Question

Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour, if she travels 2 km by rickshaw and the remaining distance by bus. On the other hand, if she travels 4 km by rickshaw and the remaining distance by bus, she takes 9 min longer. Find the speed of the rickshaw and of the bus.
(3 Marks)

Answer 1

Let the speed of the rickshaw and the bus are x and y km/h, respectively

Now, she has taken time to travel 2 km by rickshaw , $t_1=\frac{2}{x}h[\therefore Speed=\frac{distance}{time}]$
and she has taken time to travel remaining distance i.e., (14 – 2) = 12 km by bus $t_2=\frac{12}{y}h$
similarly, by Second condition;
$t_3=\frac{4}{x}h$
$t_4=\frac{10}{y}h$
$t3+t4=\frac{1}{2}+\frac{9}{60}=\frac{1}{2}+\frac{3}{20}$
$\Rightarrow \frac{4}{x}+\frac{10}{y}=\frac{13}{20}$ (1 mark)

Let $\frac{1}{x}=u\frac{1}{y}=v,$ then Eqs (i) and (ii) becomes

2u + 12v = $\frac{1}{2}$

And 4u + 10v =$\frac{13}{20}$

On multiplying in Eq. (iii) by 2 and then subtracting, we get

4u + 24v = 1

4u + 10v = $\frac{13}{20}$

_ _ _

$14v=1-\frac{13}{20}=\frac{7}{20}$
$2v=\frac{1}{20}\Rightarrow v=\frac{1}{40}$ (0.5 mark)
Now, put the value of v in Eq (iii), we get

$2u+12=\left(\frac{1}{40}\right)=\frac{1}{2}$ (0.5 mark)
$\Rightarrow 2u=\frac{1}{2}-\frac{3}{10}=\frac{5-3}{10}$
$\therefore \frac{1}{x}=u$
$\Rightarrow \frac{1}{x}=\frac{1}{10}\Rightarrow 10km/h$
$and\frac{1}{y}=v\Rightarrow \frac{1}{y}=\frac{1}{40}$
$\Rightarrow y=40km/h$ (1 mark)
Hence, the speed of rickshaw and the bus are 10 km/h and 40 km/h, respectively.