Speed distance & Time - CENTS INSIGHT

Speed, Distance and Time is a fundamental aptitude topic that tests how quickly you can understand and solve problems involving movement, travel, and time. It is important because questions from this topic often combine basic formulas with ratios, unit conversions, relative speed, and practical reasoning. To ace it, learn to switch between units quickly, use ratios and relative speed strategically, recognize shortcuts for common situations such as trains, boats, and average speed, and master the core relationship:

Speed = \frac{Distance}{Time}

As a result,

Distance= Speed\times Time

and,

TIme=\frac{Distance}{Speed}

For aptitude tests, however, the important skill is not simply remembering these formulas. Most questions are designed so that ratios and proportional reasoning allow you to solve them much faster.

There are certain conecpts that are used to test the understanding of the test taker. All discussed in this topic below:

1. The Inverse Relationship Between Speed and Time

The inverse relationship between speed and time is one of the most important ideas in speed, distance and time. When the distance remains constant, speed and time move in opposite directions: if speed increases, the time required decreases, and if speed decreases, the time required increases. when mathematically expressed:

Speed∝\frac{1}{Time}​

Therefore, for a fixed distance:

Speed×Time=Constant

A simple example might look like:

Suppose you have to travel 120 km.

At 40 km/h, time = $120/40 = 3$ hours
At 60 km/h, time = $120/60 = 2$ hours
At 80 km/h, time = $120/80 = 1.5$ hours

However, if the distance is the same, you don’t always need to calculate the actual distance or time.

Just directly use:

S_1T_1=S_2T_2

The above expression states the rearrangement of the original formula when the distance is same for the two different situations:

As,\ Distance=Speed\times TIme \rightarrow D=S_1T_!

Second situation:

D=S_2T_2

As both the situations have same distance, therefore:

S_1T_1=S_2T_2

Let’s do an example:

For example, a car takes 5 hours to complete a journey at 60 km/h. If it travels the same journey at 75 km/h, how long will it take?

Using:\quad S_1​T_1​=S_2​T_2

60(5)=75(T_2)

300=75T_2​

T_2​=4 hours

2. Percentage Change in Speed and Time

Because speed and time are inversely related, their percentage changes are not equal.

If speed increases by x%, time decreases by:

TIme \ decrease=\frac{x}{100+x}\times100\%

If speed decreases by x%, time increases by:

Time \ increase=\frac{x}{100-x}\times100\%

A simple example would be:

A person’s speed increases by 25%. By what percentage does the time required for the same journey decrease?

TIme \ decrease=\frac{25}{100+25}\times100\%

=20\%

3. Average speed

The formula:

Average\ Speed=\frac{Total\ Distance}{Total\ Time}

Do not simply average the speeds unless the time intervals are equal. If equal distances are travelled at speeds a and b:

Average \ speed=\frac{2ab}{a+b}

For example:

A car travels from A to B at 60 km/h and returns at 40 km/h. What is its average speed?

As the distances are equal so:

Average\ speed=\frac{2(60)(40)}{60+40}

\frac{4800}{100}

48km/h

However, there is an important distinction between the two means approach:

Equal distances → use harmonic mean:

\frac{2ab}{a+b}

Equal time → arithmetic mean:

\frac{a+b}{2}

4. Relative speed

Let’s understand this with an example

Two cyclists are 150 km apart and travel toward each other at 40 km/h and 35 km/h. How long will they take to meet?

Because they are travelling toward each other, we might use the formula of relative speed for objects moving towards eachother:

Speed_{Relative}\ or\ S_R=Speed_1+Speed_2

S_R=40+35=75\ km/h

Therefore:

\frac{150}{75}=2\ hours

Instead of calculating how far each cyclist travels separately, imagine one cyclist moving toward the other at their combined speed of 75 km/h. The 150 km gap is therefore closed at 75 km/h.

Similarly, for objects moving in the same direction, relative speed is found by subtracting their speeds.

For example:

A car is travelling at 80 km/h, while a bus ahead of it is travelling at 60 km/h in the same direction. If the car is initially 100 km behind the bus, how long will it take for the car to catch the bus?

Since both are moving in the same direction, we might subtract the speeds:

Speed_{Relative}=80-60=20\ km/h

The car is 100 km behind, so:

Time=\frac{Distance}{Relative\ speed}

=\frac{100}{20}

5\ hours

So the important thing to remember is:

Moving towards each other → Add speeds
Moving in the same direction → Subtract speeds

5. Length from speed

Certain questions require to calculate the total distance while using the relative speeds.

For example:

A 150 m train travels at 72 km/h and takes 20 seconds to completely cross a platform. Find the platform’s length.

As first step convert the speed:

72\text{km/h}=20\text{ m/s}

Distance travelled in 20 seconds:

20(20)=400\text{ m}

This 400 m consists of:

\text{Train length + Platform length}

Therefore:

400=150+L

L=250\ m

Practice questions