Aptitude Time and Distance Theory

Formulas

1.Speed, Time and Distance:

Speed =$\dfrac{Distance}{Time}$

Time=$\dfrac{Distance}{Speed}$

Distance = (Speed x Time).

2.km/hr to m/sec conversion:

3.m/sec to km/hr conversion:

4.Ratio of the speeds

If the ratio of the speeds of A and B is a : b, then the ratio of the times taken by then to cover the same distance is $\dfrac{1}{a}:\dfrac {1}{b}$

5.Average speed

Suppose a man covers a certain distance at x km/hr and an equal distance at y km/hr. Then,the average speed during the whole journey is $\dfrac{2xy}{x + y}$km/hr.

Basic Concepts of Time and Distance :

Most of the aptitude questions on time and distance can be solved if you know the basic correlation between speed, time and distance which you have learnt in your high school class.

Relation between time, distance and speed:

Speed is distance covered by a moving object in unit time.

Speed=$\dfrac{Distance}{Time}$

so,Time=$\dfrac{Distance}{Speed}$

Distance = (Speed x Time).

Unit conversion:

While answering multiple choice time and dinstance problems in quantitative aptitude test, double check the units of values given.
It could be in m/s or km/h.
You can use the following formula to convert from one unit to other

x km/hr = x$\dfrac{5}{18}$ m/s

x m/s = x$\dfrac{18}{5}$ km/hr

1.Given One Distance and One Time Period

Given information:
*the total distance covered by one person or vehicle
*the total time it took that person or vehicle to cover the distance.

If Ben traveled 150 miles in 3 hours, what was his speed?

Set up the formula for speed. The formula is Speed=$\dfrac{Distance}{Time}$

substitute the distance into the formula. Remember to substitute for the variable distace.
For example , if Ben drives 150 total miles, your formula will look like this:
Speed=$\dfrac{150}{Time}$

substitute the time into the formula. Remember to substitute for the variable time.
For example, if Ben drives for 3 hours, your formula will look like this:
Speed=$\dfrac{150}{3}$

Divide the distance by the time. This will give you the average speed per unit of time, usually hour.
Speed=$\dfrac{150}{3}$
S=50
So, if Ben traveled 150 miles in 3 hours, his average speed is 50 miles per hour.

Solved Examples - Easy :

A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour?
solution :
Speed =$\left(\dfrac{600}{5 \times 60}\right)$m/sec.

= 2 m/sec.
Converting m/sec to km/hr
=$\left(2 \times \dfrac{18}{5}\right)$km/hr

= 7.2 km/hr.

Exercise:

2.Given Multiple Distances in Different Amounts of Time

Average Speed

Average speed is always equal to total distance travelled to total time taken to travel that distance.
Average speed=$\dfrac{Total distance}{Total time}$

Given Information :
*multiple distances that were traveled
*the amount of time it took to travel each of those distances.

If Ben traveled 150 miles in 3 hours, 120 miles in 2 hours, and 70 miles in 1 hour, what was his average speed for the entire trip?

Set up the formula for average speed. The formula is Average Average speed=$\dfrac{Total distance}{Total time}$

Determine the total distance. To do this, add up the number of miles traveled during the entire trip. Substitute this value for distance in the formula.
For example, If Ben traveled 150 miles, 120 miles, and 70 miles, you would determine the total speed by adding the three distances together:
150+120+70=340. So, your formula will look like this: Average speed=$\dfrac{340}{Total time}$.

Determine the total time. To do this, add up the times, usually hours, that were spent traveling. Substitute this value for Total time in the formula.
For example, If Ben for 3 hours, 2 hours, and 1 hour, you would determine the total time by adding the three times together: 3+2+1=6. So, your formula will look like this: Average speed=$\dfrac{340}{6}$.

Divide the total distance traveled by the total time spent traveling. This will give you your average speed.
For example:
Average speed=$\dfrac{340}{6}$.
Average speed==56.67
So if Ben traveled 150 miles in 3 hours, 120 miles in 2 hours, and 70 miles in 1 hour, his average speed was about 57 mph.

Solved Examples - Easy :

Steve wants to walk 30 km in 3 hours,20 km in 2 hours,10 km in 1 hours. what was his average speed?

solution:
Average speed=$\dfrac{Total distance}{Total time}$
Determine the total distance : 30+20+10=60
Determine the total time :3+2+1=6
Average speed=$\dfrac{60}{6}$.
Average speed=10km/h

Exercise:

3.Given Multiple Speeds for Different Amounts of Time

Given Information :
multiple speeds used to travel
the amount of time each of those speeds was traveled for.

If Ben traveled 50 mph for 3 hours, 60 mph for 2 hours, and 70 mph for 1 hour, what was his average speed for the entire trip?

