Radha and Hema are neighbors and study in the same school. Both of them use bicycles to go to the school. Radha’s speed is 8 km/hr. whereas Hema’s speed is 10 km/hr. Hema takes 9 minutes less than Radha to reach the school. How far is the school from the locality of Radha and Hema?
6 km
Calculating Distance Using Speed and Time Difference
This problem asks us to find the distance from Radha and Hema's locality to their school, given their cycling speeds and the time difference it takes them to reach the school. This is a classic distance, speed, and time problem.
Let's break down the information given:
- Radha's speed (\( S_R \)) = 8 km/hr
- Hema's speed (\( S_H \)) = 10 km/hr
- Hema takes 9 minutes less than Radha to reach the school.
We need to find the distance to the school. Let's assume the distance from the locality to the school is \( D \) kilometers. Both Radha and Hema travel this same distance \( D \).
Setting Up the Equations for Distance, Speed, and Time
The fundamental relationship between distance, speed, and time is:
\( \text{Distance} = \text{Speed} \times \text{Time} \)
From this, we can express time as:
\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
Using this formula, we can write the time taken by Radha (\( T_R \)) and Hema (\( T_H \)) in terms of the distance \( D \):
- Time taken by Radha: \( T_R = \frac{D}{S_R} = \frac{D}{8} \) hours
- Time taken by Hema: \( T_H = \frac{D}{S_H} = \frac{D}{10} \) hours
The problem states that Hema takes 9 minutes less than Radha. The speeds are given in km/hr, so we should convert the time difference to hours.
9 minutes = \( \frac{9}{60} \) hours = \( \frac{3}{20} \) hours.
Since Hema takes less time, the difference in time is Radha's time minus Hema's time:
\( T_R - T_H = \frac{3}{20} \)
Solving for the Distance to the School
Now we can substitute the expressions for \( T_R \) and \( T_H \) into the equation:
\( \frac{D}{8} - \frac{D}{10} = \frac{3}{20} \)
To solve for \( D \), we need to find a common denominator for the fractions. The least common multiple (LCM) of 8, 10, and 20 is 40.
Multiply every term in the equation by 40:
\( 40 \times \left(\frac{D}{8}\right) - 40 \times \left(\frac{D}{10}\right) = 40 \times \left(\frac{3}{20}\right) \)
Simplify each term:
- \( 40 \times \left(\frac{D}{8}\right) = \frac{40D}{8} = 5D \)
- \( 40 \times \left(\frac{D}{10}\right) = \frac{40D}{10} = 4D \)
- \( 40 \times \left(\frac{3}{20}\right) = \frac{40 \times 3}{20} = 2 \times 3 = 6 \)
So the equation becomes:
\( 5D - 4D = 6 \)
Combine the terms with \( D \):
\( D = 6 \)
The distance \( D \) is 6 kilometers.
Verifying the Distance Calculation
Let's check if this distance satisfies the condition about the time difference.
- If the distance is 6 km, Radha's time \( T_R = \frac{6 \text{ km}}{8 \text{ km/hr}} = \frac{6}{8} \) hours = \( \frac{3}{4} \) hours.
In minutes, \( T_R = \frac{3}{4} \times 60 \) minutes = 45 minutes. - If the distance is 6 km, Hema's time \( T_H = \frac{6 \text{ km}}{10 \text{ km/hr}} = \frac{6}{10} \) hours = \( \frac{3}{5} \) hours.
In minutes, \( T_H = \frac{3}{5} \times 60 \) minutes = 36 minutes.
The difference in time is \( T_R - T_H = 45 \text{ minutes} - 36 \text{ minutes} = 9 \text{ minutes} \). This matches the information given in the problem.
Therefore, the distance from the locality of Radha and Hema to the school is 6 km.
Revision Table: Key Concepts
Reviewing the concepts used in this distance, speed, time problem.
- Distance: The total length covered during motion. Measured in units like kilometers (km) or meters (m).
