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Nov 30, 2010

Mean median - Two more questions

Have given below two more questions on Averages.

1. Consider three classes A, B and C with 20, 30 and 50 students respectively. The averages score in maths of students in class B is 16 more than that of those in class C and the average score of those in class A is 2 more than the overall average of scores in A, B and C. What is the difference between average of class C and class A

2. Natural numbers 1 to 25 (both inclusive) are split into 5 groups of 5 numbers each. The medians of these 5 groups are A, B, C, D and E. If the average of these medians is m, what are the smallest and the largest values m can take?

Have given below the solutions to the questions on mean and median .

Q. Consider 4 numbers a, b, c and d. Ram figures that the smallest average of some three of these four numbers is 30 and the largest average of some three of these 4 is 40. What is the range of values the average of all 4 numbers can take?

We can assume a, b,c d are in ascending order (with the caveat that numbers can be equal to each other)

a + b + c = 90
b + c + d = 120

We need to find the maximum and minimum value of a + b + c + d.

a + b + c + d = 120 + a. So, this will be minimum when a is minimum. Given a + b + c = 90. a is minimum when b + c is maximum. If b + c is maximum, d should be minimum. Given that b + c + d = 120, the minimum value d can take is 40 as d cannot be less than b or c. The highest value b +c can take is 80, when b = c= d = 40. When b = c = d = 40, a = 10. a + b + c + d = 130. Average = 32.5

Similarly, a + b + c + d = 90 + d. So, this will be maximum when d is maximum. Given b + c + d = 120. d is maximum when b + c is minimum. If b + c is minimum, a should be maximum. Given that a + b + c = 90, the maximum value a can take is 30 as a cannot be greater than b or c. The lowest value b +c can take is 60, when a = b= c = 30. When a = b = c = 30, d = 60. a + b + c + d = 150. Average = 37.5

So, the average has to range from 32.5 to 37.5

Q. The average of 5 distinct positive integers if 33. What are the maximum and minimum possible values of the median of the 5 numbers if the average of the three largest numbers within this set is 39?

Let the numbers be a, b, c, d, e in ascending order. a + b + c + d + e = 165. Average of the three largest numbers is 39, so c + d + e = 117, or a + b = 48.

We need to find the maximum and minimum possible values of c.

For minimum value, a and b have to be minimum. a +b = 48. Let us assume a =23, b =25, ca can be as low as 26.
23, 25, 26, 45, 46 is a possible sequence that satisfies the conditions.

For maximum value, we need d + e to be minimum as c + d + e = 117. we can have c = 38, d =39 and e =40. Or, the maximum value c can take = 38

23, 25, 38, 39, 40 is a possible sequence that satisfies the conditions specified

Q. Consider 5 distinct positive numbers a, b, c, d, and e. The average of these numbers is k. If we remove b from this set, the average drops to m (m is less than k). Average of c, b, d and e is K. We also know that c is less than d and e is less than k. The difference between c and b is equal to the difference between e and d. Average of a, b, c and e is greater than m. Write down a, d, c, d and e in ascending order.

Average of a, b, c, d and e is k, Average of b, c, d and e is also k, this implies that a = k
If we remove b, the average drops, this implies that b is higher than the average
e is less than k, or e is less than a. c is less than d.

e < a < b, c < d

Average of a, b, c and e is greater than average a, c, d and e. This tells us that d < b

From this, we get that b is the largest number

b - c = d - e
b + e = c + d
a + b + c +d + e = 5a
Or, b + c + d + e = 4a

b + e = 2a, c + d = 2a. Or, e, a, b is an AP, c, a, d is an AP.

e is the smallest number (as b is the largest number)

Have given below few questions from averages. These are slightly more challenging than the basic questions

Consider 4 numbers a, b, c and d. Ram figures that the smallest average of some three of these four numbers is 30 and the largest average of some three of these 4 is 40. What is the range of values the average of all 4 numbers can take?

The average of 5 distinct positive integers if 33. What are the maximum and minimum possible values of the median of the 5 numbers if the average of the three largest numbers within this set is 39?

Consider 5 distinct positive numbers a, b, c, d, and e. The average of these numbers is k. If we remove b from this set, the average drops to m (m is less than k). Average of c, b, d and e is K. We also know that c is less than d and e is less than k. The difference between c and b is equal to the difference between e and d. Average of a, b, c and e is greater than m. Write down a, d, c, d and e in ascending order.

