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Q.

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

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a

x=10km/h and y=40km/h

b

x=20km/h and y=40km/h

c

x=15km/h and y=40km/h

d

x=10km/h and y=50km/h

answer is A.

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Detailed Solution

Given that,
Total distance in km = 14 km.
We know that

Time = Distance speed

.
Let us assume
Speed of rickshaw = x km/ h,
Speed of bus = y km/h.
When she travels 2 km by rickshaw and the rest distance by bus, then
Total time,
T = 30 min

T = 12

hour………….(1)
Distance covered by rickshaw = 2 km.
Distance covered by bus = total distance - distance covered by rickshaw
= 14 -2
= 12 km
When she travels 4 km by rickshaw and the rest distance by bus
Total time,
t = 30 min + 9 min

t = 12 + 960

t = 12 + 320

t = 12 × 1010 + 320

t = 1020 + 320

t = 10 + 3 20

t = 1320

................(2)
Distance covered by rickshaw = 4 km.
Distance covered by bus = total distance - distance covered by rickshaw
= 14 - 4
= 10 km.
Let us find the time take by rickshaw and bus.
When she travels 2 km by rickshaw and remaining by bus,
Time taken by rickshaw

= t R

.

t R = distance speed

t R = 2 x

…………..(3)
Time taken by rickshaw =

t B

.

t B = distance speed

t B = 12 y

…………..(4)
When Ankita travel 4 km by rickshaw and remaining by bus,
Time taken by rickshaw

= t r

.

t t = distance speed

t r = 4 x

…………..(5)
Time taken by bus

= t b

.

t B = distance speed

t B = 10 y

…………..(6)
Find the relation between time taken by bus and rickshaw.
So, according to question

t R + t B = T

.
Put the values from equation (1), (3) and (4).

2 x + 12 y = 12

……………..(7) And ,

t r + t b = t

Put the values from equation (2), (5) and (6).

4 x + 10 y = 1320

……………(8)
Let,

1 x = m

and

1 y = n

So, equation (7) and (8) becomes,

2 m + 12 n = 12

………………(9)

4 m + 10 n = 1320

…………….(10)
Multiply equation (9) by 2, we get,

4 m + 24 n = 1

……………(11)
Subtract equation (11) from (10).

4 m + 24 n − 4 m + 10 n = 1 − 1320

24 n − 10 n = 20 − 13 20

14 n = 720

n = 140

…………..(12)
Put

n = 140

in equation (11).

4 m + 24 × 140 = 1

4 m + 2440 = 1

4 m + 35 = 1

4 m = 1 − 35

4 m = 5 − 3 5

m = 5 − 3 5 × 4

m = 110

………..(13)
Since,

1 x = m

1 x = 110

⇒ x = 10

And,

1 y = n

⇒ 1 y = 140

⇒ y = 40

Therefore, the speed of rickshaw, x=10

km / h

and speed of bus y=40

km / h

.
The speed of rickshaw, x=10

km / h

and speed of bus y=40

km / h

.
The correct option is (1).

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