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?
UPSC-2019
This problem involves calculating relative speeds and distances in a race scenario. We need to determine the distances covered by each contestant when the fastest contestant (X) completes the race, and then use these relative distances to find the head start Y would give to Z in a separate race.
B) 25
Ram and Shyam work on a job together for four days and complete 60% of it. Ram takes leave then and Shyam works for eight more days to complete the job. How long would Ram take to complete the entire job alone?
UPSC-2016
This problem involves work and time. We need to determine the individual work rates of Ram and Shyam to find out how long Ram would take to complete the entire job alone. We will use the concept that work done is equal to the rate of work multiplied by the time taken.
Calculate the remaining work: Ram and Shyam complete 60% of the job together. So, the remaining work is:
\[ \text{Remaining Work} = 100\% - 60\% = 40\% \]Determine Shyam's work rate: Shyam completes the remaining 40% of the job in 8 days. So, Shyam's rate of work for the entire job can be calculated:
\[ \text{Shyam's Work Rate} = \frac{\text{Work Done}}{\text{Time Taken}} = \frac{40\%}{8 \text{ days}} = \frac{0.4}{8} = 0.05 \text{ of the job per day} \]This means Shyam completes 5% of the job per day.
Calculate the total work done by Shyam in the first 4 days: Ram and Shyam worked together for 4 days. In these 4 days, Shyam would have completed:
\[ \text{Work done by Shyam in 4 days} = \text{Shyam's Work Rate} \times \text{Time} = 0.05 \times 4 = 0.20 \text{ or } 20\% \text{ of the job} \]Calculate the work done by Ram in the first 4 days: Together, Ram and Shyam completed 60% of the job in 4 days. Since Shyam did 20% of the job in those 4 days, Ram must have done the rest:
\[ \text{Work done by Ram in 4 days} = \text{Total work in 4 days} - \text{Work done by Shyam in 4 days} \] \[ \text{Work done by Ram in 4 days} = 60\% - 20\% = 40\% \text{ of the job} \]Determine Ram's work rate: Ram completed 40% of the job in 4 days. So, Ram's rate of work is:
\[ \text{Ram's Work Rate} = \frac{\text{Work Done}}{\text{Time Taken}} = \frac{40\%}{4 \text{ days}} = \frac{0.4}{4} = 0.10 \text{ of the job per day} \]This means Ram completes 10% of the job per day.
Calculate the time Ram takes to complete the entire job alone: To complete the entire job (100%), Ram would take:
\[ \text{Time for Ram} = \frac{\text{Total Work}}{\text{Ram's Work Rate}} = \frac{100\%}{10\% \text{ per day}} = \frac{1}{0.1} = 10 \text{ days} \]Correct Option: C) 10 days
Two cars start towards each other, from two places A and B which are at a distance of 160 km. They start at the same time 08:10 AM. If the speeds of the cars are 50 km and 30 km per hour respectively, they will meet each other at
UPSC-2014
This problem involves two objects moving towards each other. We need to calculate the time it takes for them to meet and then determine the meeting time based on their starting time. The key concept here is relative speed when objects are moving in opposite directions.
Identify the given information:
Calculate the relative speed:
When two objects move towards each other, their relative speed is the sum of their individual speeds.
Calculate the time taken to meet:
Time is calculated by dividing the total distance by the relative speed.
Determine the meeting time:
The cars start at 08:10 AM and will meet after 2 hours.
Correct Option: A) 10:10 AM
A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original speed of the train in km/hr?
UPSC-2013
This problem involves calculating the original speed of a train given two segments of its journey with different speeds and distances, and the total time taken. We will use the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) and set up an equation based on the total time.
Let the original speed of the train be \(x\) km/hr.
