TIME,SPEED & DISTANCE
Note: Question numbers are numbers from the actual exam in the respective years mentioned below
- In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time?(2015)
(A) 25:21
(B) 25: 20
(C) 5:3
(D) 5:7
Answer-A
Let's denote the speed of A as Sa and the speed of B as Sb.
Given:
Distance of the race = 500 meters
Initially, B starts 45 meters ahead of A.
When A finishes the race, B is still 35 meters behind.
To find the ratio of the speeds of A to B, we need to determine the time it takes for A and B to complete the race.
Let's calculate the time taken by A and B:
Distance traveled by A = 500 meters
Distance traveled by B = 500 meters - (45 meters + 35 meters) = 420 meters
Time taken by A:
Time = Distance / Speed
500 meters / Sa = Time
Time taken by B:
Time = Distance / Speed
420 meters / Sb = Time
Since both A and B start at the same time, their respective times will be equal:
500 / Sa = 420 / Sb
500 * Sb = 420 * Sa
Sa / Sb = 500 / 420
Sa / Sb = 25 / 21
Therefore, the correct option is (a) 25:21.
- Two cities A and B are 360 km apart. A car goes from A to B with a speed of 40 km/hr and returns to A with a speed of 60 km/hr. What is the average speed of the car?(2015)
(A) 45 km/hr
(B) 48 km/hr
(C) 50 km/r
(D) 55 km/hr
Answer-B
Average Speed = Total Distance / Total Time
Let's calculate the total distance covered by the car and the total time taken.
The car travels from city A to city B with a speed of 40 km/hr. The distance between the two cities is 360 km. Therefore, the time taken for this journey is:
Time = Distance / Speed = 360 km / 40 km/hr = 9 hours
The car returns from city B to city A with a speed of 60 km/hr. The distance is the same, i.e., 360 km. The time taken for this journey is:
Time = Distance / Speed = 360 km / 60 km/hr = 6 hours
Now, let's calculate the total distance covered by the car:
Total Distance = Distance to B + Distance to A = 360 km + 360 km = 720 km
Total Time = Time to B + Time to A = 9 hours + 6 hours = 15 hours
Finally, let's calculate the average speed:
Average Speed = Total Distance / Total Time = 720 km / 15 hours = 48 km/hr
Therefore, the average speed of the car is (b) 48 km/hr.
- Four friends A, B, C and D need to cross a bridge. A maximum of two persons can cross it at a time. It is night and they just have one lamp. Persons that cross the bridge must carry the lamp to find the way. A pair must walk together at the speed of a slower person. After crossing the bridge, the person having faster speed in the pair will return with the lamp each time to accompany another person in the group. Finally, the lamp has to be returned at the original place and the person who returns the lamp has to cross the bridge again without the lamp. To cross the bridge, the time taken by them is as follows: A: 1 minute, B: 2 minutes, C: 7 minutes and D: 10 minutes.
What is the total minimum time required by all the friends to cross the bridge?(2016)
(A) 23 minutes
(B) 22 minutes
(C) 21 minutes
(D) 20 minutes
Answer-A
To minimize the total time required for all friends to cross the bridge, we need to find the optimal pairing and sequence of crossings. Let's analyze the given times and solve the problem step by step.
The two fastest friends, A (1 minute) and B (2 minutes), should cross the bridge together first. This takes 2 minutes.
B (2 minutes) returns with the lamp, which takes an additional 2 minutes.
The two slowest friends, C (7 minutes) and D (10 minutes), should cross the bridge together next. This takes 10 minutes.
A (1 minute) takes the lamp and returns with C (7 minutes), which takes 7 minutes.
Finally, A (1 minute) and B (2 minutes) cross the bridge together, taking 2 minutes.
Adding up the individual times, the total minimum time required for all friends to cross the bridge is 2 + 2 + 10 + 7 + 2 = 23 minutes.
Therefore, the correct option is (a) 23 minutes.
- A class starts at 11:00 am and lasts till 2:27 pm. Four periods of equal duration are held during this interval. After every period, a rest of 5 minutes is given to the students. The exact duration of each period is :(2016)
(A) 48 minutes
(B) 50 minutes
(C) 51 minutes
(D) 53 minutes
Answer-A
The class starts at 11:00 am and lasts till 2:27 pm, which means the total duration is 3 hours and 27 minutes.
During this interval, there are four periods of equal duration, and after each period, a rest of 5 minutes is given.
Let's denote the duration of each period as "x" minutes.
