Question

ab=2 , bc=3 , cd=4 , de=5 , find e/a​

Answer 1

Question

Given $ab=2$, $bc=3$, $cd=4$, and $de=5$, find $\dfrac{e}{a}$.

Solution

• Find ways to isolate $e$ and $a$.

Multiplying altogether,

$ab^2c^2d^2e=120$

$\Longleftrightarrow\dfrac{ab^2c^2d^2e}{a^2b^2c^2d^2} =\dfrac{120}{a^2b^2c^2d^2}$

$\therefore \dfrac{e}{a} =\dfrac{120}{(abcd)^2}$ ...(I)

Finding product of $ab\times cd$,

$abcd=6$

$\therefore (abcd)^2=36$

Then (I) becomes

$\dfrac{e}{a} =\dfrac{120}{36}$

$\therefore \dfrac{e}{a} =\dfrac{10}{3}$

More information

The method of multiplying altogether is used in equation solving.

For example, when we solve

$\begin{cases} &ab= 2\\ &bc= 4\\ &ca= 8\end{cases}$

We multiply altogether to get

$(abc)^2=16$, then $abc=\pm4$ ...(I)

Dividing (I) by each equation,

$\therefore\begin{cases} & a= \pm1\\ & b= \pm\dfrac{1}{2}\\ & c= \pm2\end{cases}$