Free DI and LR Practice Test - CAT

Question 1

What is the percentage increase in the number of students going for a job for 1995 to 1997(approximately)?

22%

16%

41%

30%

SOLUTION

Solution : A

In 1995, 85% of 77% of 25000 took up a job= 16362.5
In 1997, 95% of 76% of 27500 took up a job= 19855

Percentage increase = $\frac{19855 - 16362.5}{16362.5}$ = 21.3%

Question 2

How many more students decided to opt for a job in Mumbai in 1999 than in 1997?

1181

1440

2075

2189

SOLUTION

Solution : B

Number of students who took a job in Mumbai in 1997= 18% of 95% of 76% of 27500= 3570
Number who took it up in 1999= 25% of 90% of 79% of 28200= 5010.
About 1440 million more students took up a job in Mumbai in 99 compared to 97

Question 3

What is the total number of students who opted for a job in Calcutta over the three years 1995, 1997 and 1999?

15210

10800

85200

12340

SOLUTION

Solution : B

1995 = 22% of 85% of 77% of 25000= 3600
1997= 21% of 95% of 76% of 27500= 4200
1999= 15% of 90% of 79% of 28200 = 3000
Total= 3600+4200+ 3000=10800

Question 4

What is the Compounded Annual Growth Rate of the number of students taking the Undergraduate exam from 1995 to 1999?

1.5

SOLUTION

Solution : C

We need to look for (factor multiplication) $^{1 / n}$ -1 = (1.13) $^{1 / 4}$ -1 =3%
Shortcut - Go from answer options From the equation above, ((correct answer)% + 1) $^{4}$ = 1.13
Take a middle answer option If 2 is the answer, then 1.02 $^{4}$ = 1.08... we are looking for 1.1
Thus, the answer will be slightly greater than 2. The closest answer without further calculation is 3

Question 5

LCM of two numbers A & B is 72 and HCF is 12. What is the number B?

I. A is not a factor of B.
II. B is greater than A.

A. if the question can be answered by using any of the statements alone but not by using the other statement alone.

B. if the question can be answered by using either of the statements alone

C. if the question can be answered only by using both the statements together.

D. if the question cannot be answered.

SOLUTION

Solution : C

Option c
LCM & HCF of A & B are 72 & 12 respectively.
LCM $\times$ HCF = A $\times$ B(product of the two numbers) $\Rightarrow$ 72 $\times$ 12 = A $\times$ B
Values (A, B) can take so that HCF will be 12 & LCM 72 are (12, 72) and (24, 36)
I. A is not a factor of B, which means A and B can take24 and 36 as values but not necessarily in the same order i.e.,
A = 36 and B = 24 OR
A = 24 and B = 36
II. B is greater than A. This means B can be 36.
Hence II alone is not sufficient. Combining, I & II, we get A = 24 & B = 36. Hence, the answer is (c).

Question 6

What is the age of Ram?
I. Sum of the ages of Ram and Shyam was 60 five years back.

II. Sum of the ages of Ram and Shyam would be 100 fifteen years from now.

A. if the question can be answered by using any of the statements alone but not by using the other statement alone.

B. if the question can be answered by using either of the statements alone

C. if the question can be answered only by using both the statements together.

D. if the question cannot be answered

SOLUTION

Solution : D

Option d
I. Let R & S be the ages of Ram & Shyam respectively.
Age of Ram 5 years back = R - 5 & Shyam = S - 5
(R - 5) + (S - 5) = 60 or R + S = 70 ...(1)
Hence, I alone is not sufficient.
II. Ages of Ram & Shyam fifteen years hence will be R + 15 & S + 15
(R + 15) + ( S + 15) = 100
R + S = 70 .....(2)
Hence, II alone is not sufficient .
Combining I & II,
We are getting the same equation in both case, hence we cannot conclude what are the ages of Ram & Shyam.

Question 7

What is the ratio of volume of sphere to that of the cone?
I. Radius of the cone is twice that of the sphere.