Have given a few very simple questions in mean, median. Will post more challenging questions through the week.

1. The average revenue of company A in a year was Rs.30,000 a month. During the first 5 months, the company saw an average revenue of Rs. 25,000 per month and in the last 6 months, the company saw an average of Rs.33,000 per month. What was the revenue level in the 6th month?

2. The captain of a cricket squad of 14 players is 28 years old. The wicket-keeper is 32 years old. If these two players are replaced by two younger players, the average age of the team drops by 1 year. What is the average age of the two new players.?

3. The average age of the students of class A comprising 25 students is 19 years and that of the students of class B comprising 15 students is 11 years. What is the average age of the students of the two classes taken together?

4. The average weight of a class comprising 29 students is 40 kgs. If the weight of the teacher is also included, the average weight increases by 0.5 kgs. What is the weight of the teacher?

5. 30 friends go out for dinner. 12 of them spend Rs.32.5 each for the dinner while the rest of them spend Rs.3 more than the average expense of all 30. What was the total money spent?

6. The range of the height of male members in a team is 9 cms, and that of the female members in the team is 8 cms. What is the range of the height of the team if the shortest man in the team is 3 cms taller than the shortest woman in the team?

7. The median of 68, 93, 109, x, y, and 97 is 97. What is the least possible average of x and y?

Have given below solutions to the races questions that can be found here.

1. Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

Let track length be equal to T. Time taken to meet for the first time = T / relative speed = T/(6-b) or T/(b-6)

Time taken for a lap for A = T/6 Time taken for a lap for A = T/b

So, time taken to meet for the first time at the starting point = LCM (T/6, T/b) = T / HCF (6,b)

Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b). For a more detailed discussion on this have a look at the last few slides in this presentation.

So, in essence we have to find values for b such that 6-b/ HCF(6,b) = 2 or b-6/ HCF(6,b) = 2

b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.

2. Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A,B and C?

Let track length be equal to T. When a completes a lap, let us assume B has run a distance of (t-d). At this time, C should have run a distance of (t-2d)

After three laps C would have traveled a distance of 3 * (t-2d) = 3t - 6d.

After 3 laps C is in the same position as B was at the end one lap. So, the position after 3t-6d should be the same as t-d. Or, C should be at a distance of d from the end of the lap. C will have completed less than 3 laps (as he is slower than A), so he could have traveled a distance of either t-d or 2t-d.

=> 3t-6d = t-d => 2t = 5d => d = 0.4t => The distances covered by A,B and C when A completes a lap will be t, 0.6t and 0.2t respectively. Or, the ratio of their speeds is 5:3:1

In the second scenario, 3t-6d = 2t-d => t = 5d=> d = 0.2t => The distances covered by A,B and C when A completes a lap will be t, 0.8t and 0.6t respectively. Or, the ratio of their speeds is 5:4:3

The ratio of the speeds of A, B and C is either 5:3:1 or 5:4:3

Have given below two questions on races. For a recap of the basics on Races, see questions here and the solutions here

1. Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

2. Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A,B and C?

1. In a 100m race, A can give B a start of 12 meters or 3 seconds. What is the speed of B in m/sec and how long will A take to run 100m?

2. A can give B a start of 30 in a km race and A can give C a start of 50m in a km race. How much start will B be able to give C in a km race?

3. A runs a 400m race at the speed of 8 m/sec. He gives B a start of 40m and still beats him by 10 seconds. What is B's speed?

4. If A gives B a start of 20m in a race, A beats him by 10 seconds. On the contrary, if A gives B a start of 40m, they finish the race in a dead heat. If the length of the race is 400m, how long will A take to run the race?

5. A and B start around a circular track of length 600m at speeds of 6m/s and 9 m/sec from the same point, simultaneously and in the same direction. When will they meet for the first time? How far from the starting point will they meet, when they meet for the first time?

6. A and B run around a circular track at speeds of 8m/sec and 5m/sec starting from the same point, simultaneously and in the same direction. How many points with respect to the starting point on the circular track will they meet, if the run at the above mentioned constant speeds endlessly?

