Speed, Time and Distance | Quantitative Reasoning for UCAT PDF Download

Introduction & Concept

The topic of Speed, Distance, and Time holds significant importance in the Mathematics or Quants section of competitive exams. It is widely utilized to address various types of problems encompassing motion in a straight line, circular motion, boats and streams, races, clocks, and more. Aspiring candidates should strive to comprehend the interconnectedness among the variables of speed, distance, and time.

Relationship Between Speed, Time & Distance

The equation Speed = Distance/Time illustrates the rate at which an object moves, indicating the distance traveled divided by the time taken to cover that distance. This equation reveals that speed is directly proportional to distance and inversely proportional to time. Consequently, we can derive two additional formulas: 

• Distance = Speed x Time and 
• Time = Distance / Speed. 

It is important to note that as speed increases, the time taken to cover a given distance decreases, and vice versa.
By utilizing these formulas, one can solve various basic problems related to speed, distance, and time. However, it is crucial to pay attention to the proper usage of units while applying these formulas. Units play a significant role in ensuring accurate calculations and meaningful results. Therefore, being mindful of units and making appropriate conversions, when necessary, is vital in these calculations.

Units of Speed Time & Distance

The units of speed, time, and distance commonly used in the context of speed, distance, and time calculations are as follows:

• Speed: In the International System of Units (SI), the unit of speed is meters per second (m/s).
  Other commonly used units of speed include kilometers per hour (km/h), miles per hour (mph), feet per second (ft/s), and knots (nautical miles per hour).
• Time: The unit of time in SI is seconds (s).
  Other units of time that are frequently used include minutes (min), hours (h), and days (d).
• Distance: The unit of distance in SI is meters (m).
  Other commonly used units of distance include kilometers (km), miles (mi), feet (ft), and nautical miles (nmi).

It is essential to ensure consistency in the units used when working with speed, time, and distance formulas. If necessary, conversions between units can be performed to ensure the compatibility of units in calculations.

Speed, Time & Distance Conversions

To convert from kilometers per hour (km/h) to meters per second (m/s), you multiply by 5/18. Therefore, 1 km/h = 5/18 m/s.
To convert from meters per second (m/s) to kilometers per hour (km/h), you multiply by 18/5. So, 1 m/s = 18/5 km/h = 3.6 km/h.
Additionally, you mentioned the following conversions:

• 1 km/h = 5/8 miles/hour
• 1 yard = 3 feet
• 1 kilometer = 1000 meters = 0.6214 mile
• 1 mile = 1.609 kilometers
• 1 hour = 60 minutes = 60 * 60 seconds = 3600 seconds
• 1 mile = 1760 yards
• 1 mph = (1 x 1760) / (1 x 3600) = 22/45 yards/sec
• 1 mph = (1 x 5280) / (1 x 3600) = 22/15 ft/sec

It's important to be familiar with these conversions and use them appropriately when needed.
Lastly, regarding the ratio of speeds and times, you're correct. If the ratio of speeds is a : b, then the ratio of times taken to cover the same distance will be b : a, and vice versa. This principle allows for comparisons and calculations involving different speeds and times.

Remember to apply these conversions and principles accurately to ensure correct calculations in problems involving speed, time, and distance.

Application of Speed, Time & Distance

1. Average Speed:   Average Speed = (Total distance traveled)/(Total time taken)
Case 1 - When the distance is constant:
The formula for average speed when the distance is constant is given by:
Average speed = 2xy / (x + y)
where x and y are the two speeds at which the same distance has been covered.
In this case, if an object covers a fixed distance at two different speeds, x and y, the average speed can be calculated using this formula.

Case 2 - When the time taken is constant:
The formula for average speed when the time taken is constant is given by:
Average speed = (x + y) / 2
where x and y are the two speeds at which the object traveled for the same time.
In this scenario, if an object spends an equal amount of time traveling at two different speeds, x and y, the average speed can be calculated using this formula.
These formulas are useful in situations where either the distance or the time remains constant. They allow us to determine the average speed based on the given speeds in those specific cases.

Remember to use the appropriate formula depending on the scenario you are dealing with to find the average speed accurately.

Example: Tom travels from City A to City B at a speed of 60 km/h, and then he returns from City B to City A at a speed of 40 km/h. What is Tom's average speed for the entire round trip?

In this scenario, we have two different speeds, 60 km/h and 40 km/h, and the distance traveled in each direction is the same.
To find the average speed, we can use the formula for Case 1 - When the distance is constant:
Average speed = 2xy / (x + y)
Here, x = 60 km/h and y = 40 km/h.
Average speed = 2 x 60  x  40 / (60 + 40)
Average speed = 4800 / 100
Average speed = 48 km/h
Therefore, Tom's average speed for the entire round trip is 48 km/h.
Note that in this case, the average speed is lower than the speeds at which Tom traveled individually since he spent more time at the slower speed (40 km/h) compared to the faster speed (60 km/h).

