Algebraic fraction is the term used to refer to a fraction that has an algebraic expression on either the denominator, the numerator or both. Examples of algebraic fractions include:\( \frac{(x + 2)}{3}, \space \frac{1}{(x +y) }\) and \(\frac{ (4y +2x)}{(y + 3)}\).
When subtracting or adding algebraic fractions, you should start by putting them over a common denominator.
ADDING ALGEBRAIC FRACTIONS
Addition of algebraic fractions is done in several simple steps.
For example, a∕b + c∕d =\( \frac{(ad + bc)}{bd}\)
Example 2, work out x∕2 + y∕5.
Step 1. Find a common denominator. This can be found by finding the least common divisor of the denominators. In this case, the denominators are 2 and 5. Their LCM is 10, therefore the common denominator is 10.
Step 2. Divide the common denominator by each of the denominators then multiply the answer with the numerator. For example, in x∕2, you divide 10 by 2, which gives you 5 then you multiply this by the numerator x, therefore, giving 5x. Do the same for the second equation and the answer will be 2y.
Step 3. Add the numerators and put them under the common denominator. The numerators are 5x and 2y, as found in step 2. Therefore,\(\frac{ (5x + 2y)} {10}\) is the answer.
You can also be asked to solve a more complex algebraic fraction such as\( \frac{(x + 4)}{3} + \frac{(x – 3)}{4}\).
Solution.
Step 1. Find the LCM of the denominators. This is done for the purpose of finding a common divisor. The LCM of 4 and 3 is 12, so, the common divisor is 12.
Step 2. Divide the common divisor by each numerator then multiply the answer with the numerator of the same equation. For example, in (x + 4)∕3, it will be 12 divided by three = 4. Multiply 4 by the numerator, 4x + 16. The other fraction will be 3x−9.
Step 3. Add the numerators and put them under the common denominator. \(\frac{(4x + 16) + (3x – 9) }{12}\). The answer, therefore, becomes, \(\frac{(7x + 7)}{12}\).
SUBTRACTING ALGEBRAIC FRACTIONS
The steps are the same as in the addition. For example, \(\frac{(x + 2)}{x} - \frac{x}{x} \) can be solved as shown below.
Step 1. Find the common denominator. In this case, it is already common x.
Step 2. Divide the common denominator by each denominator then multiply by the numerator. It will be 1 ⋅ (x + 2) which is equivalent to x+2. The other fraction will be x.
Step 3. \(\frac{(x + 2)- (x)}{x}\). therefore, \(\frac{2} {x }\) is the answer.
MULTIPLYING ALGEBRAIC FRACTIONS
This is the easiest. You just multiply the numerators together, and the denominators together. For example, \(\frac{3x}{x - 2} \times \frac{x}{3}\) can be solved as shown below.
Numerator: 3x⋅x and denominator: 3⋅(x−2).
Therefore, \(\frac{3x^2}{ 3(x - 2)}\). This is equivalent to x2∕x−2.
DIVIDING ALGEBRAIC FRACTIONS
It is also an easy one. Start by turning upside down the second fraction then proceed as in multiplication. For example, a∕b ÷ c/d can be solved as, \(\frac{a}{b} \times \frac{d}{c}\) which equals ad∕bc.