II. Height of the cone is equal to the radius of the sphere.

A. if the question can be answered by using any of the statements alone but not by using the other statement alone.

B. if the question can be answered by using either of the statements alone.

C. if the question can be answered only by using both the statements together.

D. if the question cannot be answered.

SOLUTION

Solution : C

Option c
We need the ratio of both, radius and height, in order to find the ratio of the volume. Thus both the statements are required.

Question 8

Who is/are the tallest among A, B, C, D and E?
I. D is the tallest among C, D and E.

II. B, who is not shorter than D, is not the shorter of A and B.

A. if the question can be answered by using any of the statements alone but not by using the other statement alone.

B. if the question can be answered by using either of the statements alone.

C. if the question can be answered only by using both the statements together.

D. if the question cannot be answered.

SOLUTION

Solution : D

I. D is the tallest among C, D & E. But, we don't know anything about A & B. Hence, I is not sufficient.
II. B is not shorter than D, means B is either taller than or equal to D, But, this is not sufficient, as nothing is known about C & E. Combining I & II,

We get that D is the tallest of C, D & E. And B not shorter of A & B. Among B & D it is not known whether they are of equal height or B is taller, which means either B is the tallest in the group or D;

Question 9

Three children, Ashu, Yacob, and Safina who stay in an orphanage in Bangalore get apples for Christmas. each gets at least one apple. Safina has more apples than Yacob, who has more than Ashu. Together, the total number of apples the three people have is 12. How many apples does Yacob have?
(1) Safina has no more than 5 apples more than Ashu.

(2) The product of the numbers of apples that Ashu, Yacob, and Safina have is less than 36.

A. IF statement (1) alone is sufficient, but statement (2) is not or If Statement (2) alone is sufficient but statement (1) is not

B. If Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient

C. If Each statement ALONE is sufficient

D. Statements (1) and (2) TOGETHER are not sufficient

SOLUTION

Solution : B

x $\to$ number of apples Ashu has
y $\to$ number of apples Yacob has
z $\to$ number of apples Safina has
$x < y < z$
x + y + z = 12
$\begin{matrix} C a s e s & x & y & z (a) & 1 & 2 & 9 (b) & 1 & 3 & 8 (c) & 1 & 4 & 7 (d) & 1 & 5 & 6 (e) & 2 & 3 & 7 (f) & 2 & 4 & 6 (g) & 3 & 4 & 5 \end{matrix}$

Statement (1): INSUFFICIENT. We are told that z - x is less than or equal to 5. This rules out scenarios (a) through (c), but the last four scenarios still work. Thus, y could be 3, 4, or 5.
Statement (2): INSUFFICIENT. We are told that xyz is less than 36. y could be 2, 3, 4, or 5.
Statements (1) and (2) together: SUFFICIENT. Only scenario (d) survives the constraints of the two statements. Thus, we know that y is 5.
The correct answer is (B): BOTH statements TOGETHER are sufficient to answer the question, but neither statement alone is sufficient.

Question 10

Arun draws a regular non-convex polygon, with spokes that extend from each vertex to the center of the board. If each spoke is 8 inches long, and spokes are used nowhere else on the board, what is the sum of the interior angles of the polygon?
(1) The sum of the exterior angles of the polygon is $360^{\circ}$ .
(2) The sum of the exterior angles is equal to five times the total length of all of the spokes used.

A. IF statement (1) alone is sufficient, but statement (2) is not or If Statement (2) alone is sufficient but statement (1) is not

B. If Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient

C. If Each statement ALONE is sufficient

D. Statements (1) and (2) TOGETHER are not sufficient

SOLUTION

Solution : A

The sum of the exterior angles of any polygon is $360^{\circ}$ . Hence, statement is of no use to us
From statement 2-
The sum of the exterior angles = 5 $\times$ length of each spoke $\times$ number of spokes.
360 = 5(8)(x)
360 = 40x
9 = x
The game board has nine sides. The sum of its interior angles is (9 - 2)(180) = 1260. Statement 2 alone is sufficient

