Distance, Time and Speed: GMAT / GRE Mathematics Tutorial Series

Solved Word Problems On Distance, Time & Speed 

Problems involving Time, Distance and Speed are solved using a simple formula:

 

Distance = Speed * Time

This also implies that:

 

    Speed = Distance / Time   

 Time = Distance / Speed

 

Let us consider some simple examples of distance, time and speed problems.

 

Example 1.  Anish walks at a speed of 4 km/h. How much time will he take to walk a distance of 20 km?

Solution:

Time taken = Distance / speed

= 20/4

= 5 hours.

 

Example 2. A motorist covers a distance of 15 miles in 2 hours. Calculate his speed.

Solution:

Speed = Distance/time

= 15/2

= 7.5 miles per hour.
 

Example 3. A truck traveling at a speed of 40mph takes 4 hours to cover a distance. What should be its speed to cover the same distance in 1.5 hours?

Solution:

Distance covered = 4*40

= 160 miles

The speed required to cover the same distance in 1.5 hours = 160/1.5

= 106.66 mph
 
Here’s another example.

 

Example 4. If Arun walks at 4 mph, he covers a certain distance. If he walks at 9 mph, he covers 7.5 miles more. What is the actual distance he covered?

This question cannot be solved with the direct application of the formula, Distance = Speed * Time or its variations. We need to carry out a few more calculations to find the actual answer.

Solution:

For questions of these types, it’s easier to arrive at a solution if the details are mentioned in a tabular form:

 

DistanceSpeedTime
d4t
d+7.59t

 

Let the distance covered be ‘d’.

Walking at 4 mph and covering a distance ‘d’ is done in a time of ‘d/4’

If he walks at 9 mph, he covers 7.5 miles more than the actual distance d, which is ‘d+7.5’.

He does this in a time of (d+7.5)/9.

Since the time is same in both the cases →

d/4 = (d+7.5)/9            →        9d = 4(d+7.5)   →        9d=4d+30        →        d = 6.

So, he covered a distance of 6 miles in 1.5 hours.
 

Example 5. A train is going at 1/3 of its usual speed and it takes an extra 30 minutes to reach its destination. Find its usual time to cover the same distance.

Solution:

Here, we see that the distance is the same. Let us assume that its usual speed is ‘s’ and time is ‘t’, then:

 

DistanceSpeedTime
dst min
dS+1/3t+30 min

 

s*t = (1/3)s*(t+30)      →        t = t/3 + 10      →        t = 15.

So the actual time taken to cover the distance is 15 minutes.

Note: Note that the time unit is in minutes. When the distance is given in miles or kilometers, time is expressed in hours. So the appropriate conversion needs to be done to arrive at the correct answer.

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Solved Word Problems on Trains

Example 1. Chicago and Washington are two stations that are 320 miles apart. A train starts at a certain time from Chicago and travels towards Washington at 70 mph. After 2 hours, another train starts from Washington and travels towards Chicago at 20 mph. At what time do they meet?

Solution:

Let the time after which they meet be ‘t’ hours.

Then the time traveled by the second train becomes ‘t-2’.

Now,

Distance covered by the first train+Distance covered by second train = 320 miles

70t+20(t-2) = 320

Solving this gives t = 4.

So the two trains meet after 4 hours.

 

Example 2. A train leaves from Faridabad station and moves at a certain speed. After 2 hours, another train leaves from the same station and moves in the same direction at a speed of 60 mph. If it catches up with the first train in 4 hours, what is the speed of the first train?

Solution:

Let the speed of the first train be ‘s’.

Distance covered by the first train in (2+4) hours = Distance covered by the second train in 4 hours

Therefore, 6s = 60*4

Solving which gives s=40.

So the slower train is moving at the rate of 40 mph. 

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Solved Word Problems On Boats / Airplanes

For problems with boats and streams

 

Upstream Boat Speed (Against Current) = Boat Speed  In Still Water – Speed Of The Stream

 

[As the stream flows against the direction of the boat, its speed has to be subtracted from the usual speed of the boat]

 

Downstream Boat Speed (along with the current) = Boat Speed In Still Water + Stream Speed

 

[As the stream flows in the same direction as the boat, speed of the stream has to be added]

 

Similarly, for airplanes traveling with/against the wind

 

Plane Speed  With Wind Direction (same direction) = Speed Of The Plane + Speed Of The Wind

Plane Speed Against The Wind (opposite direction) = Speed Of The Plane –  Speed Of Wind

 

Here are a few examples.

