Aptitude - Time and Distance - Discussion

Let's assume the original average speed of the aircraft for the 600 km trip was "x" km/hr.

When the aircraft was slowed down due to bad weather, its average speed was reduced by 200 km/hr.
So, the new average speed for the trip would be "(x - 200)" km/hr.

The time of flight increased by 30 minutes, which can be converted to hours by dividing by 60. So, the additional time is 30/60 = 0.5 hours.
We can use the formula: Time = Distance/Speed to calculate the duration of the flight.
For the original speed, the time taken would be 600 km/x km/hr = 600/x hours.
For the reduced speed, the time taken would be 600 km/(x - 200) km/hr = 600/(x - 200) hours.

Since the time taken increased by 0.5 hours, we can set up the equation:
600/x + 0.5 = 600/(x - 200).

To solve this equation, we can cross-multiply and simplify:
600(x - 200) + 0.5x(x - 200) = 600x,
600x - 120000 + 0.5x^2 - 100x = 600x,
0.5x^2 - 100x - 120000 = 0.

Dividing the equation by 0.5 to simplify further:
x^2 - 200x - 240000 = 0.

Now we can solve this quadratic equation. Using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
Where a = 1, b = -200, and c = -240000.

Plugging in the values and solving for x:

x = (-(-200) ± √((-200)^2 - 4(1)(-240000)))/(2(1)).
x = (200 ± √(40000 + 960000))/2,
x = (200 ± √1000000)/2,
x = (200 ± 1000)/2.

Now we have two possible values for x:
x1 = (200 + 1000)/2 = 600 km/hr.
x2 = (200 - 1000)/2 = -400 km/hr (rejecting this negative value).

Therefore, the original average speed of the aircraft was 600 km/hr.
To find the duration of the flight, we can substitute this value back into the equation:
Time = Distance / Speed = 600 km / 600 km/hr = 1 hour,
So, the duration of the flight is 1 hour.

Let's denote the original speed of the aircraft as "S" km/hr and the original duration of the flight as "T" hours. We're given that the flight was slowed down by 200 km/hr, so its reduced speed is (S - 200) km/hr.

We know that:

Distance = Speed × Time

So, the original duration of the flight can be expressed as:

T = 600 km / S

And the duration of the flight with the reduced speed is:

T + 0.5 hours (since it increased by 30 minutes, which is 0.5 hours).

Now, we can set up an equation for the changed situation:
600 km = (S - 200) km/hr × (T + 0.5 hours).

Now, substitute the value of T from the original duration equation:
600 km = (S - 200) km/hr × (600 km/S + 0.5 hours).

To solve for S, let's get rid of the fractions by multiplying both sides by S:
600S = (S - 200) × (600 + 0.5S).

Now, distribute on the right side of the equation:
600S = 600S - 200S + 0.5S^2 - 100S.

Combine like terms:
600S = 500S + 0.5S^2 - 100S,

Now, simplify further:
600S = 400S + 0.5S^2.

Rearrange the equation:
0.5S^2 = 200S.

Now, divide both sides by 0.5 to isolate S:
S^2 = 400S.
Divide both sides by S:
S = 400 km/hr.

Now that we have the original speed, we can find the original duration of the flight:
T = 600 km / 400 km/hr = 1.5 hours.
So, the original duration of the flight was 1.5 hours.

I don't why but this is the second time they have used the difficult method. They use the basic principle to assume the variable to find as x i.e. time of flight as x. I don't follow this always.

Let's do the opposite way. Let us assume the original (or misleadingly called average) speed as x. Now if speed is decreased.

Then the time is increased by 30 mins i.e. 30/60 hrs (since we are using kmph so time has to be in hrs).

Now using common sense logic. If less speed then more time to travel the same distance.

So here the time difference for the two speeds is 30 mins or 30/60 hrs.

i.e (Time with less speed) - (Time with faster speed) = 30/60 hrs.

Now Time = Distance/Speed.

(600/(x-200))-600/x = 30/60.

Solve this equation to get x = 600 kmph.

600 km/600 kmph = 1 hr simple as that.

Just seems lengthy because I explained the logic. But its not always good to assume the unknown as x. This method seems easier to me.

@Ben You are right.

I am just copying a comment listed above.

Let x = original speed.

And y= original time taken.

So 600/x=y-------- (1).

Now we know if speed reduces 30 minutes more will be taken.

i.e 30/60 hours.

So, 600/x-200 = y+30/60 here 30/60 means 30 minutes.

Now put the value of why as shown in (1),

So equation will be, 600/x-200 = 600/x - 30/60.

You will get, x2-200x-240000 = 0 here x2 means x square.

(x-600)(x+200) = 0.

x = 600.

In the above equation we solved x to be 600 where x is the average speed when there isn't any bad weather. But there was bad weather and the flight slowed down to 400 kmph (600kmph-200kmph).

