Chapter focus
A fraction can appear in several forms while representing the same rational number. Equivalent fractions, common units and number-line reasoning help compare and calculate accurately.
These original notes support revision; use your assigned Ganita Prakash Part II for the full diagrams, activities and exercise sequence.
1. Equivalent fractions preserve value
Multiplying or dividing the numerator and denominator by the same non-zero number does not change a fraction’s value: 3/4 = 6/8 = 9/12. These fractions name the same point on the number line, even though their numerators and denominators differ.
To simplify a fraction, divide top and bottom by a common factor. For 42/56, the greatest common factor is 14, so 42/56 = 3/4. A fraction is in simplest form when numerator and denominator have no common factor greater than 1.
Fractions can be compared by using a common denominator, by converting to decimals when appropriate, or by locating them relative to benchmarks such as 0, 1/2 and 1.
2. Add and subtract fractions
Fractions must describe like-sized parts before adding or subtracting. Use a common denominator, usually the least common multiple of the denominators.
Worked example: 2/3 + 3/4
- The least common denominator of 3 and 4 is 12.
- Rewrite: 2/3 = 8/12 and 3/4 = 9/12.
- Add the numerators: 8/12 + 9/12 = 17/12 = 1 5/12.
Check: 2/3 is about 0.67 and 3/4 is 0.75, so the sum should be between 1 and 2. The answer fits.
3. Multiply fractions
To multiply fractions, multiply numerators and multiply denominators: (a/b) × (c/d) = ac/bd, where the denominators are non-zero. Simplify before or after multiplying.
Worked example: 3/5 × 10/21
Cancel common factors across numerator and denominator: 10 ÷ 5 = 2 and 21 ÷ 3 = 7. Then 3/5 × 10/21 = 1 × 2/(1 × 7) = 2/7.
Multiplication can represent “a fraction of a quantity”: 3/4 of 20 = 3/4 × 20 = 15.
4. Divide fractions
Dividing by a non-zero fraction is equivalent to multiplying by its reciprocal. The reciprocal of a/b is b/a. Thus (a/b) ÷ (c/d) = (a/b) × (d/c), provided c/d is not zero.
Worked example: 2/3 ÷ 4/5
- Keep the first fraction and multiply by the reciprocal of the second.
- 2/3 × 5/4 = 10/12.
- Simplify: 5/6.
Reasonableness check: 4/5 is less than 1, so dividing 2/3 by 4/5 should give an answer slightly greater than 2/3; 5/6 does.
5. Fractions, decimals and number lines
A fraction with denominator 10, 100 or 1,000 converts directly to a decimal: 37/100 = 0.37. Other fractions can be converted by division: 3/8 = 3 ÷ 8 = 0.375.
A rational number can be written as a/b for integers a and b with b ≠ 0. Its decimal form either terminates or repeats. For example, 1/4 = 0.25 terminates, while 1/3 = 0.333… repeats.
To place a fraction on a number line, divide the interval into the denominator’s number of equal parts and count the numerator’s parts from zero. This makes order and equivalent forms visible.
Common errors to avoid
- For addition, do not add denominators: 1/3 + 1/3 = 2/3, not 2/6.
- When dividing, invert only the divisor, not both fractions.
- Do not cancel terms across addition: (2 + 3)/3 cannot be simplified by cancelling the 3.
- A larger denominator does not automatically mean a larger fraction; compare the values.
Practice: show each step
1. Write 5/6 with denominator 24.
Answer: Multiply numerator and denominator by 4: 20/24.
2. Calculate 2/3 + 3/4.
Answer: 8/12 + 9/12 = 17/12 = 1 5/12.
3. Calculate 7/8 − 1/3.
Answer: 21/24 − 8/24 = 13/24.
4. Find 3/5 of 45.
Answer: 3/5 × 45 = 3 × 9 = 27.
5. A jug holds 3/4 litre. How many full 1/8-litre servings can it provide?
Answer: 3/4 ÷ 1/8 = 3/4 × 8 = 6 servings.
Chapter summary
Equivalent fractions are different representations of the same value. Choose common denominators for addition and subtraction; multiply directly; divide by multiplying by the reciprocal. Use simplification, number-line sense and estimation to check every result.