Question 11

If X and Y are integers such that X $\neq$ Y, then is |X|Y $>$ 0?
(1) $| X^{Y} | > 0$

(2) $| X |^{Y}$ is a non-zero integer

A. IF statement (1) alone is sufficient, but statement (2) is not or If Statement (2) alone is sufficient but statement (1) is not

B. If Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient

C. If Each statement ALONE is sufficient

D. Statements (1) and (2) TOGETHER are not sufficient

SOLUTION

Solution : D

We need to basically find out if X is non zero and if Y is positive.
Statement (1) Insufficient. We do not know anything about Y
Statement (2) Insufficient. The expression $| X |^{Y}$ is a non-zero integer if y is greater than or equal to zero. The given expression |X|Y is equal to zero if y is zero and greater than zero if y is greater than zero.
Both statements together are also not sufficient. Answer is option (d)

Question 12

\begin{matrix} F i n a n c e & M a r k e t i n g & S o f t w a r e & O t h e r s 1992 & 12 & 36 & 19 & 33 1993 & 17 & 48 & 23 & 12 1994 & 23 & 43 & 21 & 13 1995 & 19 & 37 & 16 & 28 1996 & 32 & 32 & 20 & 16 \end{matrix}

\begin{matrix} F i n a n c e & M a r k e t i n g & S o f t w a r e 1992 & 5450 & 5170 & 5290 1993 & 6380 & 6390 & 6440 1994 & 7550 & 7630 & 7050 1995 & 8920 & 8960 & 7760 1996 & 9810 & 10220 & 8640 \end{matrix}

The number of students who got jobs in finance is less than the number of students getting marketing jobs, in the first 3 years, by

826

650

598

548

SOLUTION

Solution : C

(24% of 800)+ (31% of 600)+ (20% of 1100) = 598

Question 13

In 1994, students seeking jobs in finance earned Rs. _____ more than

those opting for software (in lakhs)

33.8

28.9

38.8

SOLUTION

Solution : B

(23% $\times$ 1100 $\times$ 12 $\times$ 7550)-(21% of 1100 $\times$ 12 $\times$ 7050)= 33.8 lakh

Question 14

The average annual rate at which the initial salary offered in Software, increases

21%

33%

16.3%

65%

SOLUTION

Solution : C

Go from answer options. Factor Multiplication= $\frac{864}{529}$ = 1.6

$(1.2)^{5}$ = 2.5 ( here we should get 1.6)

Answer is less than 20. Answer is option c

Question 15

How many students definitely scored A+ in Test 5?

SOLUTION

Solution : B

We can calculate the total points scored by each candidate in all the tests together using the formula,
Total points = GPA * 5.
Ravi's score in tests 2, 3 and 5 = 23 - 1 - 7 = 15
From the conditions given in the data, Ravi scored 1, 4, 10 or 1, 7, 7 or 4, 4 7 in tests 2, 3 and 5 respectively.
Calculating, in a similar manner, we get that Pankaj scored 0, 1, 10 or 0, 4, 7 in tests 3, 4 and 5 respectively.
Also, Akshay scored 1, 10, 10 or 4, 7, 10 in tests 2, 3 and 5 respectively.
Continuing this way, we get the table below.

$\begin{matrix} N a m e & T e s t 1 & T e s t 2 & T e s t 3 & T e s t 4 & T e s t 5 & T o t a l P o i n t s M a n i s h & 0 & 1 & 4 & 7 & 10 & 22 A l o k & 0 & 0 & 4 & 7 & 7 & 18 V i j a y & 0 & 0 & 1 & 10 & 10 & 21 R a v i & 1 & \frac{1}{4} & \frac{4}{7} & 7 & \frac{7}{10} & 23 R a j & 0 & 1 & 1 & 7 & 10 & 19 M o h i t & 1 & 1 & 7 & 7 & 10 & 26 S a g a r & 1 & 4 & 7 & 10 & 10 & 32 R a k e s h & 0 & 1 & 7 & 7 & 10 & 25 S h e k h a r & 4 & 7 & 10 & 10 & 10 & 41 P a n k a j & 0 & 0 & 0 & \frac{1}{4} & \frac{7}{10} & 11 A k s h a y & 0 & \frac{1}{4} & \frac{7}{10} & 10 & 10 & 31 P a r e s h & 0 & 4 & \frac{4}{7} & \frac{10}{7} & 10 & 28 S a l m a n & 0 & 1 & 7 & 10 & 10 & 28 V i n o d & 4 & 4 & 7 & 10 & 10 & 35 N e e r a j & 0 & 1 & 1 & 4 & 10 & 16 \end{matrix}$

