A contract on construction job specifies a penalty for delay in completion of the work beyond a certain date is as follows: ₹ 200 for the first day, ₹250 for the second day, ₹300 for the third day etc., the penalty for each succeeding day being ₹50 more than that of the preceding day. How much penalty should the contractor pay if he delays the work by 10 days?
UPSC-2011
The problem describes a penalty structure where the penalty for each succeeding day increases by a fixed amount. This indicates an arithmetic progression (AP). We need to find the total penalty for a delay of 10 days, which means calculating the sum of the first 10 terms of this AP.
B) ₹ 4,250 is the correct total penalty calculated using the sum of an arithmetic progression formula.
In a garrison, there was enough food to last 1000 soldiers for 20 days. If 1000 soldiers joined, how long would the food last?
UPSC-2013
This is a problem of inverse proportionality. The total amount of food is constant. If the number of soldiers increases, the number of days the food will last decreases proportionally. We can calculate the total 'food units' available and then divide by the new number of soldiers.
\[ \text{Total food units} = \text{Number of soldiers} \times \text{Number of days} \]
\[ \text{Total food units} = 1000 \text{ soldiers} \times 20 \text{ days} = 20000 \text{ soldier-days} \]
\[ \text{New number of soldiers} = 1000 + 1000 = 2000 \text{ soldiers} \]
\[ \text{Days food will last} = \frac{\text{Total food units}}{\text{New number of soldiers}} \]
\[ \text{Days food will last} = \frac{20000 \text{ soldier-days}}{2000 \text{ soldiers}} = 10 \text{ days} \]
D) 10 days is the correct answer. With 2000 soldiers, the food will last for 10 days.
The tank-full petrol in Arun's motor-cycle lasts for 10 days. If he starts using 25% more everyday, how many days will the tank-full petrol last?
UPSC-2013
This problem involves inverse proportionality. If the daily consumption of petrol increases, the total number of days the tank-full petrol will last will decrease. We can calculate the total amount of petrol in the tank and then determine how many days it will last with the new consumption rate.
Correct Option: D) 8
24 men and 12 women can do a piece of work in 30 days. In how many days can 12 men and 24 women do the same piece of work?
UPSC-2022
30 days
More than 30 days
Less than 30 days or more than 30 days
Data is inadequate to draw any conclusion
This problem involves the concept of work and time. To solve such problems, we typically need to know the individual work rates of men and women or a relationship between their work rates. Without this information, we cannot determine how the work capacity changes when the composition of the workforce changes.
D) Data is inadequate to draw any conclusion
As demonstrated in the step-by-step explanation, the number of days required to complete the work depends on the individual work rates of men and women. Without knowing the relationship between a man's work rate and a woman's work rate, we cannot calculate a definitive answer. Different assumptions about their efficiencies lead to different results.
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?
UPSC-2017
This problem involves relative speed and distance. We need to calculate the distance covered by the freight train before the express train starts, and then determine how long it takes for the express train to catch up, finally calculating the distance from Delhi where they meet.
The freight train travels for 2 hours before the express train starts.
\[ \text{Distance} = \text{Speed} \times \text{Time} \] \[ \text{Distance covered by freight train} = 40 \text{ km/hr} \times 2 \text{ hr} = 80 \text{ km} \]Since both trains are moving in the same direction, their relative speed is the difference between their speeds.
\[ \text{Relative Speed} = \text{Speed of Express Train} - \text{Speed of Freight Train} \] \[ \text{Relative Speed} = 60 \text{ km/hr} - 40 \text{ km/hr} = 20 \text{ km/hr} \]The express train needs to cover the 80 km head start that the freight train has.
\[ \text{Time} = \frac{\text{Distance to cover}}{\text{Relative Speed}} \] \[ \text{Time to catch up} = \frac{80 \text{ km}}{20 \text{ km/hr}} = 4 \text{ hours} \]The express train travels for 4 hours to catch up. The distance it covers in this time will be the meeting point from Delhi.
\[ \text{Distance from Delhi} = \text{Speed of Express Train} \times \text{Time to catch up} \] \[ \text{Distance from Delhi} = 60 \text{ km/hr} \times 4 \text{ hr} = 240 \text{ km} \]Alternatively, we can calculate the total distance covered by the freight train:
Total time freight train traveled = 2 hours (head start) + 4 hours (until caught) = 6 hours
\[ \text{Distance from Delhi} = \text{Speed of Freight Train} \times \text{Total Time} \] \[ \text{Distance from Delhi} = 40 \text{ km/hr} \times 6 \text{ hr} = 240 \text{ km} \]Correct Option: 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 30ᵒ 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?
UPSC-2018
We need to find the ratio of speeds of A and B. We are given their initial positions on a circular track and their meeting point. We can use the relationship between distance, speed, and time. Since they meet at the same time, the ratio of their speeds will be equal to the ratio of the distances they covered.
Let the radius of the circular track be \(R\). The circumference of the track is \(2\pi R\).
Let the starting point of A be \(P_A\). A reaches the point diametrically opposite to his starting point. This means A covers half the circumference of the circle. The distance covered by A is \(\pi R\).
Let the starting point of B be \(P_B\). B is ahead of A, and their positions make an angle of \(30^\circ\) at the center. This means the arc length between A and B initially is \[R \times \frac{30\pi}{180} = \frac{\pi R}{6}\].
Let's denote the starting point of A as \(0^\circ\). Then the point diametrically opposite to A's starting point is \(180^\circ\).
A starts at \(0^\circ\) and covers a distance corresponding to an angle of \(180^\circ\). So, A's meeting point is at \(180^\circ\).
B starts at \(30^\circ\) ahead of A. When A reaches \(180^\circ\), B also meets A at \(180^\circ\).
The distance covered by B is the arc length from \(30^\circ\) to \(180^\circ\). This corresponds to an angle of \(180^\circ - 30^\circ = 150^\circ\).
The distance covered by B is \[R \times \frac{150\pi}{180} = \frac{5\pi R}{6}\].
Let \(S_A\) be the speed of A and \(S_B\) be the speed of B. Let \(t\) be the time taken for them to meet. Since they start at the same time and meet at the same time, the time \(t\) is the same for both.
Distance covered by A: \(D_A = S_A \times t = \pi R\)
Distance covered by B: \(D_B = S_B \times t = \frac{5\pi R}{6}\)
To find the ratio of their speeds, we can divide the two equations:
\[ \frac{S_A \times t}{S_B \times t} = \frac{\pi R}{\frac{5\pi R}{6}} \]\[ \frac{S_A}{S_B} = \frac{\pi R}{1} \times \frac{6}{5\pi R} \]\[ \frac{S_A}{S_B} = \frac{6}{5} \]Therefore, the ratio of speeds of A and B is \(6:5\).
A) The ratio of speeds of A and B is \(6:5\), as calculated from the ratio of distances covered by them in the same amount of time.
