Chapter Overview
Fractions express parts, ratios and operators. Multiplication and division of fractions model scaling, sharing and measurement in many daily situations.
This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.
Learning Objectives
After studying this chapter, you should be able to:
- explain and apply multiplying fractions;
- explain and apply fraction of a quantity;
- explain and apply dividing fractions;
- explain and apply reasonableness;
- communicate the reasoning behind a solution clearly;
- check answers using estimation, substitution, or a second method.
Key Concepts
| Concept | Meaning |
|---|---|
| Multiplying fractions | Multiply numerators together and denominators together, simplifying before or after multiplication. |
| Fraction of a quantity | The word ‘of’ commonly signals multiplication: 3/5 of 40 = 3/5 × 40. |
| Dividing fractions | Dividing by a non-zero fraction is equivalent to multiplying by its reciprocal. |
| Reasonableness | Multiplying by a proper fraction makes a positive quantity smaller; dividing by a proper fraction makes it larger. |
Detailed Explanation
Multiplying fractions
Multiply numerators together and denominators together, simplifying before or after multiplication.
Fraction of a quantity
The word ‘of’ commonly signals multiplication: 3/5 of 40 = 3/5 × 40.
Dividing fractions
Dividing by a non-zero fraction is equivalent to multiplying by its reciprocal.
Reasonableness
Multiplying by a proper fraction makes a positive quantity smaller; dividing by a proper fraction makes it larger.
Do not memorise a rule without testing it on examples. Ask what each number, operation, line, or symbol represents and whether the result fits the original situation.
Important Rules and Formulae
- Convert mixed numbers to improper fractions before multiplying or dividing.
- A reciprocal exists only for a non-zero number.
- State units in word problems.
Worked Examples
Example 1
Problem: Find 3/4 × 10/9.
Solution: Cancel 3 with 9 and 10 with 4: 1/2 × 10/3 after careful simplification gives 5/6.
Example 2
Problem: How many 3/8-metre pieces can be cut from 4 1/2 metres?
Solution: 9/2 ÷ 3/8 = 9/2 × 8/3 = 12 pieces.
Show the mathematical statement, substitution or construction step before writing the final answer. Include units wherever the quantity is measured.
Common Mistakes
- Adding numerators and denominators when multiplying.
- Flipping the dividend instead of the divisor in division.
- Forgetting to convert mixed numbers.
Quick Revision
- Multiplying fractions: Multiply numerators together and denominators together, simplifying before or after multiplication.
- Fraction of a quantity: The word ‘of’ commonly signals multiplication:
3/5 of 40 = 3/5 × 40. - Dividing fractions: Dividing by a non-zero fraction is equivalent to multiplying by its reciprocal.
- Reasonableness: Multiplying by a proper fraction makes a positive quantity smaller; dividing by a proper fraction makes it larger.
Practice Questions
- Find
5/6 of 42. - Calculate
7/9 ÷ 14/27. - Is
2/3 × 5/7greater or smaller than5/7? - A recipe needs
3/4cup per batch. How much for 6 batches?
Answers and Explanations
-
3/2or1 1/2.- Smaller.
9/2 = 4 1/2cups.
Self-Check
Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.