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I am struggling to see how the following problem is simplified. Can someone include any steps that may have been skipped?

Original Equation= $\frac{T(p-b)}{(p-b+q-a)}$

Simplified Equation= $T\times\frac1{1+\frac{(q-a)}{p-b})}$

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  • $\begingroup$ Divide both numerator and denominator with $(p-b)$ and decompose into partial fractions. $\endgroup$ Jul 21, 2015 at 22:40
  • $\begingroup$ @thanasissdr thank you for your help $\endgroup$
    – Amaziah
    Jul 21, 2015 at 22:59
  • $\begingroup$ You're welcome! $\endgroup$ Jul 21, 2015 at 23:06

2 Answers 2

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Dividing top and bottom by $p-b$ gives

$$\dfrac{p-b}{p-b+q-a}=\dfrac{\dfrac{p-b}{p-b}}{\dfrac{p-b+q-a}{p-b}}=\dfrac{1}{\dfrac{p-b}{p-b}+\dfrac{q-a}{p-b}}=\dfrac{1}{1+\dfrac{q-a}{p-b}}$$

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You have $$ (p-b+q-a)=((p-b)+(q-a))=(p-b)\times\left( 1+\frac{q-a}{p-b}\right) $$ thus, simplifying by $\color{red}{(p-b)} \neq 0$, gives $$ \frac{T(p-b)}{(p-b+q-a)}=\frac{\color{red}{(p-b)}\times T }{\color{red}{(p-b)}\times\left( 1+\frac{q-a}{p-b}\right)}=\frac{T}{ 1+\frac{q-a}{p-b}}. $$

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  • $\begingroup$ @Amaziah You are welcome! $\endgroup$ Jul 21, 2015 at 23:07

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