EQUATIONS INVOLVING FRACTIONS

CONTENT

1. Clearing fractions (revision)

2. Algebraic fractions with:

(i) unknown at the numerator.

(ii) Unknown at the denominator

(iii) binomial denominators.

3. Change of subject of formulae and substitution.

CLEARING FRACTIONS/ALGEBRAIC FRACTIONS WITH UNKNOWN AT THE NUMERATOR

Equations such as \(\frac{2}{3} + x = 1,\)\(\frac{3x}{2} + \frac{3}{5} = 10,\) \(\frac{2y}{5} = \frac{3}{5}\) etc. are equations involving fractions.

To solve any of these equations, we consider the L.C.M of the denominators and multiply each term of the equation by the L.C.M to clear the fractions and solve the equation as usual.

Example 1

Solve the equation: \(\frac{x -4}{5} = 2 -\frac{x}{2}\)

Solution

The denominators of the fractions in this equation are; \(5 \text{ and } 2\).

L.C.M of \(5 \text{ and } 2 \text{ is }10\).

We multiply through by this L.C.M.

i.e. \(10 × \frac{x -4}{5} = (2 × 10) -(10 × \frac{x}{2}) \\ = 2(x -4) = 20 -5x\)

Collecting like terms we have,

\(5x  + 2x  = 20 + 8 \\ 7x  = 28\)

Thus, \(x = \frac{28}{7} = 4\)

Example 2

Solve \(\frac{2x}{3} + \frac{4}{5} = \frac{17}{15}\)

The denominators are \(3, 5 \text{ and }15\).