Question

A cyclist after riding a certain distance stopped for half an hour to repair his machine after which he completes the whole journey of 30 km at half speed in 5 hours. If breakdown had occurred 10 km further off, he would have done the whole journey in 4 hrs. Find where the breakdown occurred and his original speed.

Answer 1

A cyclist after riding a certain distance stopped for half an hour to repair his machine after which he completes the whole journey of 30Km at half speed in 5 hours.

If the breakdown has occurred 10Km further off, he would have done the whole journey in 4 hours.

Find where breakdown occurred and his original speed.

:

let s = original speed

then

.5s = half speed

:

let d = that "certain distance" where the breakdown occurred

:

He rode d at original speed and (30-d) at half speed

:

Write a time equation, time = dist/speed

:

Time at full speed + repair time + time at half speed = 5 hrs

d%2Fs + .5 + %28%2830-d%29%29%2F%28.5s%29 = 5

Multiply equation by s

d + .5s + 2(30-d) = 5s

d + .5s + 60 - 2d = 5s

combine like terms

d - 2d + .5s - 5s + 60 = 0

-d - 4.5s + 60 = 0

multiply by -1

d + 4.5s - 60 = 0

d + 4.5s = 60

Answer 2

1)Let the distance at which breakdown occurred be X.

2)Let Y be the velocity or speed of cyclist.

CASE 1:- It is given that at a certain distance.
breakdown has occurred let it be X.
Then the distance moved by him is
(30-x).

It is also said that after repairing he moves.
with half the velocity means Y/2.

Therefore equation is,


$\frac{x}{y} + \frac{30 - x}{ \frac{y}{2} } = 5$
$\frac{x + 60 - 2x}{y} = 5$
$60 = 5y + x \: \: \: \: \: \: - - - i$


CASE 2 :- It is said that if the breakdown has occurred
10km farther then equation is (x+10).

Then the distance covered by him is
30-(x+10)

Therefore equation is,


$\\ = > \frac{x + 10}{y} + \frac{30 - (x + 10)}{ \frac{y}{2} } = 4$

$= > \frac{x + 10}{y} + \frac{(20 - x)2}{y} = 4$

$= > \frac{x + 10 + 40 - 2x}{y} = 4$


$50 = 4y + x \: \: \: \: \: - - - - ii$


from i and ii we get,


$= > 60 = 5y + x \\ \: \: \: \: \: \: \: \: 50 = 4y + x$

$= > y = 10$

put this in 1st we get,


$= > 50 = 4(10) + x$
$= > x = 10$

Hence the breakdown has occurred at distance 10km.

and original speed is 10km/hr.