So the time taken for the actual flight due to slow down is 600km/(600kmph-200kmph).

i.e 1 and 1/2 hours.

Guys it is very simple.

In the problem it is given that reduced speed = 200.

The initial speed is the actual speed.

And final speed is the speed after being reduced by 200km/hr.

So "initialspeed-200=finalspeed".

Speed=distance/time.

Distance=600km (both initial and final distance).

Let x be initial time taken.

Let (x+30) in min i.e. (x+1/2) in hrs be the final time.

Initial speed=initial distance/initial time=600/x.

Final speed=final distance/final time=600/ (x+1/2).

So "initialspeed-200=finalspeed".

600/x-200=600/ (x+1/2).

600/x-600/ (x+1/2) =200.

Solving this we get.

400x2+200x-600=0.

4x2 + 2x - 6= 0.

2x2 + x - 3 = 0.

2x2 - 2x + 3x - 3 = 0.

2x (x - 1) + 3 (x- 1) = 0.

(2x + 3) (x - 1) = 0.

X=3/2 or x=1.

I solve this problem like this

First, I check the facts given:
Distance = 600km.
Original speed = we don't know,
New speed = we don't know,
The Relation between new and original speed = new speed is 200km/h slower to original speed so the diff in speed is 200km/h.
Original time = we don't know,
New time = we don't know,
Relation between new and original Time = new time is 30minutes more than original time so the diff in time is an hour.

Now, I need two separate numbers with the diff of 200 and should be able to divide 600 with a diff of .5 (means) and the numbers are 600 and 400.

600/600 = 1 (original speed 600km/h and it takes 1 hour to reach).
600/400 = 1.5 (new speed 400km/h and it takes 1 hour 30 min to reach).

Hey guys!

Actually x=3/2 is the correct answer.
if new time = 1.5 hrs(i.e 3/2 hrs)
then original time=1.5-0.5(i.e 30 minutes)
=1 hr.

Then original speed = 600 kmph (since 600 km in an hr)
new speed = 600-200
= 400 kmph.

New time = 600/400
= 1.5 hrs (fits correctly with x=1.5 i.e our initial assumption).
Now,
x=1
original time =30 minutes= 0.5 hr
original speed = 600 kmph * 05 hr
= 1200 kmph
new speed = 1200-200
= 1000 kmph.

new time = 600/1000
= 36 minutes.

Since x=30 minutes; x is not equal new time.
so conclusion:- time =1.5 hrs (answer choices are incorrect).

Distance: 600km
Let the original speed: s.

(avg speed reduced) so: s - 200.
let original time: T.
(time increased by 30 min => 30/60 => 1/2) so: T+1/2
s = d/t;
s = 600/t ----> (1);

New speed is
s - 200 = 600/(T +1/2) -----> (2)
Sub (1) in (2);
600/T - 200 = 600 / (T+1/2);
(600 - 200T)/ T = (600 *2)/(2T+1);

Cross multiply:
(2T+1)(600-200T) = (1200)*T;
1200T - 400T^2 + 600 -200T -1200T = 0;
400T^2 + 200T - 600 = 0;
2T^2 + T - 3 = 0;
(2T + 3 ) (T-1) 0
T-1 = 0 => T = 1.
2T+3 =0 => T=-3/2(neglect the negative value)
Hence time is 1 HOUR.

@Aarman and @Swetha.

(600/x)-(600/(x+1/2)) = 200
=> (600/x)-((600*2)/2x+1)) = 200
=> (600/x)-((1200)/2x+1)) = 200;

==>(600(2x+1)-1200x)/(x(2x+1)) = 200;
==>(1200x+600-1200x)/(2x^2+x) = 200;

==> (600)/(2x^2+x) = 200;
==> 600 = 400x^2+200x;

==> 400x^2+200x-600 = 0;
==> 200(2x^2+x-3) = 0;

==> 2x^2+x-3 = 0;
==> 2x^2+3x-2x-3 = 0;

==> x(2x+3)-1(2x+3) = 0;
==> (2x+3)(x-1) = 0;

So x=-3/2; x=1;

Since time can never be negative, Therefore by leaving the negative sign we get, x=1 hr and x=3/2 hr.

distance = 600km, speed x kmph.

t = 600/x-----Actual time taken by journey.

t + 30 = 600/x - 200------> this due to reduce speed, x-200, 200kmph speed reduce from actual speed.
t = (600/x-200) - 30.

600/x = (600/x-200)-30/60------30min convert to hrs
600/x = (600/x - 200)-1/2,
600/x - 200 - 600/x = 1/2,
[600x - (x - 200)600]/x(x - 200) = 1/2,
x^2-200x - 240000 = 0,
x^2-600x + 400x - 240000 = 0,
x(x-600) + 400(x-600) = 0,
x-600 = 0,&x + 400 = 0 +ve values only consider.

x = 600kmph.
Actual journey time = 600/600 = 1hr.