# IIM CAT Preparation Tips

IIM CAT Preparation Tips: December 2010

## Dec 28, 2010

### Pipes Cisterns - Questions

Have given below a few questions on pipes and cisterns.

1) Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. Find t?

2) Pipe A fills a tank at the rate of 100lit/min, Pipe B fills at the rate of 25 lit/min, pipe C drains at the rate of 50 lit/min. The three pipes are kept open for one minute each, one after the other. If the capacity of the tank is 7000 liters, how long will it take to fill the tank if

i) A is kept open first, followed by B and then C
ii) B first, followed by A, and then C
iii) B first, followed by C, and then A

3) 4 men and 6 women complete a task in 24 days. If the women are at least half as efficient as the men, but not more efficient than the men, what is the range of the number of days for 6 women and 2 men to complete a task twice as difficult as this one?

4) A fill pipe can fill a tank in 20 hours, a drain pipe can drain a tank in 30 hours. If a system of n pipes (fill pipes and drain pipes put together) can fill the tank in exactly 5 hours, which of the following are possible values of n (More than one option could be correct)?
1) 32 2) 54 3) 29 4) 40

Labels: ,

## Dec 24, 2010

### CAT Pipes and Cisterns - Basic Questions

Have given below a few basic questions on Pipes and Cisterns.

1. Pipe A can fill a tank in 6 hours. On account of a crack at the bottom of the tank through which water leaks, it takes an hour more for the tank to overflow. How long will the crack take to empty a full tank when pipe A is kept shut?

2. Pipe A that can fill a tank in 40 minutes and pipe B that can fill a tank in 20 minutes are opened when the tank is empty. Pipe B is shut 16 minutes before the tank overflows. How long did it take for the tank to overflow?

3. Pipe A can fill a tank in 36 minutes. Pipe B can fill the same tank in 72 minutes. There is a drain pipe, pipe C, at the bottom of the tank. A man opens pipe A and pipe B when the tank is empty and returns back to shut them when the tank should overflow. However, he finds that he had kept pipe C also open and shuts it immediately. The tank overflows in the next 6 minutes. How long will pipe C take to drain the tank, if pipes A and B are kept shut?

4. A tank is fitted with 22 pipes. Some of these are fill pipes that fill the tank and others waste pipes that drain the tank. If each fill pipe can fill the tank in 24 hours and each drain pipe can drain the tank in 16 hours, and if all the pipes are kept open simultaneously when the tank is empty, the tank overflows in 12 hours. How many of these are waste pipes?

5. Adrain completes 60% of a task in 72 days. He realizes that to meet his deadline, he needs additional help and gets Allen who is thrice as efficient as Adrain to help him. They complete the task with four days to spare. What was the deadline for the project from the day Adrain started work on it?

6. Working together A and B can complete a task in 18 days. If A worked alone, completed the first half of the task and then B took over from A, the task will be completed in 48 days. If A is more efficient than B, how long will A alone take to complete the task?

7. Working alone, Shiv will take 27 more hours to complete a task than if he worked with George to complete the task. Working alone, George will take 3 more hours to complete the task than if he worked with Shiv. How long will George take to complete the task, if he worked alone?

8. 5 men and 7 women can complete a task in 12 days. However, if 11 men and 9 women worked on the same task, the work will be completed only in 8 days. How long will a man and woman take to complete the task?

## Dec 17, 2010

### Percentages Solutions

Have given below solution to the final question on percentages .

6. A, B, C and D share a loot. A gets a% of the total. B gets b% of the remaining (after A has taken his share). C gets c% of the remaining and D gets the rest. D gets a% less than what A gets. B and C get equal amounts. b = 2a.

i) What percentage of what A got did C get?
ii) If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?

Let total amount = T.

A would have got T * a. After A takes his share, there would be T (1-a) left
B would have got T (1-a) * b
C would have got T (1-a)(1-b) * c
D would have got T (1-a)(1-b)(1-c)

D gets a % less than what A gets, or, T *a (1-a) = T (1-a)(1-b)(1-c), or

a = (1-b)(1-c) -------- Eqn 1

B and C get the same amounts, or (1-a) * b = (1-a)*(1-b)*c, or b = (1-b) c, or, b = c - cb, or

b = c/c+1 ------------ Eqn 2
b = 2a -----------------Eqn 3

Substituting, eqn 2 in and 3 in eqn 1, we get

c/2(c+1) = (1-c)/(c+1)
Or, c = 2 -2c, or c = 2/3

b = c/(c+1) = 2/5, a = 1/5
a gets 20% of the loot, b gets 40% of the remaining 80%, or 32% of the loot. C gets 66.6% of 48%, or 32% of the overall loot. And d gets the final 16% of the overall amount (20% lesser than A's share)

Now, to the questions

i) What percentage of what A got did C get?

A got 20% of overall, C got 32% of overall. Or, C got 160% of A

ii) If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?
A would have got Rs. 200, D Rs. 160. A would have got Rs. 40 more than D.