2. Inverse Proportionality of Speed & Time
Speed is inversely proportional to Time when the Distance is constant. S is inversely proportional to 1/T when D is constant. If the Speeds are in the ratio m:n then the Time taken will be in the ratio n:m.
There are two methods to solve questions:

• Using Inverse Proportionality
• Using Constant Product Rule

Example: John is driving from Town A to Town B. If he travels at a speed of 80 km/h, he reaches Town B in 4 hours. How long will it take for John to reach Town B if he increases his speed to 100 km/h?

According to the inverse proportionality between speed and time, as the speed increases, the time taken will decrease.
Let's denote the initial time taken as T₁ (when John is traveling at 80 km/h) and the time taken at the increased speed as T₂ (when John is traveling at 100 km/h).
We can set up the following proportion to represent the relationship between speed and time:
80 km/h / T₁ = 100 km/h / T2
To find the value of T₂, we can rearrange the equation:
T₂ = (T₁ x 100 km/h) / 80 km/h
Given that T1 = 4 hours (the time taken at 80 km/h):
T₂ = (4 hours x 100 km/h) / 80 km/h
T₂ = (400 km/h) / 80 km/h
T₂ = 5 hours
Therefore, if John increases his speed to 100 km/h, it will take him 5 hours to reach Town B.
This example showcases the inverse proportionality between speed and time. As the speed increases, the time taken decreases, maintaining the overall distance traveled constant.

3. Meeting Point Questions

• When two individuals travel from two different points, A and B, towards each other and eventually meet at a point P, the total distance covered by both individuals on their way to the meeting point is equal to the distance between points A and B, denoted as d.
• Since they meet at the same time, the time taken by both individuals to reach point P is identical. As a result, the distances AP and BP are in the ratio of their respective speeds.
• In this scenario, it can be observed that when they meet for the first time, their combined distance covered is d. When they meet for the second time, the combined distance covered is 3d. Similarly, for the third time they meet, the combined distance covered is 5d, and so on.

In summary, as the individuals continue to meet multiple times while traveling towards each other, the distances covered during each meeting follow a sequence where the combined distance is an odd multiple of d (1d, 3d, 5d, etc.).

Example: Alex and Sarah start running from two different points, A and B, towards each other. Alex runs at a speed of 10 km/h, while Sarah runs at a speed of 8 km/h. The distance between points A and B is 60 kilometers. At what point will they meet?

Since they are traveling towards each other, the total distance covered by both of them on the meeting will be equal to the distance between points A and B, which is 60 kilometers.
We can determine the time taken by each person to reach the meeting point using the formula Time = Distance / Speed.
For Alex:
Time taken by Alex = Distance / Speed = 60 km / 10 km/h = 6 hours
For Sarah:
Time taken by Sarah = Distance / Speed = 60 km / 8 km/h = 7.5 hours
Since both Alex and Sarah start running at the same time, the meeting point will be reached after the same amount of time. In this case, it will be 6 hours.
To find the meeting point, we can use the formula Distance = Speed x Time.
Distance covered by Alex = Speed x Time = 10 km/h x 6 hours = 60 kilometer
Distance covered by Sarah = Speed x Time = 8 km/h x 6 hours = 48 kilometers
Therefore, Alex and Sarah will meet at a point that is 48 kilometers from point A and 12 kilometers from point B.
In summary, Alex and Sarah will meet 48 kilometers away from point A and 12 kilometers away from point B after 6 hours of running.
Meeting point questions involve determining the point at which two individuals or objects meet while traveling towards each other. By considering their speeds, distances, and the time taken, we can calculate the meeting point accurately.

Sample Questions on Speed, Time & Distance

Q.1. A train travels a distance of 300 kilometers in 4 hours. What is its average speed?

Average speed = Total distance / Total time
Average speed = 300 km / 4 hours
Average speed = 75 km/h
Answer: The average speed of the train is 75 km/h.

Q.2. Mary can cycle at a speed of 12 km/h. How long will it take her to cycle a distance of 48 kilometers?

Time = Distance / Speed
Time = 48 km / 12 km/h
Time = 4 hours
Answer: It will take Mary 4 hours to cycle a distance of 48 kilometers.

Q.3. A car covers a distance of 450 miles in 6 hours. What is its average speed in miles per hour?

Average speed = Total distance / Total time
Average speed = 450 miles / 6 hours
Average speed = 75 miles/hour
Answer: The average speed of the car is 75 miles per hour.

Q.4. A bus travels from Town A to Town B, a distance of 180 kilometers, at an average speed of 60 km/h. How long will it take for the bus to reach Town B?

Time = Distance / Speed
Time = 180 km / 60 km/h
Time = 3 hours
Answer: It will take the bus 3 hours to reach Town B.

Q.5. Rachel runs at a speed of 8 m/s. How far can she run in 40 seconds?

Distance = Speed × Time
Distance = 8 m/s × 40 s
Distance = 320 meters
Answer: Rachel can run a distance of 320 meters in 40 seconds.