Aptitude - Time and Distance - Discussion

I am not a mathematician. But, a question arises in me that is :

First half of 10 hours = 5 hours.
Second half of 10 hours = 5 hours.

Then, 5 Hours @ 21 KM Per hour = 21x5 = 105 km.
And nest 5 hours @ 24 KM per hour = 24x5 = 120 km.

Total (21x5) + (24x5) = 105 + 120 = 225.

So, why the answer is not 225 KM.

What would be the mistake here?

If there was no mention of first half and 2nd half of 10 hours, (i.e. 10 hours / 2 = 5 hours,) if it was only mentioned that first the person traveled @ 21 KM/hour and then 24 Km/per, then may be it would be solved using any time & distance formula.

Being a ordinary class student, this question has arisen in my mind.

May I kindly be helped with proper answer ?

If the answer is 224,(DISTANCE BASED).

If you think total journey 224, then half of the journey is 112 KM, travels at the speed 21 KMPH means time taken to complete 1st half is 112/21=5.33333.. hrs.

For 2nd half journey, i.e 112 KM travels at the speed 24 KMPH, the time taken for this is 112/24=4.666666...hrs.

If we add both times it will be so much near to 10 hours, but not exactly 10 hours.

If we go through following way,(TIME-BASED).its gonna be right.

Total journey is 10 hours.half of the journey is 5 hours for the 1st half journey at the speed 21 KMPH means distance travelled in 1st half is 21*5=105, in the same way, distance travelled in 2nd half is 24*5=120.

So, total distance 105+120=225km.

Here you need to know first that average speed is the uniform speed.

That is if the person travelled with the uniform speed then he would complete the journey in 10 hrs and cover the given distance as well. We don't know the distance and we are given two different speeds for half the journey.

Now as we can see the last formula in the important formula section for this chapter. We can get the average speed as: 2xy/(x+y).

So average speed = 2x21x24/(21+24).

Now not getting the exact average speed I reduce the above by basic division as.

Average speed = (2x7x8)/5.

But Distance = Speed*Time.

So, D = [(2x7x8)/5]*10 (Now 10/5 is 2 so the equation becomes).

2*7*8*2 = 224 km that's it!

[ ................................10hrs........................................]
5hrs 5hrs
/-------------------------------/-----------------------------------/

[speed21kmph...............] [....speed24kmph..................]


So, here clearly we know time and speed.

V = D/T
D = VT

Take 1st half D = 21 * 5 = 105kmph----------> 1
Take 2nd half D = 24 * 5 = 120kmph----------> 2


Total journey means add of 1st and 2nd half.

Total distance D = 105 + 120.
= 225kmph.

Following is explanation:

Total Travel Time = 10 hours.

Let's assume Total Distance = d kms and 1st half (i.e d/2) was completed in x hours at speed of 21 kmph.

=> d/2 = 21x.

=> d/42 = x.....equation (1).

=> 2nd half (i.e d/2) was completed in (10-x) hours @ speed of 24 kmph.

=> d/2 = 24*(10-x).
=> d/48 = 10-x.

=> x = 10-(d/48)....equation (2).

From equation (1) and (2),

=> d/42 = 10-(d/48).
=> d/42 + d/48 = 10.
=> d = 2*5*2*3*7*8/(3*5).
=> d = 224 kms.

d=s*t.

Let the total distance be 2d.
for first part,
distance=d
time taken be=t
speed=21 km/hr.

for the second part
dis=d
time=10-t
speed=24 km/hr

Since distance covered is equal
so,
d for 1st half=d for the 2nd half
t*21=(10-t)*24

Using this I calculated the time for the individual half's through get the distance.

but I am not able to get the correct answer. Can anyone tell me the logic behind that?

WKT,

Average speed over two different speeds in the journey is;
Avg speed= (2XY/X+Y)km/h or M/S.
Where X is the speed during the first half (km/hr or m/s).
And Y is the speed during the second half (km/hr or m/s).

And WKT.
s= d/t and d=s*t.

Here Given
X=21 km/hr.
Y =24 km/hr.

And time over the whole jouneny is 10hrs,
Avg speed = (21*24)/(21+24),
= 22.4 km/hr.
And Total Distance =s*t.
22.4 * 10 = 224 km.

Let, distance be 'x'
The for the first half, distance = x * 1/2.
For the second half also, distance = x * 1/2.
Then , time = distance/speed.
Therefore, time taken for first half = (x*1/2)/21 = x/42.
Time taken for second half = (x*1/2)24 = x/48.
Therefor total time = x/42 +x/48 and that is given as 10 minutes.
So, x/42+x/48 = 10.
Solving this, you get (48x + 42x) = 48*42*10.
On solving , 90x = 48*42*10.
We get x = 224.

@Muhammed way is incorrect because, you can't just assume that first half equals second half equals 5 hours. Your speed is different in the two halves of the journey. You are going slower at first half but faster at second half. So you are taking a bit more than 5 hours in first half, and a bit less than 5 hours in second half. We have to weight that first. 224 km is the correct answer, 225 km is not.

Rati, your method for getting average of speed is wrong,

Because formla of average of speed is = Total distance traveled/Total time taken.

Now let me solve the problem.

Let the total distance = X
& 1st half means X/2
2nd half means also X/2

Again time 4 1st half= X/2*21 =X/42, time 4 2nd half = X/48

According to question: X/42 X/48 = 10hrs.
or, 7x+8x/336=10
or, 15x=3360

So X = 224 Answer.