Working with Fractions
Class 6 introduced fractions; now we operate with them. This Class 7 Ganita Prakash chapter covers multiplying fractions by whole numbers and by other fractions, what 'of' means, the idea of a reciprocal, and how to divide fractions — turning fractions into a flexible tool.
Learning objectives
- Multiply a fraction by a whole number.
- Multiply a fraction by a fraction.
- Understand 'of' as multiplication.
- Divide fractions using reciprocals.
Key concepts
Multiplying by a whole number
To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator. So 3 × 2/5 = 6/5. This is just repeated addition: 2/5 + 2/5 + 2/5 = 6/5.
Multiplying a fraction by a fraction
To multiply two fractions, multiply the numerators together and the denominators together: 2/3 × 4/5 = 8/15. Multiplying by a proper fraction gives a smaller result, since we are taking only a part of a part.
The meaning of 'of'
In fractions, 'of' means multiply. So 'half of 8' is 1/2 × 8 = 4, and '3/4 of 20' is 3/4 × 20 = 15. Recognising 'of' as multiplication makes many word problems straightforward.
Reciprocals and dividing fractions
The reciprocal of a fraction is found by flipping it: the reciprocal of 2/3 is 3/2. To divide by a fraction, multiply by its reciprocal: 4/5 ÷ 2/3 = 4/5 × 3/2 = 12/10 = 6/5. Dividing by a fraction less than 1 gives a larger result.
Important formulas
Fraction × fraction
a/b × c/d = (a×c)/(b×d)
'of'
'of' means ×
Dividing fractions
a/b ÷ c/d = a/b × d/c
Key definitions
- Reciprocal
- The fraction obtained by swapping numerator and denominator.
- Product
- The result of multiplying numbers or fractions.
- Proper fraction
- A fraction less than one (numerator below denominator).
- Of
- In fraction problems, the word 'of' means multiply.
Solved examples
Q1. Find 3 × 2/5.
Solution: Multiply the numerator: 6/5.
Q2. Find 2/3 × 4/5.
Solution: (2 × 4)/(3 × 5) = 8/15.
Q3. Find 3/4 of 20.
Solution: 3/4 × 20 = 60/4 = 15.
Common mistakes to avoid
- Adding denominators when multiplying fractions.
- Forgetting that 'of' means multiply.
- Dividing without flipping (multiplying by the reciprocal).
- Expecting multiplication by a proper fraction to give a bigger answer.
Working with Fractions — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
3 × 1/4 equals:
Practice questions
Short answer
What does 'of' mean in fraction problems?
It means multiply.
What is the reciprocal of 3/7?
7/3.
How do you multiply two fractions?
Multiply numerators together and denominators together.
Long answer
Explain how to multiply fractions, including by a whole number, with examples.
To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same — for example, 3 × 2/5 = 6/5, which is the same as adding 2/5 three times. To multiply a fraction by another fraction, multiply the numerators together and the denominators together — for example, 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15. A useful observation is that multiplying by a proper fraction (one less than 1) gives a smaller result, because we are taking only a part of the original amount.
Explain how to divide fractions using reciprocals, with an example.
Dividing by a fraction is done by multiplying by its reciprocal, which is the fraction turned upside down. So to compute 4/5 ÷ 2/3, we replace dividing by 2/3 with multiplying by its reciprocal 3/2: 4/5 × 3/2 = 12/10 = 6/5. The reason this works is that dividing asks 'how many of the divisor fit into the number', and flipping converts that question into a multiplication. Notice that dividing by a fraction smaller than 1 gives a larger answer, since many small pieces fit into the whole.
HOTS (Higher Order Thinking)
Without calculating exactly, is 8 × 3/4 more or less than 8? Why?
Less, because multiplying by 3/4 (a proper fraction) takes only three-quarters of 8, giving 6.
A recipe needs 2/3 cup of sugar, but you make half the recipe. How much sugar?
Half of 2/3 = 1/2 × 2/3 = 2/6 = 1/3 cup.
Quick revision
Revision notes
- Fraction × whole: multiply numerator, keep denominator.
- Fraction × fraction: multiply tops, multiply bottoms.
- 'of' means multiply.
- Divide by a fraction = multiply by its reciprocal.
Key takeaways
- Multiplying fractions multiplies tops and bottoms.
- 'of' is multiplication.
- Divide by flipping the divisor.
Frequently asked questions
How do I multiply two fractions?
Multiply the numerators together and the denominators together.
What is a reciprocal?
A fraction flipped upside down; the reciprocal of 2/3 is 3/2.
Does multiplying by a fraction always make a number bigger?
No — multiplying by a fraction less than 1 makes it smaller.