Set up the formula for average speed. The formula is average speed=$\dfrac{Total distance}{Total time}$

Determine the total distance. To do this, separately multiply each speed by each time period. This will give you the distance traveled for each section of the trip. Add up these distances. Substitute this sum for Total distance in the formula.
For example:
50 mph for 3 hours = 50 * 3=150miles
60 mph for 2 hours = 60 * 2=120miles
70 mph for 1 hour =70 * 1=70miles
So, the total distance is 150+120+70=340miles.
So, your formula will look like this:
average speed=$\dfrac{340}{Total time}$

Determine the total time. To do this, add up the times, usually hours, that were spent traveling. Substitute this value for Total time in the formula.
For example, If Ben for 3 hours, 2 hours, and 1 hour, you would determine the total time by adding the three times together: 3+2+1=6. So, your formula will look like this:average speed=$\dfrac{340}{6}$

Divide the total distance traveled by the total time spent traveling. This will give you your average speed.
For example:
average speed=$\dfrac{340}{6}$
average speed=56.67.
So if Ben traveled 50 mph for 3 hours, 60 mph for 2 hours, and 70 mph for 1 hour, his average speed was about 57 mph.

Solved Examples - Easy :

John drove for 3 hours at a rate of 50 miles per hour and for 2 hours at 60 miles per hour. What was his average speed for the whole journey?
As we know that, Distance = Speed × Time

So, in 3 hour, distance covered = 50 × 3 = 150 miles

In next 2 hours, distance covered = 60 × 2 = 120 miles

Total distance covered = 150 + 120 = 270 miles

Total Time = 3 + 2 = 5 hrs

We know that,Average speed=$\dfrac{Total distance}{Total time}$

⇒ Avg. Speed = 270/5 = 54 mph
The average speed is 54 miles per hour.

Exercise:

4.Given Two Speeds for Half the Time

Given Information :
*two or more different speeds
*that those speeds were traveled for the same amount of time.

if Ben drives 40 mph for 2 hours, and 60 mph for another 2 hours, what is his average speed for the entire trip?

Set up the formula for average speed given two speeds used for the same amount of time. The formula is S=$\dfrac {a+b}{2}$, where S equals the average speed, a equals the speed for the first half of the time, and b equals the speed for the second half of the time.
*In these types of problems, It doesn’t matter for how long each speed is driven, as long as each speed is used for half the total duration of time.
*You can modify the formula if you are given three or more speeds for the same amount of time. For example, S=$\dfrac {a+b+c}{3}$ or S=$\dfrac {a+b+c+d}{4}$. As long as the speeds were used for the same amount of time, your formula can follow this pattern.

substitute the speeds into the formula. It doesn’t matter which speed you substitute for "a" and which you substitute for "b".
For example, if the first speed is 40 mph, and the second speed is 60 mph, your formula will look like this: S=$\dfrac {40+60}{2}$.

Add the two speeds together. Then, divide the sum by two. This will give you the average speed for the entire trip.
For example:
S=$\dfrac {40+60}{2}$
S=$\dfrac {100}{2}$
S=50
So, if Ben traveled 40 mph for 2 hours, then 60 mph for another 2 hours, his average speed is 50 mph.

5.Given Two Speeds for Half the Distance

Given Information :
*two different speeds
*that those speeds were used for the same distance.
For example
if Ben drives the 160 miles to the waterpark at 40 mph, and returns the 160 miles home driving 60 mph, what is his average speed for the entire trip?

Set up the formula for average speed given two speeds used for the same distance. The formula is S=$\dfrac {2ab}{a+b}$, where S equals the average speed, "a" equals the speed for the first half of the distance, and {\displaystyle b} equals the speed for the second half of the distance.
*Often problems requiring this method will involve a question about a return trip.
*In these types of problems, it doesn’t matter how far each speed is driven, as long as each speed is used for half the total distance.
*You can modify the formula if given three speeds for the same distance. For example, S=$\dfrac {3abc}{ab+bc+ca}$

substitute the speeds into the formula. It doesn’t matter which speed you substitute for "a" and which you substitute for "b".
For example, if the first speed is 40 mph, and the second speed is 60 mph, your formula will look like this: S=$\dfrac {(2)(40)(60)}{40+60}$.

Multiply the product of the two speeds by 2. This number should be the numerator of your fraction.
For example:
S=$\dfrac {(2)(40)(60)}{40+60}$
S=$\dfrac {4800}{40+60}$

Add the two speeds together. This number should be the denominator of your fraction.
For example:
S=$\dfrac {4800}{40+60}$
S=$\dfrac {4800}{100}$

Simplify the fraction. This will give you the average speed for the entire trip.
For example:
S=$\dfrac {4800}{100}$
S=48. So, if Ben drives 40 mph for 160 miles to the waterpark, then 60 mph for the 160 miles home, his average speed for the trip is 48 mph.

6.Relative Speed

If two objects are moving in same direction with speeds a and b then their relative speed is |a−b|
If two objects are moving is opposite direction with speeds a and b then their relative speed is (a+b)

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