Considering the time taken by the periods and the rest intervals, we can set up the following equation:
4x + 5(4-1) = 3 hours and 27 minutes
4x + 15 = 3 hours and 27 minutes
To convert the hours and minutes to minutes:
3 hours = 3 * 60 = 180 minutes
27 minutes = 27 minutes
So, the equation becomes:
4x + 15 = 180 + 27
4x + 15 = 207
Now, subtract 15 from both sides of the equation:
4x = 207 - 15
4x = 192
Divide both sides by 4 to solve for x:
x = 192 / 4
x = 48
Therefore, the exact duration of each period is 48 minutes.
Hence, the correct option is (a) 48 minutes.
- A and B walk around a circular park. They start at 8 a.m. from the same point in the opposite directions. A and B walk at a speed of 2 rounds per hour and 3 rounds per hour respectively. How many times shall they cross each other after 8.00 a.m. and before 9.30. a.m.?(2016)
(A) 7
(B) 6
(C) 5
(D) 8
Answer-A
Let's first calculate the time it takes for A and B to complete one round of the circular park.
A walks at a speed of 2 rounds per hour, which means it takes him 1/2 hour (or 30 minutes) to complete one round.
B walks at a speed of 3 rounds per hour, which means it takes him 1/3 hour (or 20 minutes) to complete one round.
Now, let's calculate the time between 8:00 a.m. and 9:30 a.m., which is 1 hour and 30 minutes.
During this time, A and B will cross each other whenever their combined distance traveled is equal to one complete round (360 degrees).
The relative speed between A and B is 2 + 3 = 5 rounds per hour.
Let's calculate the number of times they cross each other:
Total time between 8:00 a.m. and 9:30 a.m. = 1 hour + 30 minutes = 1.5 hours
Number of times they cross each other = (Total time * Relative speed) / Complete rounds
= (1.5 * 5) / 1
= 7.5
Since they can only cross each other as whole rounds, the actual number of times they cross each other will be the integer part of 7.5, which is 7.
Therefore, the correct option is (a) 7.
- In a race, a competitor has to collect 6 apples which are kept in a straight line on a track and a bucket is placed at the beginning of the track which is a starting point. The condition is that the competitor can pick only one apple at a time, run back with it and drop it in the bucket. If he has to drop all the apples in the bucket, how much total distance he has to run if the bucket is 5 meters from the first apple and all other apples are placed 3 meters apart ?(2016)
(A) 40 m
(B) 50 m
(C) 75 m
(D) 150 m
Answer-D
To calculate the total distance the competitor has to run, we need to consider the distances between each apple and the bucket.
The first apple is 5 meters away from the bucket, and all the remaining apples are placed 3 meters apart.
Let's calculate the distance for each apple:
Apple 1: 5 meters
Apple 2: 5 + 3 = 8 meters
Apple 3: 5 + 3 + 3 = 11 meters
Apple 4: 5 + 3 + 3 + 3 = 14 meters
Apple 5: 5 + 3 + 3 + 3 + 3 = 17 meters
Apple 6: 5 + 3 + 3 + 3 + 3 + 3 = 20 meters
Now, let's calculate the total distance by summing up the distances for each apple and multiplying by 2 since he has to run back and drop the apple in the bucket.
Total distance = (5 + 8 + 11 + 14 + 17 + 20) *2 = 75 *2 meters = 150m
Therefore, the total distance the competitor has to run to drop all the apples in the bucket is (d) 150 meters.
- A freight train left Delhi for Mumbai at an average speed of 40 km/hr. Two hours later, an express train left Delhi for Mumbai, following the freight train on a parallel track at an average speed of 60 km/hr. How far from Delhi would the express train meet the freight train?(2017)
(A) 480 km
(B) 260 km
(C) 240 km
(D) 120 km
Answer-C
To solve this problem, we can set up a simple equation using the concept of relative speed.
Let's assume the time it takes for the express train to meet the freight train is 't' hours.
The freight train travels for 't + 2' hours because it left two hours earlier than the express train.
Now, we can calculate the distance traveled by each train and set them equal to each other, as they meet at the same point:
Distance traveled by the freight train = Speed of the freight train × Time taken by the freight train
Distance traveled by the express train = Speed of the express train × Time taken by the express train
Since both trains meet at the same point, these distances are equal:
40 × (t + 2) = 60 × t
Now, let's solve the equation to find the value of 't':
40t + 80 = 60t
80 = 20t
t = 4
The express train will meet the freight train after 4 hours.
To find the distance from Delhi where they meet, we can substitute the value of 't' into either equation. Let's use the equation for the distance traveled by the express train:
Distance traveled by the express train = Speed of the express train × Time taken by the express train
Distance traveled by the express train = 60 × 4 = 240 km
Therefore, the express train will meet the freight train 240 km from Delhi.
