Multiplication and division of fractions do not require a common denominator. The main tasks are converting mixed numbers, cancelling factors, and simplifying the final answer.
Multiplication rule
a/b × c/d = ac/bd
For 2/3 × 9/10, cancel 3 with 9 and 2 with 10 before multiplying. The result is 3/5.
Division rule
Keep the first fraction, change division to multiplication, and invert the second fraction. For 3/4 ÷ 2/5, calculate 3/4 × 5/2 = 15/8 = 1 7/8.
Mixed-number example
Convert 1 1/2 to 3/2 and 2 1/3 to 7/3. Then 3/2 × 7/3 simplifies to 7/2 or 3 1/2.
Common errors
- Inverting both fractions during division.
- Multiplying a mixed number without converting it.
- Cancelling terms across addition.
- Leaving a reducible answer.
The Scientific Calculator can verify decimal values. Keep the exact fraction when precision matters; a rounded decimal may lose information.
Frequently asked questions
Why use the reciprocal?
Dividing by a non-zero number is equivalent to multiplying by its multiplicative inverse.
Can zero be a denominator?
No. Division by zero is undefined.
Should I cancel before multiplying?
It is optional but reduces the size of intermediate numbers and makes simplification easier.
Fraction-of-a-quantity example
If a recipe uses 2/3 of a cup per batch and you make 1 1/2 batches, convert 1 1/2 to 3/2. Then 2/3 × 3/2 = 1 cup. Cancelling before multiplication makes the result visible and reduces arithmetic.
For division, ask how many groups of the divisor fit into the first amount. If 3/4 litre is poured into 2/5-litre portions, 3/4 ÷ 2/5 = 15/8, so there are 1 7/8 portions. The reciprocal rule is not merely a memorised trick; it answers that grouping question.
Check the direction of the answer
Multiplying by a positive fraction smaller than one should make a positive quantity smaller. Dividing by that fraction should make it larger. This reasonableness test often catches an inverted reciprocal or an accidental multiplication before you rely on the result.