Out of 15 candidates, except Alok, Ravi and Pankaj, all the others scored 10 points or A+ grade in test 5. Hence , [b].

Question 16

The number of candidates who got A+ grade in at least two out of the five tests is:

not more than 5

not more than 6

not more than 7

not more than 4

SOLUTION

Solution : C

Vijay, Sagar, Shekhar, Akshay, Salman and Vinod i.e. 6 candidates got A+ in at least two of the five tests. Paresh may or may not have scored 10 points in at least two out of the five tests. Hence, [c].

Question 17

Which of the following statements cannot be true?

Only two students secured the same grade in three consecutive tests.

B. Exactly five students did not get D grade in any of the tests.

C. The average score of the fifteen students in Test 2 is equal to 2.

D. The average score of the fifteen students in Test 3 is equal to 5.

SOLUTION

Solution : D

Shekhar scored A+ in tests 3, 4 and 5 and Pankaj scored D grade in tests 1, 2 and 3. Ravi may or may not have scored the same grades in three consecutive tests. Thus option [a] may or may not be true.
Ravi, Mohit, Sagar, Shakhar and Vinod did not score D grade in any of the tests.
Thus, option [b] is also true.
The average score of the 15 candidates in tests 2 is $\frac{27}{15} = 1.8$ or $\frac{30}{15} = 2$ or $\frac{33}{15} = 2.2$

Thus, option [c] can be true.
The total score of all the 15 candidates in Test 3, except Ravi, Akshay and Paresh, is 56. Considering all the possibilities of the points scored by these three in Test 5, none of them add up to 19. Thus the average of 5 cannot be achieved.
$(∴ 56 + 19 = 75$ and $\frac{75}{15} = 5)$

Thus option [d] cannot be true. Hence, [d].

Question 18

If Tom lives in room no. 22, then which two students live in room no. 26?

Sachin and Mahesh

Rohan and Chirag

Mohit and Salil

Bharat and Mahesh

SOLUTION

Solution : A

From the $1^{s t}, 2^{n d}, 3^{r d}$ the $6^{t h}$ statements, we can determine where Bharat and Mahesh will stay

Mahesh cannot stay in room 22 (i) nor can he stay in room 21 or 23 (ii) nor can he stay in room 25(vi). Thus Mahesh stays in room 26. Similarly, we can deduce that Bharat stays in room 21.

Now Rohan and Chirag can stay in Room 23 or 25. But If they stay in Room 25, then Chetan will stay in room 24 (which will violate condition ii). Hence, Rohan and Chirag stay on room 23 and there are totally 2 possible cases

Sachin and Mahesh

Question 19

Who among the following cannot live in the house adjacent to that of Chirag?

Bhavin

Tom

Salil

Mahesh

SOLUTION

Solution : D

(d) Mahesh

Question 20

Who among the following definitely lives in room no. 21?

A. Tarun

B. Bharat

C. Tom

D. Salil

SOLUTION

Solution : B

(b) Bharat

Question 21

Among them who scored the second highest marks?