The correct answer is (c) 240 km.
- Two persons, A and B, are running on a circular track. At the start, B is ahead of A and their positions make an angle of 30° at the centre of the circle. When A reaches the point diametrically opposite to his starting point, he meets B. What is the ratio of speeds of A and B, if they are running with uniform speeds?(2018)
(a) 6:5
(b) 4:3
(c) 6:1
(d) 4:2
Answer-A
Let's assume that the length of the circular track is 'L' and the speeds of A and B are 'vA' and 'vB', respectively. Since they are running with uniform speeds, we can assume that they cover equal distances in equal times.
When A reaches the point diametrically opposite to his starting point, he has covered half the circumference of the circle. This means that he has covered a distance of L/2. At the same time, B has covered a distance of L/2 - x, where 'x' is the distance by which A was behind B at the start.
Since A and B meet at this point, they have covered the same distance. Therefore, we can set up an equation based on the distances covered by A and B:
L/2 = vA × t
L/2 - x = vB × t
where 't' is the time taken by both A and B to cover the distances.
We can eliminate 't' by equating the right-hand sides of these equations:
vA = (L/2) / t
vB = (L/2 - x) / t
vA / vB = (L/2) / (L/2 - x)
We know that the angle between A and B at the start is 30 degrees. Therefore, the distance between their starting positions is L/12 (since the circumference of the circle is 2πr = L, and the angle between A and B is 30 degrees, their starting positions are L/12 apart).
Now, we can substitute this value of 'x' into the equation for the ratio of speeds:
vA / vB = (L/2) / (L/2 - x)
vA / vB = (L/2) / (L/2 - L/12)
vA / vB = 6/5
Therefore, the ratio of speeds of A and B is 6:5.
The correct answer is 6:5.
- A train 200 metres long is moving at the rate of 40 kmph. In how many seconds will it cross a man standing near the railway line?(2018)
(A) 12
(B) 15
(C) 16
(D) 18
Answer-D
To solve this problem, we need to determine the time it takes for the train to cross the man standing near the railway line.
First, let's convert the speed of the train from km/h to m/s:
40 km/h = (40 * 1000) m/3600 s = 40000/3600 m/s = 100/9 m/s
Now, we can calculate the time it takes for the train to cover a distance of 200 meters:
Time = Distance / Speed
Time = 200 meters / (100/9 m/s) = 200 * (9/100) seconds = 18 seconds
Therefore, the train will cross the man standing near the railway line in 18 seconds.
- X, Y and Z are three contestants in a race of 1000 m. Assume that all run with different uniform speeds. X gives Y a start of 40 m and X gives Z a start of 64 m. If Y and Z were to compete in a race of 1000 m, how many metres start will Y give to Z?(2019)
(A) 20
(B) 25
(C) 30
(D) 35
Answer-B
Given that X gives Y a start of 40 m and X gives Z a start of 64 m, we can set up the following equations:
Distance covered by X = 1000 m
Distance covered by Y = 1000 m - 40 m = 960 m (since X gives Y a start of 40 m)
Distance covered by Z = 1000 m - 64 m = 936 m (since X gives Z a start of 64 m)
So, when Y runs 1000 m, Z runs
= 936/960 × 1000 = 975 m
So, start Y gives to Z = 1000 - 975 = 25 m
- A man takes half the time in rowing a certain distance downstream than upstream. What is the ratio of the speed in still water to the speed of current?(2020)
(A) 1:2
(B) 2: 1
(C) 1: 3
(D) 3: 1
Answer-D
Let's assume the speed of the man in still water is 'v' and the speed of the current is 'c'.
When the man rows downstream, his effective speed is increased by the speed of the current, so his speed is 'v + c'. Similarly, when the man rows upstream, his effective speed is decreased by the speed of the current, so his speed is 'v - c'.
Given that the man takes half the time in rowing downstream than upstream, we can set up the following equation:
Time taken downstream = 1/2 * Time taken upstream
The time taken for rowing can be expressed as the distance divided by the speed:
Distance / (v + c) = (1/2) * (Distance / (v - c))
We can simplify this equation by canceling out the distance:
(v - c) / (v + c) = 1/2
Now, let's solve this equation to find the ratio of the speed in still water to the speed of the current:
2(v - c) = v + c
2v - 2c = v + c
v = 3c
Therefore, the ratio of the speed in still water to the speed of the current is 3:1.
The correct answer is (d) 3:1.