- Speed: The rate at which an object moves. It is the distance covered per unit of time. Measured in units like km/hr or m/s. \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
- Time: The duration of the motion. Measured in units like hours (hr) or seconds (s). \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
- Units Conversion: Important to use consistent units for distance, speed, and time when solving problems. Minutes often need to be converted to hours or seconds depending on the speed units.
Additional Information: Solving Related Speed Problems
Understanding how to handle speed, distance, and time is crucial. Here are some related types of problems:
- Same Direction Travel: When two objects move in the same direction, the relative speed is the difference between their speeds. This is useful for finding the time taken for one object to overtake another.
- Opposite Direction Travel: When two objects move towards each other (or away from each other) in opposite directions, the relative speed is the sum of their speeds. This helps calculate the time until they meet or the distance between them after a certain time.
- Average Speed: The total distance traveled divided by the total time taken. Note that it is not simply the average of different speeds if the time or distance for each part of the journey is different.
- Problems with Ratios: Sometimes the speeds or times are given as ratios, requiring setting up proportional equations.
Practice with different variations of these problems will improve your understanding of distance, speed, and time calculations.
Similar Questions
A passenger train departs from Delhi at 6 p.m. for Mumbai. At 9 p.m., an express train, whose average speed exceeds that of the passenger train by 15 km/hour leaves Mumbai for Delhi. Two trains met each other mid-route. At what time do they meet, given that the distance between the cities is 1080 km?
Two men, A and B run a 4 km race on a course 0.25 km round. If their speeds are in the ratio 5 : 4, how often does the winner pass the other?
X, Y and Z travel from the same place with uniform speeds 4 km/hr, 5 km/hr and 6 km/hr respectively. Y starts 2 hours after X. How long after Y must Z start in order that they overtake X at the same instant?
A car did a journey in t hours. Had the average speed been x kmph greater, the journey would have taken y hours less. How long was the journey?
In a race of 1000 m, A beats B by 150 m, while in another race of 3000 m, C beats D by 400 m. Speed of B is equal to that of D. (Assume that A, B, C and D run with uniform speed in all the events). If A and C participate in a race of 6000 m, then which one of the following is correct?
A thief is spotted by a policeman from a distance of 100 m. When the policeman starts the chase, the thief also starts chasing. If the speed of the thief is 8 km/hr and that of the policeman is 10 km/hr, then how far will the thief have to run before he is overtaken?
When the speed of a train is increased by 20%, it takes 20 minutes less to cover the same distance. What is the time taken to cover the same distance with the original speed?
The speeds of three cars are in the ratio 2 : 3 : 4. What is the ratio between the time taken by these cars to travel the same distance?
A bike consumes 20 mL of petrol per kilometre, if it is driven at a speed in the range of 25 – 50 km/hour and consumes 40 mL of petrol per kilometre at any other speed. How much petrol is consumed by the bike in travelling a distance of 50 km, if the bike is driven at a speed of 40 km/hour for the first 10 km, at a speed of 60 km/hour for the next 30 km and at a speed of 30 km/hour for the last 10 km?
In a race of 1000 m, A beats B by 100 m or 10 seconds. If they start a race of 1000 m simultaneously from the same point and if B gets injured after running 50 m less than half the race length and due to which his speed gets halved, then by how much time will A beat B?
Important Questions from Relative Speed
In a circular race of 2500 m, a man and a woman start from a point towards opposite directions with speeds of 37 km/h and 35 km/h, respectively. After how much time from the start of the race will they meet for the first time?
- X and Yrun a 3 km race along a circular course of length 300 m. Their speeds are in the ratio 3 : 2. If they start together in the same direction, how many times would the first one pass the other (the start-off is not counted as passing)?
A train of length 600 meters passes another train of length 1000 meters moving in the opposite direction in 96 seconds. If the speed of both the trains is the same, then what is the speed of each train?
A train, running at a speed of 36 kmph, takes 60 seconds to cross an electric pole. How much time it will take to cross a platform of length 250 meter?
A truck covered a distance of 780 km in 12 hours. What distance would a car cover in 11 hours travelling at a speed which is seven kmph more than that of the truck?
