What this chapter is about
This chapter helps you become comfortable with fractions in everyday situations. You already know what fractions are from earlier classes. Now you will learn to add, subtract, multiply and divide fractions, including mixed numbers. These skills let you solve problems about sharing food, measuring cloth, mixing ingredients and many other daily tasks.
By the end of this chapter, you should be able to perform all four operations on fractions and mixed numbers. You will also learn when to use which operation. For example, finding how much pizza is left uses subtraction, while finding half of a rope's length uses multiplication.
Fractions appear everywhere: in recipes, in distances, in time and in money. Mastering them now prepares you for algebra, ratio, percentage and geometry in higher classes.
Key ideas
- A fraction has a numerator (top number) and a denominator (bottom number). The denominator tells how many equal parts make a whole; the numerator tells how many parts we have.
- To add or subtract fractions, first make their denominators the same (find a common denominator), then add or subtract the numerators.
- To multiply two fractions, multiply the numerators together and multiply the denominators together: (a/b) × (c/d) = (a × c) / (b × d).
- To divide by a fraction, multiply by its reciprocal. The reciprocal of c/d is d/c. So (a/b) ÷ (c/d) = (a/b) × (d/c).
- A mixed number like 2 3/4 can be written as an improper fraction: 2 3/4 = (2 × 4 + 3)/4 = 11/4. Convert before multiplying or dividing.
- Always simplify your final answer by dividing numerator and denominator by their highest common factor.
- Multiplication of a fraction by a whole number means repeated addition: 3 × 2/5 = 2/5 + 2/5 + 2/5 = 6/5.
Formulas and facts to remember
- Addition: a/b + c/b = (a + c)/b (same denominator). For different denominators, convert to a common denominator first.
- Subtraction: a/b − c/b = (a − c)/b.
- Multiplication: (a/b) × (c/d) = (a × c) / (b × d).
- Division: (a/b) ÷ (c/d) = (a/b) × (d/c).
- Reciprocal: the reciprocal of n is 1/n; the reciprocal of a/b is b/a.
- Mixed to improper: whole part × denominator + numerator, all over the same denominator.
Worked examples
Example 1: Adding fractions with different denominators
Ravi ate 1/4 of a roti and Meena ate 2/3 of the same roti. How much did they eat together?
Step 1: Find the LCM of 4 and 3. LCM = 12.
Step 2: Convert each fraction. 1/4 = 3/12 (multiply numerator and denominator by 3). 2/3 = 8/12 (multiply numerator and denominator by 4).
Step 3: Add the numerators: 3/12 + 8/12 = 11/12.
Answer: Together they ate 11/12 of the roti.
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Example 2: Multiplying a fraction by a fraction
A ribbon is 3/4 metre long. Sita uses 2/5 of it for a gift box. How long is the piece she used?
Step 1: Multiply the fractions. (2/5) × (3/4) = (2 × 3) / (5 × 4) = 6/20.
Step 2: Simplify. Divide numerator and denominator by 2. 6/20 = 3/10.
Answer: Sita used 3/10 metre of ribbon.
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Example 3: Dividing a mixed number by a fraction
A jug holds 2 1/2 litres of juice. Each glass holds 1/4 litre. How many glasses can be filled?
Step 1: Convert the mixed number to an improper fraction. 2 1/2 = (2 × 2 + 1)/2 = 5/2.
Step 2: Divide by 1/4. Multiply by the reciprocal. (5/2) ÷ (1/4) = (5/2) × (4/1) = 20/2 = 10.
Answer: 10 glasses can be filled.
Common mistakes
- Adding numerators and denominators separately, like 1/4 + 2/3 = 3/7 → Find a common denominator first; add only the numerators.
- Forgetting to convert mixed numbers before multiplying or dividing → Always change 2 3/4 to 11/4 first.
- Inverting the wrong fraction when dividing → Flip only the divisor (the second fraction), not the first.
- Not simplifying the final answer → Divide numerator and denominator by their HCF.
- Thinking multiplication always makes a number larger → Multiplying by a proper fraction gives a smaller result.
Quick revision
- Same denominator: just add or subtract numerators.
- Multiply straight across: top × top, bottom × bottom.
- Divide means multiply by the reciprocal.
- Convert mixed numbers to improper fractions before multiplying or dividing.
- Always simplify your answer.