A. Ninja

B. Mahoud

C. Chandru

D. Raul

SOLUTION

Solution : B

Let the marks scored by Charles, David, Hanish, Kedar, Mahoud, Ninja and Raul be represented by C, D, H, K, M, N and R respectively.
From (a), all scored distinct marks:
From (b), C $>$ K $>$ D
From (c), H $>$ M, N $>$ R
From (d), Kedar scored the fourth highest marks.
From (e), M $>$ C
From (c), R is not the least marks scored.
D is the least.
From (b), (c) and (e), H $>$ M $>$ C $>$ K $>$ D
As K is the fourth highest score and D the lowest score and as N $>$ R, N and R must be the fifth and sixth highest scores respectively.

H $>$ M $>$ C $>$ K $>$ N $>$ R $>$ D

Mahoud scored the second highest marks.

Question 22

Among them who scored the second lowest marks?

A. Charles

B. Raul

C. Charles

D. Ninja

SOLUTION

Solution : B

Raul scored the second lowest marks.

Question 23

How many people scored more marks than Charles?

A. 5

B. 4

C. 3

D. 2

SOLUTION

Solution : D

Only Hanish and Mahoud scored more marks than Charles

Question 24

Who entered first?

A. A

B. C

C. D

D. E

SOLUTION

Solution : B

From (a), (c) and (d), and as there are only five boys.
C and D are sitting together, and B is two places away from E. This is possible only when one of B and E is sitting in the middle of the row and the other sitting at any of the extreme ends of the row and A is sitting between B and E.
From (b) and (f), we can say that B cannot sit at any of the extreme ends of the row.
B is sitting at the middle of the row and E is sitting at one of the extreme ends of the row.
We have two possibilities:
Case (1) CDBAE
Case (2) EABCD
Consider case (1):
From (e), C entered second, from (f) A and D would have entered third and fourth respectively.
E entered first, which violates condition (f).
Case (a) is not possible
Consider case (2):
From (e) E entered second.
From (f), A and D would have entered third and fourth respectively. Hence, C entered first.

C entered first.

Question 25

Who is sitting at the middle of the row?

A. A

B. B

C. C

D. D

SOLUTION

Solution : B

B is sitting at the middle of the row.

Question 26

Who is to the immediate left of C?

A. Q

B. P

C. R

D. S

SOLUTION

Solution : A

From the given information, boys and girls should be sitting alternately.
From (a) and (c), we can say that C and B are opposite each other, and A and D are opposite each other.
From (b) and (c), and the above results, we have the following arrangement:

Using (d) since D is sitting three places away to the right of P, we can decide the location of P.

From (c), R must be to the immediate left of A and C is to the immediate left of R.
The final arrangement will be as follows:

Q is to the immediate left of C.

Question 27

Who is two places away to the left of D?

A. A

B. B

C. C

D. Either (a) or (b)

SOLUTION

Solution : B

B is two places away to the left of D.

Question 28

Who is sitting to the immediate right of Q?

A. A

B. B

C. C

D. D

SOLUTION

Solution : C

C is to the immediate right of Q.

Question 29

How many trees did Suresh plant in month 4 ?___

SOLUTION

Solution : $\begin{matrix} N a m e & M o n t h 1 & M o n t h 2 & M o n t h 4 & M o n t h 4 & M o n t h 5 & A v e r a g e S u r e s h & 12 & 27 & 37 & 47 & 57 & 36 V i r a t & 11 & 21 & 31 & 46 & 56 & 33 R o h i t & 9 & 24 & 34 & 44 & 54 & 33 A j i n k y a & 10 & 20 & 35 & 45 & 55 & 33 M a h e n d r a & 15 & 25 & 35 & 50 & 60 & 37 S h i k a r & 21 & 31 & 46 & 56 & 66 & 44 \end{matrix}$

Had each of the six planted 10 trees more in a month than in the previous month for all the five months, their averages would be:

Suresh: 32, Virat: 31, Rohit: 29, Ajinkya: 30, Mahendra: 35, Shikhar: 41.

But the actual averages are: Suresh: 36, Virat: 33, Rohit: 33, Ajinkya: 33, Mahendra: 37, Shikhar: 44.

Since there is a difference of 4 between the expected average and actual average, Suresh planted 4 × 5 = 20 more trees than expected. Similarly, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 10, 20, 15, 10 and 15 more trees respectively than expected.

