How can we quickly find t in $\frac xt = \frac x a + \frac x b$ for larger integer values of a and b?

Question

If $$\frac x t = \frac x a + \frac x b$$ is there a faster way of finding the value of t instead of finding a common denominator?

Answer 1

Yes, definitely. We can rearrange the equation like so: $$\frac{x}t=\frac{x}a+\frac{x}b$$ $$\frac{x}t=\frac{bx+ax}{ab}$$ $$\frac{x}t=\frac{x(a+b)}{ab}$$ $$\frac1t=\frac{a+b}{ab}$$ $$t = \frac{ab}{a+b}$$

For example, if $x=2,\ a=4,\ b=6$, then $t=\frac{24}{10}=\frac{12}5$
This is proved by the long way too:

$$\frac2t=\frac24+\frac26$$ $$\frac2t=\frac{12+8}{24}=\frac{20}{24}$$ $$\frac{t}2=\frac{24}{20}\; \to\ t=\frac{12}5$$

Answer 2

$$\frac{x}{a}+\frac{x}{b}=\frac{x(a+b)}{ab}$$

Hence $$x\bigg(\frac{1}{t}\bigg)=x\bigg(\frac{a+b}{ab}\bigg)$$ $$\to t=\frac{ab}{a+b}$$

e.g. $a=3, b=6$ leads to $t=\frac{18}{9}=2$, as we'd expect.