Station X of length 900 meters has two station masters A and B. But as the station is not a busy one, they are mostly jobless and decide to conduct an experiment. They stand at either end of the station and decide to note the exact time when trains cross the stationmasters. They synchronize their watches and proceed to either end of the station. Two trains P and Q go past the station (neither train stops here), and after having taken down their readings, the station masters sit down to have a chat

1. A: Train P entered the station at exactly 8:00:00 2. B: Train Q entered the station at exactly 8:00:10 (10 seconds past 8) 3. A: The last carriage of train P crossed me by at 8:00:20, and precisely two seconds after this, the engines of the two trains went past each other. (Engines are at the front of the train) 4. B: The last carriage of train Q crossed me 22 seconds after the engine of P went past me. 5. A: After the last carriage of train P crossed by me, it took 35 seconds for the engine of train Q to cross me. 6. B: I got bored and I came here.

Now, let us try to jot down the points in a slightly different format and see if we can make any inferences. Let us assume speed of train P = p, speed of train Q = q, length of train P = L, length of train Q = M
Take statements 1 and 3
Train P crosses the stationmaster A entirely in 20 seconds (enters station at 8:00:00 and last carriage passes at 8:00:20) => Length of train = 20p => L = 20p

Statement 5 tells us that engine of train Q crosses station master A at 8:00:55. Train Q enters the station at 8:00:10, so the train takes 45 seconds to cross the entire station. Or, it takes 45 seconds to travel 900 meters => Speed to train Q, q = 20m/s.

Statement 3 tells us that the two trains cross each other at 8:00:22. This implies train P has traveled for 22 seconds since entering the station and train Q has traveled for 12 seconds since entering the station before they cross each other. The cumulative distance traveled by the two trains should be equal to the length of the station = 900m.

=> 900 = 22p + 12q
q = 20m/s => p = 30m/s.

So, train P_ takes 900/p = 30 seconds to cross the station. So, engine of train P will cross stationmaster B at 8:00:30.

Statement 4 states that the last carriage of Q went past 22 seconds after the engine of P went by. Or, the last carriage of Q went by at 8:00:52.
Engine of train Q went by at 8:00:10, last carriage went by at 8:00:52, or train Q took 42 seconds to cross station master B. Train Q travels at 20m/s. => Length of train Q = 840m

Now, to the questions

1. What is the length of train Q?
Train Q is 840m long

2. At what time do the rear ends of the two trains cross each other?
The engines cross each other at 8:00:22. The relative speed of the two trains = 20+30 = 50m/s. The relative distance traveled by two trains from the time the engines cross each other to the times the rear ends cross each other = Sum of the two lengths = 600 + 840 =1440. Time taken = 1440/50 = 28.8 seconds past 8:00:22, or at 8:00:50.8 seconds.

3. How far from station master A do the rear ends of the two trains cross each other?
At 8:00:50.8, P would have traveled 50.8 * 30m/s post entering the station. Or, train P would have traveled 1524m. The rear end of train P would be at a point 1524-600m = 924m from stationmaster A (or 24 meters from stationmaster B and outside the station)

4. You are told that the two trains enter the station at the same times mentioned and the length of the two trains are unchanged. Furthermore, train P continues to travel at the same speed (as computed above). At what minimum speed should train Q travel such that the rear ends of the two trains cross each other at a point within the length of the platform?

Rear end of train P crosses the station completely at 8:00:50. (Train P takes 30 seconds to travel the station and 20 seconds to travel a distance equal to its length). Train Q should have traveled 840m by this time. => Train Q should travel 840 within 40 seconds.

Station X of length 900 meters has two station masters A and B. But as the station is not a busy one, they are mostly jobless and decide to conduct an experiment. They stand at either end of the station and decide to note the exact time when trains cross the stationmasters. They synchronize their watches and proceed to either end of the station. Two trains P and Q go past the station (neither train stops here), and after having taken down their readings, the station masters sit down to have a chat

A: Train P entered the station at exactly 8:00:00 B: Train Q entered the station at exactly 8:00:10 (10 seconds past 8) A: The last carriage of train P crossed me by at 8:00:20, and precisely two seconds after this, the engines of the two trains went past each other. (Engines are at the front of the train) B: The last carriage of train Q crossed me 22 seconds after the engine of P went past me. A: After the last carriage of train P crossed by me, it took 35 seconds for the engine of train Q to cross me. B: I got bored and I came here.

