Example 14

One cake needs ¾ cup of sugar.

How much sugar is needed for 3 cakes?

3 × ¾

3/1 × 3/4 = 9/4

9/4 = 2 ¼

The baker needs 2 ¼ cups of sugar.

🌍 Real-Life Example: Fuel Remaining

Example 15

A vehicle begins a journey with a full tank.

It uses ⅜ of the fuel in the morning and ¼ in the afternoon.

What fraction of the tank remains?

Find the total fraction used.

3/8 + 1/4

1/4 = 2/8

3/8 + 2/8 = 5/8

Subtract from one full tank.

1 − 5/8

8/8 − 5/8 = 3/8

3/8 of the tank remains.

🌍 Real-Life Example: Time Spent Studying

Example 16

Nyasha spends:

⅓ of an hour revising Mathematics;
¼ of an hour revising Science.

How much time does she spend studying altogether?

1/3 + 1/4

LCM of 3 and 4 = 12

1/3 = 4/12

1/4 = 3/12

4/12 + 3/12 = 7/12

Convert the fraction of an hour into minutes.

7/12 × 60 = 35

Nyasha studies for 35 minutes.

🧠 The StudyNest Fraction Strategy

Step 1: Identify the operation.

Step 2: Check whether mixed numbers must first be converted.

Step 3: For addition or subtraction, check the denominators.

Step 4: For multiplication, multiply across and cancel where possible.

Step 5: For division, multiply by the reciprocal of the second fraction.

Step 6: Simplify the final answer.

Step 7: Convert to a mixed number if the question requires it.

Step 8: Check whether the answer makes sense in context.

🎯 Exam Tips

Always simplify your final fraction unless the question says otherwise.

When adding or subtracting, use the LCM of the denominators as the common denominator where practical.

Do not convert fractions to decimals unless it makes the question easier or the question requests a decimal answer.

Read carefully to see whether the final answer should be a fraction, improper fraction or mixed number.

⚠ Common Mistakes

Mistake 1:

Adding both the numerators and denominators.

For example, 1/3 + 1/4 is not 2/7.

Mistake 2:

Using a common denominator but forgetting to change the numerator.

Mistake 3:

Forgetting to simplify the final answer.

Mistake 4:

Adding the whole numbers and fractions separately without checking whether regrouping is needed.

Mistake 5:

Taking the reciprocal of the first fraction instead of the second when dividing.

Mistake 6:

Multiplying mixed numbers before converting them to improper fractions.

Mistake 7:

Cancelling terms across addition or subtraction.

Cancelling applies to factors in multiplication, not separate terms being added.

🎮 Your Turn!

Identify the numerator and denominator of 7/12.
State whether 5/8 is proper, improper or mixed.
State whether 11/6 is proper, improper or mixed.
Simplify 20/30.
Convert 13/5 to a mixed number.
Convert 2 3/7 to an improper fraction.
Which is greater: 5/6 or 7/9?
Calculate 3/8 + 1/4.
Calculate 7/10 − 1/5.
Calculate 4/9 × 3/8.
Calculate 5/6 ÷ 10/9.
Calculate 1 ½ + 2 ¼.
Calculate 2 ⅓ × 1 ½.
A water tank is 5/6 full. During the day, 1/4 of the full tank is used. What fraction of the tank remains?
A recipe uses 2/3 cup of flour for one loaf. How much flour is required for 4 loaves?

✅ Show Answers

Numerator = 7, denominator = 12
Proper fraction
Improper fraction
20/30 = 2/3
13/5 = 2 3/5
2 3/7 = 17/7
5/6 = 15/18 and 7/9 = 14/18

Therefore, 5/6 is greater.
3/8 + 2/8 = 5/8
7/10 − 2/10 = 1/2
4/9 × 3/8 = 12/72 = 1/6
5/6 × 9/10 = 45/60 = 3/4
1 2/4 + 2 1/4 = 3 3/4
7/3 × 3/2 = 7/2 = 3 1/2
5/6 − 1/4

= 10/12 − 3/12

= 7/12
4 × 2/3 = 8/3 = 2 2/3 cups

🧠 Remember This

Equivalent fractions have the same value.

Simplify using the HCF.

Add and subtract only after making denominators equal.

Multiply numerators together and denominators together.

To divide fractions, multiply by the reciprocal of the second fraction.

Always simplify your final answer.

🎯 Before Moving On...

✅ Can I identify the numerator and denominator?
✅ Can I recognise proper, improper and mixed fractions?
✅ Can I find equivalent fractions?
✅ Can I simplify a fraction fully?
✅ Can I convert between mixed numbers and improper fractions?
✅ Can I compare fractions?
✅ Can I add and subtract fractions?
✅ Can I multiply and divide fractions?
✅ Can I solve a word problem involving fractions?

🎉 Well Done!

You can now understand, compare and calculate confidently with fractions.

Remember:

Identify the operation → Choose the correct fraction method → Simplify the answer

📈 Your Progress

Lesson 6 of 17

Real Numbers Progress 35%

Fractions

🎬 See a fraction fill the whole

🎯 By the end of this lesson...

🤔 Think About This...

Who ate the most?

1️⃣ What Is a Fraction?

2️⃣ Types of Fractions

Proper Fraction

Improper Fraction

Mixed Number

Unit Fraction

3️⃣ Equivalent Fractions

Example 1: Find an equivalent fraction for 3/5

4️⃣ Simplifying Fractions

Example 2: Simplify 18/24

🧠 StudyNest Method: Simplifying Fractions

5️⃣ Improper Fractions and Mixed Numbers

Improper Fraction to Mixed Number

Example 3: Convert 11/4 to a mixed number

Mixed Number to Improper Fraction

Example 4: Convert 3 2/5 to an improper fraction

🧠 StudyNest Conversion Methods

Improper → Mixed

Mixed → Improper

6️⃣ Comparing Fractions

Same Denominator

Example 5: Compare 3/8 and 5/8

Different Denominators

Example 6: Compare 2/3 and 3/5

🧠 StudyNest Method: Comparing Fractions

7️⃣ Adding Fractions

Fractions with the Same Denominator

Example 7

Fractions with Different Denominators

Example 8

8️⃣ Subtracting Fractions

Example 9

🧠 StudyNest Method: Adding or Subtracting

9️⃣ Multiplying Fractions

Example 10

✂️ Cancelling Before Multiplying

Example 11

🔟 Dividing Fractions

Example 12

🧠 StudyNest Method: Dividing Fractions

1️⃣1️⃣ Calculations with Mixed Numbers

Example 13

🌍 Real-Life Example: Baking

Example 14

🌍 Real-Life Example: Fuel Remaining

Example 15

🌍 Real-Life Example: Time Spent Studying

Example 16

🧠 The StudyNest Fraction Strategy

🎯 Exam Tips

⚠ Common Mistakes

🎮 Your Turn!

🧠 Remember This

🎯 Before Moving On...

🎉 Well Done!

📈 Your Progress