Now If any of these people had planted 15 trees more than the previous month in month 5, then the number of excess trees planted would only have been 5. Similarly, had the person planted 15 trees more than the previous month in month 4, the number of excess trees planted would be 5 + 5 = 10. Thus, if a person planted 15 trees more than the previous month in month 2, month 3, month 4 or month 5, the corresponding number of total excess trees would be 20, 15, 10 and 5 respectively. Thus, Suresh, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 2, month 4, month 2, month 3, month 4 and month 3 respectively.

Thus, the number of trees planted by them in each month can be calculated as shown in the table above.

Thus, Suresh planted 47 trees in month 4.

Alternatively,

From month 2 onwards, each person plants 10 more trees in every subsequent month and 15 more trees than the previous month in exactly one month. If the person plants x trees in month 1 and had planted exactly 10 more trees in each subsequent month, the total number of trees planted by that person would be x + (x+ 10) + (x+ 20) + (x+ 30) + (x+ 40) = 5x+ 100.

Now, if the person had planted 15 trees more than the previous month in month 2, the total number of trees would be x + (x+ 15) + (x+ 25) + (x+ 35) + (x+ 45) = 5x+ 100 + (5 × 4). On the other hand, if the person had planted 15 trees more than the previous month in month 5, the total number of trees would be x+ (x+ 10) + (x+ 20) + (x+ 30) + (x+ 45) = 5x+ 100 + (5 × 1). Thus, if a person had planted 15 more trees than the previous month in month, the total number of trees planted by that person = 5x+ 100 + (5 × No. of months to go from month of planting to end of tenure (p)).

Total number of trees planted by Suresh = 36 × 5 = 180

∴(5 × 12) + 100 + 5p= 180 ⇒p= 4

This simply means that Suresh planted 15 more trees than the previous month in month 4.

Using this approach, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 4, month 2, month 3, month 4 and month 3 respectively.
Thus, Suresh planted 47 trees in month 4.

Question 30

In which month did Shikhar plant 15 more trees than in the previous month? Just enter the numeric value for e.g. if the answer is Month 3, write 3.
___

SOLUTION

Solution :
$\begin{matrix} N a m e & M o n t h 1 & M o n t h 2 & M o n t h 4 & M o n t h 4 & M o n t h 5 & A v e r a g e S u r e s h & 12 & 27 & 37 & 47 & 57 & 36 V i r a t & 11 & 21 & 31 & 46 & 56 & 33 R o h i t & 9 & 24 & 34 & 44 & 54 & 33 A j i n k y a & 10 & 20 & 35 & 45 & 55 & 33 M a h e n d r a & 15 & 25 & 35 & 50 & 60 & 37 S h i k a r & 21 & 31 & 46 & 56 & 66 & 44 \end{matrix}$

Had each of the six planted 10 trees more in a month than in the previous month for all the five months, their averages would be:

Suresh: 32, Virat: 31, Rohit: 29, Ajinkya: 30, Mahendra: 35, Shikhar: 41.

But the actual averages are: Suresh: 36, Virat: 33, Rohit: 33, Ajinkya: 33, Mahendra: 37, Shikhar: 44.

Since there is a difference of 4 between the expected average and actual average, Suresh planted 4 × 5 = 20 more trees than expected. Similarly, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 10, 20, 15, 10 and 15 more trees respectively than expected.

Now If any of these people had planted 15 trees more than the previous month in month 5, then the number of excess trees planted would only have been 5. Similarly, had the person planted 15 trees more than the previous month in month 4, the number of excess trees planted would be 5 + 5 = 10. Thus, if a person planted 15 trees more than the previous month in month 2, month 3, month 4 or month 5, the corresponding number of total excess trees would be 20, 15, 10 and 5 respectively. Thus, Suresh, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 2, month 4, month 2, month 3, month 4 and month 3 respectively.

Thus, the number of trees planted by them in each month can be calculated as shown in the table above.

