Question

Divide$1400$ into three parts such that the $1^{st}$ part is $\frac{2}{3}$of the $2^{nd}$part and the ratio between the $2^{nd}$ and the $3^{rd}$ part is$4:5$.

Find each part.

Answer 1

Let the common multiples of these ratios be $x$.

The ratio of the $2^{nd}$part and the $3^{rd}$ part $=4:5$

So, $2^{nd}$ part$=4x$ and $3^{rd}$ part $=5x$

Step 1: Find the value of $x$

It is given that, the $1^{st}$ part is $\frac{2}{3}$of the $2^{nd}$part.

$\Rightarrow$ $1^{st}$ part$=\frac{2}{3}\times 4x$

$\Rightarrow$$1^{st}$ part $=\frac{8x}{3}$

Also, the sum of all the parts is $1400$.

$\Rightarrow$ $\frac{8x}{3}+4x+5x=1400$

$\Rightarrow$$\frac{8x+12x+15x}{3}=1400$ [ take the least common multiple of the denominators, change them into like fractions and add them]
$\Rightarrow$ $8x+12x+15x=4200$

$\Rightarrow$ $35x=4200$

$\Rightarrow$ $x=120$

Step 2: Find all the parts by substituting the value of $x$

$1^{st}$ part $=\frac{8x}{3}$

$=\frac{8\times 120}{3}$

$=320$

$2^{nd}$ part $=4x$

$=4\times 120$

$=480$

$3^{rd}$ part $=5x$

$=5\times 120$

$=600$

Hence, the required parts are $320$, $480$ and $600$.