- A car travels from a place X to place Y at an average speed of v km/hr, from Y to X at an average speed of 2v km/hr, again from X to Y at an average speed of 3v km/hr and again from Y to X at an average speed of 4v km/hr. Then the average speed of the car for the entire journey(2020)
(A) is less than v km/hr
(B) lies between v and 2v km/hr
(C) lies between 2v and 3v km/hr
(D) lies between 3v and 4v km /hr
Answer-B
To find the average speed for the entire journey, we need to consider the total distance traveled and the total time taken.
Let's assume the distance from X to Y is D.
The time taken to travel from X to Y at a speed of v km/hr is D/v.
The time taken to travel from Y to X at a speed of 2v km/hr is D/(2v).
The time taken to travel from X to Y at a speed of 3v km/hr is D/(3v).
The time taken to travel from Y to X at a speed of 4v km/hr is D/(4v).
The total distance traveled is 2D (going from X to Y and then from Y to X).
The total time taken is D/v + D/(2v) + D/(3v) + D/(4v).
To find the average speed, we divide the total distance by the total time:
Average speed = Total distance / Total time
Average speed = 2D / (D/v + D/(2v) + D/(3v) + D/(4v))
Simplifying the expression:
Average speed = 2D / (D/v(1 + 1/2 + 1/3 + 1/4))
Average speed = 2D / (D/v(24/24 + 12/24 + 8/24 + 6/24))
Average speed = 2D / (D/v(50/24))
Average speed = (48/25)v
The average speed of the car for the entire journey is 1.92v km/h.
Therefore the average speed of the car for the entire journey,lies between v and 2v km/hr
- A person X from a place A and another person Y from a place B set out at the same time to walk towards each other. The places are separated by a distance of 15 km. X walks with a uniform speed of 1.5 km/hr and Y walks with a uniform speed of 1 km/hr in the first hour, with a uniform speed of 1.25 km/hr in the second hour and with a uniform speed of 1.5 km/hr in the third hour and so on. Which of the following is/are correct?(2021)
- They take 5 hours to meet.
- They meet midway between A and B.
Select the correct answer using the code given below:
(A) 1 only
(B) 2 only
(C) Both 1 and 2
(D) Neither 1 nor 2
Answer-C
To solve this problem, let's analyze the situation hour by hour:
In the first hour, X walks 1.5 km, and Y walks 1 km.
The remaining distance between them is 15 km - 1.5 km - 1 km = 12.5 km.
In the second hour, X walks 1.5 km, and Y walks 1.25 km.
The remaining distance between them is 12.5 km - 1.5 km - 1.25 km = 9.75 km.
In the third hour, X walks 1.5 km, and Y walks 1.5 km.
The remaining distance between them is 9.75 km - 1.5 km - 1.5 km = 6.75 km.
In the fourth hour, X walks 1.5 km, and Y walks 1.75 km.
The remaining distance between them is 6.75 km - 1.5 km - 1.75 km = 3.5 km.
In the fifth hour, X walks 1.5 km, and Y walks 2 km.
The remaining distance between them is 3.5 km - 1.5 km - 2 km = 0 km.
Now, let's analyze the conclusions:
They take 5 hours to meet.
From our analysis above, we can see that they meet in the fifth hour when the remaining distance between them becomes zero. Therefore, conclusion 1 is correct.
They meet midway between A and B.
Since the total distance between A and B is 15 km and they meet when the remaining distance is zero, it means they meet at the midpoint between A and B, which is 7.5 km from both A and B. Therefore, conclusion 2 is correct.
Therefore, the correct answer is (c) Both 1 and 2.
- X and Y run 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)?(2022)
(A) 2
(B) 3
(C) 4
(D) 5
Answer-C
To determine how many times the first runner passes the other, we need to compare their relative speeds and the number of laps they complete.
The length of the circular course is 300 meters, and the distance of the race is 3 kilometers. Therefore, the number of laps for the race is 3000 meters / 300 meters = 10 laps.
The ratio of their speeds is 3:2, so the first runner completes 3 laps while the second runner completes 2 laps in the same time.
To find the number of times the first runner passes the second, we need to compare their number of laps completed.
Let's list the number of laps completed by each runner:
First runner: 3 laps, 6 laps, 9 laps, ...
Second runner: 2 laps, 4 laps, 6 laps, ...
We can see that the first runner passes the second runner once every 3 laps.
The total number of laps for the race is 10, so the first runner passes the second runner 10 / 3 = 3.33 times.
However, since the start-off is not counted as passing, we need to round down the number of passes. So, the first runner passes the second runner three times during the race.
The correct answer is (b) 3.
Therefore, the first runner passes the second runner three times during the race.