Example 1. A man travels at 3 mph in still water. If the current’s velocity is 1 mph, it takes 3 hours to row to a place and come back. How far is the place?

Solution:

Let the distance be ‘d’ miles.

Time taken to cover the distance upstream + Time taken to cover the distance downstream = 3

Speed upstream = 3-1 = 2 mph

Speed downstream = 3+1 = 4 mph

So, our equation would be d/2 + d/4 = 3 → solving which, we get d = 4 miles.
 

Example 2. An airplane covers a distance of 2400 kms in 4 hours when it flies in the direction of the wind and against the wind in 6 hours. What is the speed of the plane and that of the wind?

Solution:

Let the speed of the plane be ‘p’ and that of the wind be ‘w’. Our table looks like this:
 

DistanceSpeedTime
With the wind2400p+w4
Against the wind2400p-w6

 

 

4(p+w) = 2400 and 6(p-w) = 2400

Expressing one unknown variable in terms of the other makes it easier to solve, which means

p+w = 600 → w=600-p

Substituting the value of w in the second equation,

p-w = 400

p-(600-p) = 400 → p = 500

The speed of the plane is 500 km/h and that of the wind is 100 km/h.

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More solved examples on Speed, Distance and Time

Example 1. Ajay traveled by train to Ajmer which moved at the speed of 30 mph. He then boarded a bus from Ajmer, which moved at the speed of 40 mph and reached his hometown in Jaipur. The entire distance covered was 100 miles and the entire duration of the journey was 3 hours. Find the distance he traveled by bus.

Solution:

DistanceSpeedTime
Traind30t
Bus100-d403-t

 

 

Let the time taken by the train be ‘t’. Then that of the bus is ‘3-t’.

The entire distance covered was 100 miles

So, 30t + 40(3-t) = 100

Solving which gives t=2.

Substituting the value of t in 40(3-t), we get the distance traveled by bus is 40 miles.

Alternatively, we can add the time and equate it to 3 hours, which directly gives the distance.

d/30 + (100-d)/40 = 3

Solving which gives d = 60, which is the distance traveled by train. 100-60 = 40 miles is the distance traveled by bus.

 

Example 2. A plane flying to Doha covered a distance of 630 miles in 6 hours. For the first part of the trip, the average speed was 100 mph and for the second part of the trip, the average speed was 110 mph. what is the time it flew at each speed?

Solution

Our table looks like this.

 

DistanceSpeedTime
1st part of the journeyd100t
2nd part of the journey630-d1106-t

 

Assuming the distance covered in the 1st part of the journey to be ‘d’, the distance covered in the second half becomes ‘630-d’.

Assuming the time taken for the first part of the journey to be ‘t’, the time taken for the second half becomes ‘6-t’.

From the first equation, d=100t

The second equation is 630-d = 110(6-t).

Substituting the value of d from the first equation, we get

630-100t = 110(6-t)

Solving this gives t=3.

So the plane flew the first part of the journey in 3 hours and the second part in 3 hours.
 

Example 2. Two friends are walking towards each other on a walking path that is 20 miles long. One is walking at the rate of 3 mph and the other at 4 mph. After how much time will they meet each other?

Solution:

 

DistanceSpeedTime
First-persond3t
Second person20-d4t

 

Assuming the distance traveled by the first person to be ‘d’, the distance traveled by the second person is ’20-d’.

The time is ‘t’ for both of them because when they meet, they would have walked for the same time.

Since time is the same, we can equate as

d/3 = (20-d)/4

Solving this gives d=60/7 miles (8.5 miles approximately)

Then t = 20/7 hours

So the two persons meet after 2 6/7 hours.

Practice Question

1) A boat covers a certain distance in 2 hours, while it comes back in 3 hours. If the speed of the stream is 4 kmph, what is the boat’s speed in still water?

A) 30 kmph
B) 20 kmph
C) 15 kmph
D) 40kmph

Answer B.

Explanation

Let the speed of the boat be ‘s’ kmph.

Then, 2(s+4) = 3(s-4) → s=20

2) A cyclist travels for 3 hours, traveling for the first half of the journey at 12 mph and the second half at 15 mph. Find the total distance he covered.

  1. A) 30 miles
    B) 35 miles
    C) 40 miles
    D) 180 miles

Answer 2

C.

Explanation

Since it is mentioned, that the first ‘half’ of the journey is covered in 12 mph and the second in 15, the equation looks like

(d/2)/12 + (d/2)/15 = 3

Solving this gives d=40 miles

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