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?
UPSC-2022
To determine how many times the faster runner (X) passes the slower runner (Y) on a circular track, we need to find the relative speed and the time it takes for X to gain one full lap on Y. The total distance of the race is irrelevant to the number of times one passes the other, as long as the race continues until the passes occur. The key is the relative distance covered per lap.
Identify Given Information:
Determine Relative Speed:
Let the speeds of X and Y be \(3v\) and \(2v\) respectively. Since they are running in the same direction, their relative speed is the difference between their individual speeds.
Calculate Time for X to gain one lap on Y:
X passes Y when X gains one full lap (300 m) on Y. The time taken for X to gain one lap is:
Calculate Total Time for the Race:
The race is 3 km (3000 m). X is the faster runner, so X will finish the race first. We calculate the time taken by X to complete the race.
Determine Number of Passes:
The number of times X passes Y is the total race time divided by the time it takes for X to gain one lap on Y.
B) The number of times X passes Y is 3.
A man completes 7/8 of a job in 21 days. How many more days will it take him to finish the job if quantum of work is further increased by 50%?
UPSC-2021
This problem involves work and time. We need to calculate the remaining work and then the additional work due to the increase in quantum. Finally, we calculate the time required to complete this new total remaining work.
Calculate the total work:
A man completes \(\frac{7}{8}\) of a job in 21 days.
This means that for \(\frac{7}{8}\) of the work, he takes 21 days.
So, for the full job (1 unit of work), the time taken would be:
\[ \text{Time for full job} = 21 \text{ days} \times \frac{8}{7} = 3 \times 8 = 24 \text{ days} \]Calculate the remaining work initially:
The work completed is \(\frac{7}{8}\).
The remaining work is \[1 - \frac{7}{8} = \frac{1}{8}\].
Calculate the time needed for the initial remaining work:
If the full job takes 24 days, then \(\frac{1}{8}\) of the job would take:
\[ \text{Time for remaining work} = \frac{1}{8} \times 24 \text{ days} = 3 \text{ days} \]Calculate the new quantum of work:
The quantum of work is further increased by 50%. This means the original full job is now 1.5 times its initial size.
Original total work = 1 unit.
New total work = \(1 + 0.50 = 1.5\) units.
Calculate the new total remaining work:
The man has already completed \(\frac{7}{8}\) of the original job.
So, the completed work in terms of the new quantum is still \(\frac{7}{8}\) of the original 1 unit of work. The new total work is 1.5 units.
The total work that needs to be done now is 1.5 units.
The work already completed is \(\frac{7}{8}\) of the original job.
So, the new remaining work is:
\[ \text{New remaining work} = 1.5 - \frac{7}{8} \]\[ \text{New remaining work} = \frac{3}{2} - \frac{7}{8} \]\[ \text{New remaining work} = \frac{12}{8} - \frac{7}{8} = \frac{5}{8} \text{ of the original job's equivalent} \]Calculate the additional days needed:
We know that the man completes 1 unit of the original job in 24 days.
Therefore, to complete \(\frac{5}{8}\) of the original job's equivalent (which is the new remaining work), he will take:
\[ \text{Days needed} = \frac{5}{8} \times 24 \text{ days} = 5 \times 3 = 15 \text{ days} \]D) 15 days. This is the calculated time required to finish the new remaining quantum of work.
W can do 25% of a work in 30 days, X can do 1/4 of the work in 10 days, Y can do 40% of the work in 40 days and Z can do 1/3 of the work in 13 days. Who will complete the work first?
UPSC-2016
To determine who will complete the work first, we need to calculate the total number of days each person would take to complete 100% of the work. The person who takes the fewest days will complete the work first.
W does 25% of the work in 30 days.
\[ \text{Time for W to do 100% work} = \frac{30 \text{ days}}{25\%} \times 100\% = \frac{30}{0.25} \text{ days} = 120 \text{ days} \]X does \(1/4\) of the work in 10 days. Note that \(1/4 = 25\%\).
\[ \text{Time for X to do 100% work} = \frac{10 \text{ days}}{1/4} = 10 \times 4 \text{ days} = 40 \text{ days} \]Y does 40% of the work in 40 days.
\[ \text{Time for Y to do 100% work} = \frac{40 \text{ days}}{40\%} \times 100\% = \frac{40}{0.40} \text{ days} = 100 \text{ days} \]Z does \(1/3\) of the work in 13 days.
\[ \text{Time for Z to do 100% work} = \frac{13 \text{ days}}{1/3} = 13 \times 3 \text{ days} = 39 \text{ days} \]Comparing these values, Z takes the least number of days (39 days) to complete the work.
Correct Option: D) Z
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. ?
UPSC-2016
This problem involves relative speed in a circular track. When two people move in opposite directions on a circular track, their relative speed is the sum of their individual speeds. The number of times they cross each other is determined by how many full rounds their relative distance covers within the given time frame.
A) 7
As calculated, they complete 7 crossings within the 1.5-hour period, with the 8th crossing not yet completed by 9:30 a.m.
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?
UPSC-2018
To find the time taken for a train to cross a man, we need to consider the distance the train travels as its own length. The speed of the train is given in kmph, which needs to be converted to metres per second (m/s) to be consistent with the length given in metres and time required in seconds. Then, we can use the basic formula: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).
To convert kmph to m/s, multiply by \(\frac{5}{18}\).
\[ \text{Speed in m/s} = 40 \times \frac{5}{18} \text{ m/s} \]\[ \text{Speed in m/s} = \frac{200}{18} \text{ m/s} \]\[ \text{Speed in m/s} = \frac{100}{9} \text{ m/s} \]When a train crosses a man (or a pole, or a point object), the distance covered by the train is equal to its own length.
Distance = 200 metres
Using the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\):
\[ \text{Time} = \frac{200 \text{ m}}{\frac{100}{9} \text{ m/s}} \]\[ \text{Time} = 200 \times \frac{9}{100} \text{ seconds} \]\[ \text{Time} = 2 \times 9 \text{ seconds} \]\[ \text{Time} = 18 \text{ seconds} \]Thus, the train will cross the man in 18 seconds.
D) 18 is the correct answer, as calculated above.
P works thrice as fast as Q, whereas P and Q together can work four times as fast as R. If P, Q and R together work on a job, in what ratio should they share the earnings?
UPSC-2017
The earnings of individuals working together on a job are shared in the ratio of their efficiencies or the amount of work they do. We need to find the ratio of the efficiencies of P, Q, and R based on the given information.
Correct Option: A) 3:1:1 is the correct ratio for sharing earnings, as it directly reflects the ratio of their efficiencies.
