Question

A rail engine is moving at a uniform speed of 30 km per hour towards a place A. When the engine is still 20 km away from A, an insect starts at A and shuttles between A and engine at a uniform speed of 42 km per hour. The total distance travelled by the insect, by the time engine reaches A is, in km.

• A. 28
• B. 30
• C. 20
• D. 42

Answer 1

The correct option is A 28

Relative speed $=30+42=72km/hr$.
$\therefore$ The rail Engine and the insect will meet for the first time after
$=\frac{20}{72}hrs$
$\therefore$ Distance covered by the insect in this interval
$=\frac{20}{72}\times 42=\frac{35}{3}km$
In this time engine covered a distance
$=\frac{20}{72}\times 30=\frac{25}{3}km$
As the insect will return to A, in that time engine will again cover a distance
$=\frac{25}{3}km$
So,the distance left $=20-\frac{25}{3}-\frac{25}{3}=\frac{10}{3}km$
Again the rail engine and the insect will meets for the second time after
$=\frac{10}{3\times 72}=\frac{5}{108}hrs.$
$\therefore$ Distance covered in this time by the insect
$=\frac{5\times 42}{108}=\frac{35}{18}km$
In returning it will describe a distance
$=\frac{35}{18}$
$\therefore$ Distances covered by the insect
$=\frac{70}{3}+\frac{70}{18}+\dots$
$=\frac{70}{3}(1+\frac{1}{6}+\dots )$
$=\frac{70}{3}\times \frac{1}{1-\frac{1}{6}}$
$=\frac{70}{3}\times \frac{1}{\frac{5}{6}}==\frac{70}{3}\times \frac{6}{5}$
$=28km.$