Labels: ,

## Dec 14, 2010

### Solutions to questions on percentages

Have given below solutions to the questions 4 and 5 on percentages .

4. In a class, if 50% of the boys were girls, then there would be 50% more girls than boys. What percentage of the overall class is girls?

For questions of this type, it is best to go from the final step. In the final state, the number of girls should be 1.5 * the number of boys.

When 50% of the boys are taken as girls, let the number of boys = x
Number of girls = 1.5x
total number of students = 2.5x

Original number of boys = 2x (50% of boys = x)
Original number of girls = 0.5x

Girls form 20% of the overall class.

5. Class B has 50% more students than class A. Number of girls in class A is equal to number of boys in class B. The percentage of girls is the same in both classes. What percentage of the student group are boys?

Let number of girls in class A = x
Let number of boys in class A = y
Total number of students = x + y
Proportion of girls = x/(x+y)

Number of boys in class B = x.
Total number of students in class B = 1.5( x + y)
Proportion of girls = 1 - x/1.5(x+y)

Percentage of boys in the overall student community = (x+y)/2.5* (x + y) * 100 = 40%

Labels: ,

## Dec 10, 2010

### Percentages - Solutions

Have given below solutions to the first three questions on Percentages

1. a is x % of b, b is x% more than a. Find x?

a = bx,
b = a (1+x), substituting this in the previous equation, we get
x(x+1) = 1
x^2 + x -1 = 0

or x is approximately 0.62 or 62%

2. A two digit number ab is 60% of x. The two-digit number formed by reversing the digits of ab is 60% more than x. Find x?

10a + b = 0.6x
10b + a = 1.6x

Subtracting one from the other, we have

9b - 9a = x or x = 9( b-a)
x should be a multiple of 9.

10a + b = 3x/5, this implies that x should also be a multiple of 5.Or, x should be a multiple of 45

x should be equal to 45, ab = 27, ba = 72

3. A is x % more than B. B is y% less than C. If A,B and C are positive and A is greater than C, find the relation between x and y

a = (1 +x)b
b = c(1-y)
c = b /(1-y)

a is greater than c
Or, b (1+x) > b/(1-y)
1 + x > 1/(1-y)
(1+x)(1-y) > 1
1 + x -y - xy > 1
x > y (1+x)
y < x/(1+x)

## Dec 9, 2010

### Percentages - Solutions to basic questions

Have given below solutions to the basic questions on Percentages.

## Dec 8, 2010

### Percentages - Questions

1. a is x % of b, b is x% more than a. Find x?

2. A two digit number ab is 60% of x. The two-digit number formed by reversing the digits of ab is 60% more than x. Find x?

3. A is x % more than B. B is y% less than C. If A,B and C are positive and A > C, find the relation between x and y

4. In a class, if 50% of the boys were girls, then there would be 50% more girls than boys. What percentage of the overall class is girls?

5. Class B has 50% more students than class A. Number of girls in class A is equal to number of boys in class B. The percentage of girls is the same in both classes. What percentage of the student group are boys?

6. A, B, C and D share a loot. A gets a% of the total. B gets b% of the remaining (after A has taken his share). C gets c% of the remaining and D gets the rest. D gets a% less than what A gets. B and C get equal amounts. b = 2a.

i) What percentage of what A got did C get?
ii) If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?

## Dec 6, 2010

### Percentages - Basic Questions

1. The price of a mobile phone decreased by 25% to Rs. 100. What was the price before the decrease and by what amount did it decrease?

2. The price of a mobile phone decreased by 25% from Rs. 100. What was the price after the decrease and by what amount did it decrease?

3. The price of a mobile phone decreased by 25% or Rs. 100. What was the price before the decrease and what is after the decrease?

4. In an election contested by two candidates, the winning candidate got 24% of the total votes more than the losing candidate. If the losing candidate got 76,000 votes, how many votes did the winning candidate get? Assume that there were no invalid votes

5. The price of Ford Fusion is 25% less than that of Honda City. By what percentage is a Honda City more expensive than a Ford Fusion?

6. 40% of the people in a gathering speak English. Of the remaining people, 60% speak Hindi. If 96 people speak neither Hindi nor English, how many people were there in the gathering?

7. Ann and Ron are a married couple and Ann earned twice as much as Ron in the month of January. In Feb Ann got a rise of 20% while Ron got a rise of only 10%. If both of them saved half their wages and the combined savings of the couple in the month of February is Rs.52,500, what is the salary of Ron in February?

8. 50% of the students in a class that has 40% women students have got jobs during the recently concluded placement week. If 60% of the men in the class have got placements, what percentage of the women in class are yet to be placed?

9. A sample of 40 liters of diesel is found to be adulterated to the extent of 20%. How many liters of pure diesel be added to the above mixture so that level of impurity stands reduced at 5%?

10. What is the percentage change in the perimeter of a rectangle whose length increases by 5% and width decreases by 15%?

Labels: ,

## Dec 3, 2010

### 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.

Labels: ,

## Dec 2, 2010

### 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.

Labels: ,

### 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.

Labels: ,