1. What is the length of train Q?

2. At what time do the rear ends of the two trains cross each other?

3. How far from station master A do the rear ends of the two trains cross each other?

4. You are told that the two trains enter the station at the same times mentioned and the length of the two trains are unchanged. Furthermore, train P continues to travel at the same speed (as computed above). At what minimum speed should train Q travel such that the rear ends of the two trains cross each other at a point within the length of the platform?

Have given below the solutions to the two questions on Speed Time Distance .

1. Two friends A and B leave City P and City Q simultaneously and travel towards Q and P at constant speeds. They meet at a point in between the two cities and then proceed to their respective destinations in 54 minutes and 24 minutes respectively. How long did B take to cover the entire journey between City Q and City P?

Let us assume Car A travels at a speed of a and Car B travels at a speed of b. Further, let us assume that they meet after t minutes.

Distance traveled by car A before meeting car B = a*t. Likewise distance traveled by car B before meeting car A = b * t.

Distance traveled by car A after meeting car B = a *54. Distance traveled by car B after meeting car A = 24* b

Distance traveled by car A after crossing car B = distance traveled by car B before crossing car A (and vice versa)

=> at = 54b -------- 1 and bt = 24a -------- 2

Multiplying equations 1 and 2 we have ab * t^{2} = 54 * 24 * ab

=> t^{2} = 54 * 24 => t = 36.

So, both cars would have traveled 36 minutes prior to crossing each other. Or, B would have taken 36 + 24 = 60 minutes to travel the whole distance.

2. Car A trails car B by 50 meters. Car B travels at 45km/hr. Car C travels from the opposite direction at 54km/hr. Car C is at a distance of 220 meters from Car B. If car A decides to overtake Car B before cars B and C cross each other, what is the minimum speed at which car A must travel?

To begin with, let us ignore car A. Car B and car C travel in opposite directions. Their relative speed = Sum of the two speeds = 45 + 54 kmph. = 99kmph. = 99 * 5/18 m/s = 55/2 m/s = 27.5m/s.

The relative distance = 220m. So, time they will take to cross each other = 220/27.5 = 8 seconds.

Now, car A has to overtake car B within 8 seconds. The relative distance = 50m => Relative speed should be at least 50/8m/s. = 6.25m/s = 6.25 * 18/5 kmph = 22.5kmph.

Car B travels at 45kmph, so car A should travel at at least 45 + 22.5 = 67.5kmph.

Two more questions from Speed Time Distance. Reasonably straightforward ones.

1. Two friends A and B leave City P and City Q simultaneously and travel towards Q and P at constant speeds. They meet at a point in between the two cities and then proceed to their respective destinations in 54 minutes and 24 minutes respectively. How long did B take to cover the entire journey between City Q and City P?

2. Car A trails car B by 50 meters. Car B travels at 45km/hr. Car C travels from the opposite direction at 54km/hr. Car C is at a distance of 220 meters from Car B. If car A decides to overtake Car B before cars B and C cross each other, what is the minimum speed at which car A must travel?

1. City A to City B is a downstream journey on a stream which flows at a speed of 5km/hr. Boats P and Q run a shuttle service between the two cities that are 300 kms apart. Boat P, which starts from City A has a still-water speed of 25km/hr, while boat Q, which starts from city B at the same time has a still-water speed of 15km/hr. When will the two boats meet for the first time? (this part is easy) When and where will they meet for the second time?

When boat P travels downstream, it will effectively have a speed to 30kmph. Likewise, Q will have an effective speed of 10kmph. The relative speed = 40kmph. So, the two boats will meet for the first time after 300/40 hours (Distance/relative speed) = 7.5 hours (Actually, for this part we do not need the speed of the stream)

The second part is more interesting, because the speed of the boats change when they change direction. Boat P is quicker, so it will reach the destination sooner. Boat P will reach City B in 10 hours (300/30). When boat P reaches city B, boat Q will be at a point 100kms from city B.

After 10 hours, both P and Q will be traveling upstream

P will catch up with Q after 10 more hours (Relative Distance/relative speed - 100/10).

So, P and Q will meet after 20 hours at a point 200 kms from city B

2. Cities M and N are 600km apart. Bus A starts from city M towards N at 9AM and bus B starts from city N towards M at the same time. Bus A travels the first one-third of the distance at a speed of 40kmph, the second one-third at 50kmph and the third one-third at 60km/hr. Bus B travels the first one-third of the total time taken at a speed of 40kmph, the second one-third at 50kmph and the third one-third at 60km/hr. When and where will the two buses cross each other?

