Second number

Let the first and second numbers be $a$ and $b$ respectively. Then we have

$\begin{cases} \dfrac 15 a = \dfrac 58 b &...(1) \\ a + 35 = 4b &...(2) \end{cases}$

From (1): $a = \dfrac {25}8b$ , substituting into (2):

$\begin{aligned} \frac {25}8b + 35 & = 4b & \small \color{#3D99F6} \text{Multiplying both sides by }8 \\ 25b + 280 & = 32b \\ 7b & = 280 \\ \implies b & = \boxed{40} \end{aligned}$

Suppose the first number is $a$ and the second number is $b$ .

$\Rightarrow$ $\dfrac{1}{5}$ $a$ $=$ $\dfrac{5}{8}$ $b$

$8a = 25b \Rightarrow 8a - 25b = 0 ..........................(1)$

$\Rightarrow a + 35 = 4b$

$8a - 25b = 0 ......................(1)$

$a - 4b = -35...........................(2)$

$\Rightarrow -8a + 32b = 280 ...............(2) \times (-8)$

Adding the above two equations,

$7b = 280 \Rightarrow b = 40$ $;$ Since $8a = 25b, 8a = 25 \times 40 = 1000, a = 125$

Therefore , the second number is $40$ .