Seventy-eight is divided into two parts such that four times the first part and five times the second part are in the ratio 14 : 15. The first part is:

Let's assume the first part is 'y' and the second part is 'z'.

Seventy-eight is divided into two parts such that four times the first part and five times the second part are in the ratio 14 : 15.

y+z = 78    Eq.(i)

$$\frac{4y}{5z}\ =\ \frac{14}{15 }$$

$$\frac{2y}{z}\ =\ \frac{7}{3}$$

$$\frac{y}{z}\ =\ \frac{7}{6}$$

So y = 7a and z = 6a.    Eq.(ii)

Put Eq.(ii) in Eq.(i).

7a+6a = 78

13a = 78

a = 6

First part = y = 7a

= $$7\times6$$

= 42