Algebraic fractions
An algebraic fraction has an algebraic expression in its numerator, denominator, or both, such as x/3 or (x + 1)/2. You work with them using the same rules as number fractions. Always simplify by cancelling common factors when you can.
Key idea / 要点
To add or subtract, use a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal (flip the second fraction).
Multiplying and dividing
Multiply straight across: (x/2) × (3/x) = 3x/(2x) = 3/2 after cancelling x. To divide, flip: (x/4) ÷ (x/2) = (x/4) × (2/x) = 2x/(4x) = 1/2.
Worked example / 例题
Simplify x/3 + x/6.
Common denominator is 6: x/3 = 2x/6.
2x/6 + x/6 = 3x/6 = x/2.
Adding and subtracting
Fractions must share a denominator before you add or subtract the numerators. For 2/x + 3/x the denominator is already x, so 2/x + 3/x = 5/x. For different denominators, rewrite each fraction first, as in the worked example above.
Remember / 记住
- Only add or subtract fractions with the same denominator.
- Dividing means multiplying by the reciprocal.
- Cancel common factors to simplify the final answer.