Few more questions - assorted topics

Have given below a few more questions from assorted topics

1. A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.

2. [x] is the greatest integer less than or equal to x. Find the number of positive integers n such that [n/11] = [n/13].

3. Positive numbers 1 to 55, inclusive are placed in 5 groups of 11 numbers each. What is the maximum possible average of the medians of the 5 groups?

Solutions to questions on mean median

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

1. The median of n distinct numbers is greater than the average, does this mean that there are more terms above the average than below it?

No.

If there are an odd number of terms, say, 2n+1, then the median is the middle term. And if average is lesser than the middle term, there will at least be n +1 terms greater than the average. So, there will be more terms above the average than below it.

However, this need not be the case when there are an even number of terms. When there are 2n distinct terms, n of them will be greater than the median and n will be lesser than the median. The average of these two terms can be such that there are n terms above the average and n below it.

For instance, if the numbers are 0, 1, 7, 7.5. The median is 4, average is 3.875. Average is less than the median. And there are more 2 numbers above the average and 2 below the average.

2. In a sequence of 25 terms, can 20 terms be below the average? Can 20 terms be between median and average?

Yes. We can have 24 zeroes and 1500 as the 25 numbers. In this case, there are 24 numbers below the average.

No. In a sequence of 25 numbers, 12 will be greater than or equal to the median and 12 will be lesser than or equal to the median. We cannot have 20 terms in between the average and median

3. From 10 numbers, a,b,c,...j, all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the average of the 10 numbers?

Yes. All the terms appear the same number of times when we select them 4 at a time. So, the average of averages will be equal to the overall average.

Mean Median - Final few questions

Few more questions on Mean, median. Have just tried to construct a few Yes/No questions to try to build some intuition. These questions are better-answered by constructing examples/counter-examples.

1. The median of n distinct numbers is greater than the average, does this mean that there are more terms above the average than below it?

2. In a sequence of 25 terms, can 20 terms be below the average? Can 20 terms be between median and average?

3. From 10 numbers, a,b,c,...j, all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the average of the 10 numbers?

There is one further question that does not come under this classification, but an interesting one nevertheless

Q. The 64 squares of a chessboard are filled with natural numbers from 1 to 100. Multiples cells can have the same number (Any number of cells can have the same number). If the value in any square is equal to the average of the numbers in all the squares around it (either 3, 5 or 8 squares, depending on the position of the square), in how many ways can the chessboard be filled?

This is a fairly difficult question, and not exactly a question for CAT prep, but a fun question nevertheless. (This question is an adaptation from one of the Olympiad questions. Have not been able to locate the exact one yet.Will send in a post once I locate the correct Olympiad)

Happy cracking.

Mean median - Solutions

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

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

Let average in class C = x. Average of class B = x +16. Average of class A = y
y = 2 + (20y + 30(x+16) + 50x) /100

(y - 2) * 10 = 2y + 3x + 48 + 5x
10y -20 = 2y + 8x + 48
8y - 8x = 68
y - x = 8.5. This is the difference between average of class A and class C

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?

Let us try to construct a scenario for getting the smallest value of the average. We need to have the minimum value for each of the 5 medians.

Now, what is the lowest value the median of any of the 5 groups can take?

The median is the middle term among the 5 terms. So, it cannot be 1 or 2, but it can be 3. If the set were 1, 2, 3, 4, 5, the median would be 3. If we choose a sub-group such as this, the smallest median fro the next group will be 8. The next group could be 6, 7, 8, 9, 10. The idea is to create sub-groups in such a way so that the second group can have a smaller median than 8.

Now, even a sub-group that has the numbers 1, 2, 3, 24, 25 would have the median as 3. So, if we want to keep the medians as small as possible, we should choose sub-groups that have small numbers till the median, and then very large numbers. Because, these large numbers do not affect the median, we will still have small numbers to choose from for the next group. So, choose sub-groups such that the first 3 numbers are small, the final two are as large as possible.

1, 2, 3, 24, 25
4, 5, 6, 22, 23
7, 8, 9, 20, 21
10, 11, 12, 18, 19
13, 14, 15, 16, 17

would be an ideal set of groups. The medians would be 3, 6, 9, 12, 15, and the average of the medians would be 9.

Similar set of groups can be found to find the highest value of the average. The medians would be 23, 20, 17, 14, 11 and the average would be 17.

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?

Mean Median (Averages) - Solutions

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)

Or, the ascending order should be e c a d b

Mean Median - Solutions to Basic Questionsn

Have given below the solutions to the basic questions in Mean Median .

Averages online

View more presentations from rajeshb1980.

Averages - Questions

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

1. 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?
2. 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?
3. 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.

Mean Median Basic questions

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?

Happy cracking