If a sum of ₹7,13,700 is divided into 3 parts so that their amounts after 2, 3 and 4 years, respectively, be equal, the rate of interest being $5\%$ per annum at simple interest, then the first part (in ₹) is:

Let the three parts be $P_1,P_2,\text{ and }{P}_3$.The rate of interest is $R=5\%$.The amounts after 2, 3, and 4 years are equal.
$$\text{Amount}=P\left(1+\frac{RT}{100}\right)$$
So,
$$P_1\left(1+\frac{5\times 2}{100}\right)=P_2\left(1+\frac{5\times 3}{100}\right)=P_3\left(1+\frac{5\times 4}{100}\right)$$
$$P_1\left(\frac{110}{100}\right)=P_2\left(\frac{115}{100}\right)=P_3\left(\frac{120}{100}\right)$$
$$110P_1=115P_2=120P_3$$
Dividing by 5:
$$22P_1=23P_2=24P_3$$
The ratio $P_1:P_2:P_3$ is:
$$P_1:P_2:P_3=\frac{1}{22}:\frac{1}{23}:\frac{1}{24}$$
To eliminate fractions, multiply by $22\times 23\times 24$:
$$P_1:P_2:P_3=(23\times 24):(22\times 24):(22\times 23)$$
$$P_1:P_2:P_3=552:528:506$$
Dividing by 2:
$$P_1:P_2:P_3=276:264:253$$
Sum of the ratio units = $276+264+253=793$ units.Total amount = ₹7,13,700.
$$1\text{ unit}=\frac{7,13,700}{793}=900$$
The first part ($P_1$):
$$P_1=276\times 900=\text{₹}2,48,400$$