How to quickly solve $ \frac{x}{58.5} + \frac{0.118-x}{74.5} = \frac{0.2451}{143.5} $?

Question

How do I quickly solve $ \frac{x}{58.5} + \frac{0.118-x}{74.5} = \frac{0.2451}{143.5} $ or a similar equation? I was actually doing chemistry questions for practice for JEE and while solving one of the questions I encountered this equation. This is only a part of the entire problem and I can't spend more than one or two minutes on this at the most. Trying to take the LCM is difficult and very time consuming. How do I solve this quickly without spending too much time on this? The answer I'm supposed to get is x = 0.0338.

Answer 1

This is easy. For a start multiply the equation by 58.5 and then 74.5 to get: $$74.5x+58.5(0.118-x)=0.2451*58.5*74.5/143.5=7.444$$ This can be re-arranged to give: $$74.5x-58.5x+58.5*0.118=7.444$$

The next line will read: $$16x=7.444 - 6.903=0.541$$

That is, x = 0.0338

Answer 2

We want a quick way to solve $\frac{x}{58.5} + \frac{0.118-x}{74.5} = \frac{0.2451}{143.5}$.

Since we want an answer with three significant digits we should round intermediate numbers to four or five significant digits and round the answer off to three.

We begin by separating out the $x$ terms: \begin{eqnarray} \left(\frac{1}{58.5}-\frac{1}{74.5}\right)x+\frac{0.118} {74.5}&=&\frac{0.2451}{143.5}\\ &&\\ \text{Convert to decimal notation}&&\\ &&\\ 3.6712\times10^{-3}x+1.5839\times10^{-3}&=&1.7080\times10^{-3}\\ &&\\ \text{Multiply through by }10^{3}\text{ to get}&&\\ &&\\ 3.6712x+1.5839&=&1.7088\\ &&\\ x&=&\frac{1.7088-1.5839}{3.6712}\\ &&\\ x&=&0.03380 \end{eqnarray}