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 minutes longer. Find the speed of the rickshaw and of the bus.

Answer 1

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

Now, time taken to travel 2 km by rickshaw,

t₁ = `2/x`hours   ....`[because  "speed" = "distance"/"time"]`

And time taken to travel remaining distance

i.e., (14 – 2) = 12 km by bus,

t₂ = `12/y` hours

By first condition,

t₁ + t₂ = `1/2`

⇒ `2/x + 12/y = 1/2`  ....(i)

Now, time taken to travel 4 km by rickshaw,

t₃ = `4/x` hours

And time taken to travel remaining distance i.e., (14 – 4) = 10 km by bus,

t₄ = `10/y`hours

By second condition,

t₃ + t₄ = `1/2 + 9/60 = 1/2 + 3/20`

⇒ `4/x + 10/y = 13/20`  .....(ii)

Let `1/x` = u and `1/y` = v,

Then equation (i) and equation (ii) becomes

2u + 12ν = `1/2`  .....(iii)

And 4u + 10ν = `13/20`  ......(iv)

On multiplying equation (iii) by 2 and then subtracting equation (iv) from it, we get

(4u + 24v) – (4u + 10v) = `1 - 13/20`

⇒ 14v = `7/20`

⇒ 2v = `1/20`

⇒ v = `1/40`

Now, put the value of v in equation (iii), we get

`2u + 12(1/40) = 1/2`

⇒ 2u = `1/2 - 3/10 = (5 - 3)/10`

⇒ 2u = `2/10`

⇒ u = `1/10`

Now, `1/x` = u

⇒ `1/x = 1/10`

⇒ x = 10

And `1/y` = v

⇒ `1/y = 1/40`

⇒ y = 40

Hence, the speed of rickshaw and the bus are 10 km/h and 40 km/h, respectively.