# Which expression is equivalent to the following complex fraction – 2/x + (5/y)/(3/y) – 2/x ?

The purpose of this question is to practice the algebraic simplification and reduction process with reference to algebraic expressions.

An algebraic expression is nothing but a combination of constant numerals and variables joined through some mathematical operator. For example, 50x + 3 and    x – 322. Such expressions are widely used in almost all the branches and applications of mathematics.

Simplifying algebraic expressions involves finding the most efficient (least no. of terms) form without affecting the underlying mathematical function. This process is explained in the solution to the above question and the example given at the end of this article.

Given:

$– \ \dfrac{ 2 }{ x } \ + \ \dfrac{ \dfrac{5}{y} }{ \dfrac{3}{y} } \ – \ \dfrac{ 2 }{ x }$

Multiply the second term with $\dfrac{ y }{ y }$:

$– \ \dfrac{ 2 }{ x } \ + \ \dfrac{ \dfrac{5}{y} }{ \dfrac{3}{y} } \times \dfrac{ y }{ y }\ – \ \dfrac{ 2 }{ x }$

$– \ \dfrac{ 2 }{ x } \ + \ \dfrac{ \dfrac{5}{y} \times y }{ \dfrac{3}{y} \times y} \ – \ \dfrac{ 2 }{ x }$

$– \ \dfrac{ 2 }{ x } \ + \ \dfrac{ 5 }{ 3 } \ – \ \dfrac{ 2 }{ x }$

Rearranging:

$– \ \dfrac{ 2 }{ x } \ – \ \dfrac{ 2 }{ x } \ + \ \dfrac{ 5 }{ 3 }$

Adding the numerators of the first two terms with same denominator:

$\dfrac{ (-2) + (-2) }{ x } \ + \ \dfrac{ 5 }{ 3 }$

$- \ \dfrac{ 4 }{ x } \ + \ \dfrac{ 5 }{ 3 }$

This is the required simplified expression.

## Numerical Result

$– \ \dfrac{ 2 }{ x } \ + \ \dfrac{ \dfrac{5}{y} }{ \dfrac{3}{y} } \ – \ \dfrac{ 2 }{ x } \ = \ – \ \dfrac{ 4 }{ x } \ + \ \dfrac{ 5 }{ 3 }$

## Example

Which expression is equivalent to the following complex fraction $– \dfrac{ 10/x }{ 9/x } – \dfrac{ 50/y }{ 9/y } – \dfrac{ 2 }{ x }$ ?

Multiplying first term with $\dfrac{ x }{ x }$ and second term with $\dfrac{ y }{ y }$:

$=\ – \ \dfrac{ 10/x }{ 9/x } \ – \ \dfrac{ 50/y }{ 9/y } \ – \ \dfrac{ 2 }{ x }$

$=\ – \ \dfrac{ \dfrac{ 10 }{ x } \times x }{ \dfrac{ 9 }{ x } \times x } \ – \ \dfrac{ \dfrac{ 50 }{ y } \times y }{ \dfrac{ 9 }{ y } \times y } \ – \ \dfrac{ 2 }{ x }$

$=\ – \ \dfrac{ 10 }{ 9 } \ – \ \dfrac{ 50 }{ 9 } \ – \ \dfrac{ 2 }{ x }$

$=\ \dfrac{ – 10 – 50}{ 9 } \ – \ \dfrac{ 2 }{ x }$

$=\ – \ \dfrac{ 60}{ 9 } \ – \ \dfrac{ 2 }{ x }$

This is the required simplified expression.