A problem by Kiran Nivedh

Assume the total distance is $S$ . We have: $t=1.5h$ and $t=\frac {S}{V}=\frac {S}{3 \times 6}+\frac {2\times S}{3 \times 15}=1.5$ Solve this very easy equation give us the answer $S=15km$ .

$$ Let $t_1$ and $t_2$ be the time to cover the first and second distance, respectively. Let $v_1$ and $v_2$ be the velocity to cover the first and second distance, respectively. Let $x$ be the total distance. Therefore $$ $\begin{aligned} \\ t_1 + t_2 = 1.5 \text{ h} & &(1)\\ \frac{1}{3} x = v_1 t_1 = 6t_1 & &(2)\\ \frac{2}{3} x = v_2 t_2 = 15t_2 & &(3)\\ \end{aligned}$ Divide $(2)$ by $(3)$ , we obtain $\begin{aligned} \\ \frac{1}{2} &= \frac{2t_1}{5t_2}\\ t_2 &= \frac{4}{5}t_1 & & & & &(4)\\ \end{aligned}$ Substitute $(4)$ to $(1)$ , we get $\begin{aligned} \\ t_1 + \frac{4}{5}t_1 &= 1.5\\ \frac{9}{5}t_1 &= \frac{3}{2}\\ t_1 &= \frac{5}{6}\\ \end{aligned}$ Finally, $\frac{1}{3} x= 6t_1 = 6 \cdot \frac{5}{6} = 5 \Rightarrow x = \boxed{15 \text{ km}}$ . $$ $\text{\# }\mathbb{Q}.\mathbb{E}.\mathbb{D}.\text{\#}$

If the total distance=x, then [(x/3)/6] +[(2x/3)/15]= 3/2, On solving x=15