For the first part of the journey:
For the second part of the journey:
The total time taken for the complete journey is 3 hours. So, \(T_1 + T_2 = 3\).
\[ \frac{63}{x} + \frac{72}{x+6} = 3 \]To solve for \(x\), find a common denominator and combine the fractions:
\[ \frac{63(x+6) + 72x}{x(x+6)} = 3 \] \[ 63x + 378 + 72x = 3x(x+6) \] \[ 135x + 378 = 3x^2 + 18x \]Rearrange the terms to form a quadratic equation:
\[ 3x^2 + 18x - 135x - 378 = 0 \] \[ 3x^2 - 117x - 378 = 0 \]Divide the entire equation by 3 to simplify:
\[ x^2 - 39x - 126 = 0 \]Solve the quadratic equation. We can use factorization or the quadratic formula. For factorization, we need two numbers that multiply to -126 and add to -39. These numbers are -42 and 3.
\[ (x - 42)(x + 3) = 0 \]This gives two possible values for \(x\): \(x = 42\) or \(x = -3\).
Since speed cannot be negative, we discard \(x = -3\).
Therefore, the original speed of the train is \(x = 42\) km/hr.
C) 42 — The calculation shows that the original speed of the train is 42 km/hr.
If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds, then what is the ratio of the distances travelled by them in one hour?
UPSC-2011
To find the ratio of distances travelled in one hour, we first need to calculate the speed of the bus and the train. Speed is defined as distance divided by time. Once we have their respective speeds, which represent the distance travelled in one hour, we can form the ratio.
The bus travels 160 km in 4 hours.
\[ \text{Speed of bus} = \frac{\text{Distance}}{\text{Time}} = \frac{160\ \text{km}}{4\ \text{hours}} = 40\ \text{km/hour} \]The train travels 320 km in 5 hours.
\[ \text{Speed of train} = \frac{\text{Distance}}{\text{Time}} = \frac{320\ \text{km}}{5\ \text{hours}} = 64\ \text{km/hour} \]Since speed is the distance covered per unit of time, the distance travelled by the bus in one hour is 40 km, and by the train in one hour is 64 km.
Ratio of (distance by bus in 1 hour) : (distance by train in 1 hour)
\[ \text{Ratio} = 40 : 64 \]Divide both sides by their greatest common divisor, which is 8.
\[ \text{Ratio} = \frac{40}{8} : \frac{64}{8} = 5 : 8 \]B) 5:8 is the correct ratio of the distances travelled by the bus and the train in one hour.
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?
UPSC-2015
To find the average speed, we need to calculate the total distance traveled and the total time taken for the entire journey. The average speed is then the total distance divided by the total time.
The distance between city A and city B is given as 360 km. So, the distance from A to B is 360 km, and the distance from B to A is also 360 km.
The car goes from A to B and then returns to A. Therefore, the total distance traveled is:
\[ \text{Total Distance} = \text{Distance (A to B)} + \text{Distance (B to A)} \\ \text{Total Distance} = 360 \text{ km} + 360 \text{ km} = 720 \text{ km} \]The speed from A to B is 40 km/hr.
\[ \text{Time (A to B)} = \frac{\text{Distance}}{\text{Speed}} = \frac{360 \text{ km}}{40 \text{ km/hr}} = 9 \text{ hours} \]The speed from B to A is 60 km/hr.
\[ \text{Time (B to A)} = \frac{\text{Distance}}{\text{Speed}} = \frac{360 \text{ km}}{60 \text{ km/hr}} = 6 \text{ hours} \]Average speed is total distance divided by total time.
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{720 \text{ km}}{15 \text{ hours}} = 48 \text{ km/hr} \]Alternatively, for a journey where the distance is the same for both parts (to and fro) but speeds are different, the average speed can be calculated using the formula:
\[ \text{Average Speed} = \frac{2 \times S_1 \times S_2}{S_1 + S_2} \]Where \(S_1\) is the speed in one direction and \(S_2\) is the speed in the other direction.
Given \(S_1 = 40 \text{ km/hr}\) and \(S_2 = 60 \text{ km/hr}\).
\[ \text{Average Speed} = \frac{2 \times 40 \times 60}{40 + 60} = \frac{2 \times 40 \times 60}{100} = \frac{4800}{100} = 48 \text{ km/hr} \]B) 48 km/hr
