2.8.3 Solving algebraic fractions
To solve an equation that contains algebraic fractions
Find the least common denominator
Multiply both sides of equation with the least common denominator
Cancel the common factors/brackets
Expand the brackets
Simplify by collecting the like terms
You can swap the sides if it makes solving the equation easier
Worked example:
\( \frac{3}{m+4}-\frac{4}{m}=6 \)
Show that this equation can be written as \( 6m^2+25m+16=0 \)
The least common denominator is \( m(m+4) \)
Multiply both sides by \( m(m+4) \)
\( \frac{3}{m+4}\times m(m+4)-\frac{4}{m}\times m(m+4)=6 \times m(m+4) \)
Cancel the common factors/brackets
Expand the brackets
\( 3m-4(m+4)=6\times m(m+4) \)
Collect the like terms
\( 3m-4m-16=6m^2+24m \)
\( 6m^2+24m+m+16=0 \)
\( 6m^2+25m+16=0 \)
Hence the result.
Test yourself:
Question 1:
Solve.
\( \frac{1}{x}-\frac{2}{x+1}=3 \)
Show all your working and give your answers correct to \( 2 \) decimal places.
[7]
Question 2:
A car completes a \( 200 \) km journey at an average speed of \( x \) km/h.
The car completes the return journey of \( 200 \) km at an average speed of \( x+10 \) km/h.
Show that the difference between the time taken for each of the two journeys is
\( \frac{2000}{x(x+10)} \) hours.
[3]
Find the difference between the time taken for each of the two journeys when \( x=80 \).
Give your answer in minutes and seconds.
[3]