20. It takes 4 hours for a motorboat to travel downstream from $\mathrm{X}$ to $\mathrm{Y}$. To return upstream from $\mathrm{Y}$ to $\mathrm{X}$ it takes the motorboat 6 hours. How many hours would it take a wooden log to be carried from $\mathrm{X}$ to $\mathrm{Y}$ by the current, assuming it is unhindered by any obstacles? [Assume that the current flows at a constant rate, and that the motorboat moves at a constant speed relative to the water.]
A 5
B 10
C 12
D 20
E 24
20. E Let $D$ be the distance from $X$ to $Y$, and let $u$ be the speed of the boat and $v$ be the speed of the current. When travelling downstream the overall speed of the boat in the current is $u+v$, and travelling upstream against the current it is $u-v$. Then using time $=\frac{\text { distance }}{\text { speed }}$, we get $\frac{D}{u+v}=4$ for the journey downstream and $\frac{D}{u-v}=6$ for the journey upstream. Inverting the equations gives $\frac{u+v}{D}=\frac{1}{4}$ or $4 u+4 v=D \ldots$ (1) and $\frac{u-v}{D}=\frac{1}{6}$ or $6 u-6 v=D \ldots$ (2). Multiplying (1) by 3 and (2) by 2 , we get $12 u+12 v=3 D \ldots$ (3) and $12 u-12 v=2 D \ldots$ (4). Subtracting, (3) - (4) gives $24 v=D$ which rearranges to $\frac{D}{v}=24$ (hours), which is the time taken for the log to float downstream at the speed of the current alone.