**Previous CAT Questions on Averages**

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**Question 1: **DSBO Company produces Z units of output at a total cost of Rs. R, where $R=\frac{1}{10}Z^{3}-5Z^{2}+10Z+5$ At what level of output will the average variable cost attain its minimum?

a) 20

b) 33

c) 25

d) None of the above

**Question 2: **The average of 7 consecutive numbers is P. If the next three numbers are also added, the average shall

a) remain unchanged

b) increase by 1

c) increase by 1.5

d) increase by 2

**Question 3: **2 years ago, one-fifth of Amita’s age was equal to one-fourth of the age of Sumita, and the average of their age was 27 years. If the age of Paramita is also considered, the average age of three of them declines to 24. What will be the average age of Sumita and Paramita 3 years from now?

a) 25 years

b) 26 years

c) 27 years

d) cannot be determined

**Question 4: **Prof. Suman takes a number of quizzes for a course. All the quizzes are out of 100. A student

can get an A grade in the course if the average of her scores is more than or equal to 90.Grade B is awarded to a student if the average of her scores is between 87 and 89 (both included). If the average is below 87, the student gets a C grade. Ramesh is preparing for the last quiz and he realizes that he will score a minimum of 97 to get an A grade. After the quiz, he realizes that he will score 70, and he will just manage a B. How many quizzes did Prof. Suman take?

a) 6

b) 7

c) 8

d) 9

e) None of these

**Question 5: **The average marks of a student in 10 papers are 80. If the highest and the lowest scores are not considered, the average is 81. If his highest score is 92, find the lowest.

a) 55

b) 60

c) 62

d) Cannot be determined

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**Instructions**There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.

**Question 6: **What is the average weight of the students in group D?

a) More than the average weight of A

b) More than the average weight of C

c) Less than the average weight of C

d) Cannot be determined

**Question 7: **Consider a sequence of seven consecutive integers. The average of the first five integers is n. The average of all the seven integers is:

[CAT 2000]

a) n

b) n+1

c) kn, where k is a function of n

d) n+(2/7)

**Question 8: **Three classes X, Y and Z take an algebra test.

The average score in class X is 83.

The average score in class Y is 76.

The average score in class Z is 85.

The average score of all students in classes X and Y together is 79.

The average score of all students in classes Y and Z together is 81.

What is the average for all the three classes?

a) 81

b) 81.5

c) 82

d) 84.5

**Question 9: **A college has raised 75% of the amount it needs for a new building by receiving an average donation of Rs. 600 from the people already solicited. The people already solicited represent 60% of the people the college will ask for donations. If the college is to raise exactly the amount needed for the new building, what should be the average donation from the remaining people to be solicited?

a) Rs. 300

b) Rs. 250

c) Rs. 400

d) 500

**Question 10: **Consider the set S = {2, 3, 4, …., 2n+1}, where n is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X – Y ?

a) 0

b) 1

c) (1/2)*n

d) (n+1)/2n

e) 2008

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**Answers & Solutions:**

**1) Answer (C)**

$Z$ is the number of items produced by the company.

Total cost, $R=\frac{1}{10}Z^{3}-5Z^{2}+10Z+5$

As we can see, the term ‘5’ does not vary with the number of quantities produced. Therefore, 5 is the fixed cost.

Variable cost = $\frac{1}{10}Z^{3}-5Z^{2}+10Z$

Average variable cost = Total variable cost/ number of quantities.

Average variable cost =$\frac{1}{10}Z^{2}-5Z+10$

=$\frac{Z^2-50Z+100}{10}$

=$\frac{(Z^2-50Z+625)- 525}{10}$

=$\frac{(Z-25)^2-525}{10}$

As we can see, the least value of the expression will be obtained at $Z=25$

Therefore, option C is the right answer.

**2) Answer (C)**

Let the 7 consecutive numbers be a-3, a-2, a-1, a, a+1, a+2 and a+3.

Sum of the numbers = 7a and the average of these numbers = a

If next 3 numbers a+4, 4+5 and a+6 are also added then the average of these 10 numbers = $\dfrac{7a+a+4+a+5+a+6}{10} = a+1.5$

Thus, the average increases by 1.5

Hence, option C is the correct answer.

**3) Answer (B)**

Let ‘A’, ‘S’ and ‘P’ be Amita’s, Sumita’s and Paramita’s present age.

It is given that 2 years ago, one-fifth of Amita’s age was equal to one-fourth of the age of Sumita, and the average of their age was 27 years.

$\dfrac{(A-2)+(S-2)}{2} = 27$

$A+S = 58$ … (1)

Also, $\dfrac{A-2}{5} = \dfrac{S-2}{4}$

$4A-8 = 5S-10$

$5S – 4A = 2$ … (2)

From equation (1) and (2) we can say that S = 26, A = 32.

Average age of Amita, Sumita and Paramita before 2 years = 24.

$\dfrac{(A-2)+(S-2)+(P-2)}{3} = 24$

$A+S+P = 78$. Hence, P = 20.

Therefore, the average age of Sumita and Paramita 3 years from now? = $\dfrac{(S+3)+(P+3)}{2}$ = $\dfrac{(26+3)+(20+3)}{2}$ = 26 years.

Hence, option B is the correct answer.

**4) Answer (D)**

Grade A $\geq$ 90 and Grade B = 87 to 89

If Ramesh scores 70 instead of 97, => Change of marks = 97 – 70 = 27

It creates a change from grade A to B, this means an overall change in average by

= Minimum marks for grade A – Minimum marks for Grade B = 90 – 87 = 3

$\therefore$ Number of subjects = $\frac{27}{3} = 9$

**5) Answer (B)**

Total marks = 80 x 10 = 800

Total marks except highest and lowest marks = 81 x 8 = 648

So Summation of highest marks and lowest marks will be = 800 – 648 = 152

When highest marks is 92, lowest marks will be = 152-92 = 60

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**6) Answer (D)**

As data regarding weights of people is not given, hence we can’t determine the avg. weight of people in group D

**7) Answer (B)**

The first five numbers could be n-2, n-1, n, n+1, n+2. The next two number would then be, n+3 and n+4, in which case, the average of all the 7 numbers would be $\frac{(5n+2n+7)}{7}$ = n+1

**8) Answer (B)**

Let x , y and z be no. of students in class X, Y ,Z respectively.

From 1st condition we have

83*x+76*y = 79*x+79*y which give 4x = 3y.

Next we have 76*y + 85*z = 81(y+z) which give 4z = 5y .

Now overall average of all the classes can be given as $\frac{83x+76y+85z}{x+y+z}$

Substitute the relations in above equation we get,

$\frac{83x+76y+85z}{x+y+z}$ = (83*3/4 + 76 + 85*5/4)/(3/4 + 1 + 5/4) = 978/12 = 81.5

**9) Answer (A)**

Let there be total 100 people whom the college will ask for donation. Out of these 60 people have already given average donation of 600 Rs. Thus total amount generated by 60 people is 36000. This is 75% of total amount required . so the amount remaining is 12000 which should be generated from remaining 40 people. So average amount needed is 12000/40 = 300

**10) Answer (B)**

The odd numbers in the set are 3, 5, 7, …2n+1

Sum of the odd numbers = 3+5+7+…+(2n+1) = $n^2 + 2n$

Average of odd numbers = $n^2 + 2n$/n = n+2

Sum of even numbers = 2 + 4 + 6 + … + 2n = 2(1+2+3+…+n) = 2*n*(n+1)/2 = n(n+1)

Average of even numbers = n(n+1)/n = n+1

So, difference between the averages of even and odd numbers = 1

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