Bus A Travels 200km at 40kmph the next 200km @ 50kmph and the final 200km @ 60kmph

So, Bus A will be at a distance of 200km from city M after 5 hours, and at a distance of 400km after 9 hours, and reach N after 12 hours and 20 mins

Bus B Travels at an overall average speed of 50kmph, so will take 12 hours for the entire trip So, Bus B will travel 160kms in the first 4 hours 200 kms in the next 4 and 240 in the final 4

So, both buses cross each other when they are in their middle legs.

After 5 hours, bus A will be at a position 200kms from city M. At the same time, bus B will be at a distance 210kms from city N (4*40+50).

The distance between them will be 190kms (600-200-210). Relative speed = Sum of the the two speeds = 50+50 = 100 kmph.

Time taken = 190/100 = 1.9 hours. = 1 hour and 54 minutes. So, the two buses will meet after 6 hours and 54 minutes. Bus B will have travelled 210 + 95 = 305 kms. So, the two buses will meet at a point that is 305 kms from City N and 295 kms from city A.

Have given below two questions on Speed Time Distance. These are slightly difficult questions. For a recap of the basics, have a look at questions here and solutions here .

1. City A to City B is a downstream journey on a stream which flows at a speed of 5km/hr. Boats P and Q run a shuttle service between the two cities that are 300 kms apart. Boat P, which starts from City A has a still-water speed of 25km/hr, while boat Q, which starts from city B at the same time has a still-water speed of 15km/hr. When will the two boats meet for the first time? (this part is easy) When and where will they meet for the second time?

2. Cities M and N are 600km apart. Bus A starts from city M towards N at 9AM and bus B starts from city N towards M at the same time. Bus A travels the first one-third of the distance at a speed of 40kmph, the second one-third at 50kmph and the third one-third at 60km/hr. Bus B travels the first one-third of the total time taken at a speed of 40kmph, the second one-third at 50kmph and the third one-third at 60km/hr. When and where will the two buses cross each other?

Have given below the solutions to the basic questions on Speed Time Distance. Have also added some basic theory on Speed Time distances in the PPT. Will start posting tougher questions on Speed Time Distance beginning next week.

The answer key for the questions is as follows

1. 80 minutes (There was an error in this question. Have now fixed the error) 2. 6 km 3. 50 km/hr 4. 45km/hr 5. 200km from Bangalore 6. 520m 7. 350m 8. 18kmph

Have given below, a few basic questions in Speed Time Distance. These are fairly straightforward questions and should be reasonable practice for revising basic concepts. Over the next week, let us have a go at tougher questions from this topic. Just thought it helps to have a re-cap before we go on to the more difficult questions.

1.Traveling at 9/8 of his usual speed, Arvind reaches the restaurant 10 minutes early. How long did he take to reach the restaurant this time?

2.Cycling at a speed of 10 km/h from his residence, Rajesh reached the railway station 3 minutes late. Had he cycled at a speed of 12 km/h he would have reached 3 minutes early. What is the distance between his house and the railway station?

3.Anand travels the first 30 minutes of the journey at a speed of 80 kmph and the balance 90 minutes of his journey at a speed of 40 kmph. What is the average speed of his journey in kmph?

4.A car covers a third of the distance between two cities at a speed of 30 kmph and the balance distance at a speed of 60 kmph. What is the average speed of the car?

5.Ram leaves Bangalore at 4 PM in his car and travels to Chennai at 50 kmph. Shyam leaves Chennai at 5 PM and travels in his bike at 60 kmph to Bangalore along the same route. If the distance between Bangalore and Chennai is 380 kms, when will Ram and Shyam meet and how far from Bangalore will they meet?

6.A train travelling at a speed of 72 mph crosses a platform of length 1400 meters completely in 60 seconds. What is the length of the train in meters. Use conversion: 1 mile = 1600 meters

7.A passenger train traveling at 70 kmph crosses a freight train, of length 400 m traveling in the opposite direction at a speed of 38 kmph, completely in 25 seconds. What is the length of the passenger train in meters?

8.A boat travels from city A to city B, a distance of 72 km. The onward journey is a downstream journey and the return journey is an upstream one. If the round trip took the boat 7 hours and the speed of the stream is 3 kmph, what was its speed in the return journey?