Thus, Suresh planted 47 trees in month 4.

Alternatively,

From month 2 onwards, each person plants 10 more trees in every subsequent month and 15 more trees than the previous month in exactly one month. If the person plants x trees in month 1 and had planted exactly 10 more trees in each subsequent month, the total number of trees planted by that person would be x + (x+ 10) + (x+ 20) + (x+ 30) + (x+ 40) = 5x+ 100.

Now, if the person had planted 15 trees more than the previous month in month 2, the total number of trees would be x + (x+ 15) + (x+ 25) + (x+ 35) + (x+ 45) = 5x+ 100 + (5 × 4). On the other hand, if the person had planted 15 trees more than the previous month in month 5, the total number of trees would be x+ (x+ 10) + (x+ 20) + (x+ 30) + (x+ 45) = 5x+ 100 + (5 × 1). Thus, if a person had planted 15 more trees than the previous month in month, the total number of trees planted by that person = 5x+ 100 + (5 × No. of months to go from month of planting to end of tenure (p)).

Total number of trees planted by Suresh = 36 × 5 = 180

∴(5 × 12) + 100 + 5p= 180 ⇒p= 4

This simply means that Suresh planted 15 more trees than the previous month in month 4.

Using this approach, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 4, month 2, month 3, month 4 and month 3 respectively.

Shikhar planted 15 more trees than in the previous month in month 3.

Question 31

Had Ajinkya not planted any tree in month 5, by how much would his average change?
___

SOLUTION

Solution :
$\begin{matrix} N a m e & M o n t h 1 & M o n t h 2 & M o n t h 4 & M o n t h 4 & M o n t h 5 & A v e r a g e S u r e s h & 12 & 27 & 37 & 47 & 57 & 36 V i r a t & 11 & 21 & 31 & 46 & 56 & 33 R o h i t & 9 & 24 & 34 & 44 & 54 & 33 A j i n k y a & 10 & 20 & 35 & 45 & 55 & 33 M a h e n d r a & 15 & 25 & 35 & 50 & 60 & 37 S h i k a r & 21 & 31 & 46 & 56 & 66 & 44 \end{matrix}$

Had each of the six planted 10 trees more in a month than in the previous month for all the five months, their averages would be:

Suresh: 32, Virat: 31, Rohit: 29, Ajinkya: 30, Mahendra: 35, Shikhar: 41.

But the actual averages are: Suresh: 36, Virat: 33, Rohit: 33, Ajinkya: 33, Mahendra: 37, Shikhar: 44.

Since there is a difference of 4 between the expected average and actual average, Suresh planted 4 × 5 = 20 more trees than expected. Similarly, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 10, 20, 15, 10 and 15 more trees respectively than expected.

Now If any of these people had planted 15 trees more than the previous month in month 5, then the number of excess trees planted would only have been 5. Similarly, had the person planted 15 trees more than the previous month in month 4, the number of excess trees planted would be 5 + 5 = 10. Thus, if a person planted 15 trees more than the previous month in month 2, month 3, month 4 or month 5, the corresponding number of total excess trees would be 20, 15, 10 and 5 respectively. Thus, Suresh, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 2, month 4, month 2, month 3, month 4 and month 3 respectively.

Thus, the number of trees planted by them in each month can be calculated as shown in the table above.

Thus, Suresh planted 47 trees in month 4.

Alternatively,

From month 2 onwards, each person plants 10 more trees in every subsequent month and 15 more trees than the previous month in exactly one month. If the person plants x trees in month 1 and had planted exactly 10 more trees in each subsequent month, the total number of trees planted by that person would be x + (x+ 10) + (x+ 20) + (x+ 30) + (x+ 40) = 5x+ 100.

Now, if the person had planted 15 trees more than the previous month in month 2, the total number of trees would be x + (x+ 15) + (x+ 25) + (x+ 35) + (x+ 45) = 5x+ 100 + (5 × 4). On the other hand, if the person had planted 15 trees more than the previous month in month 5, the total number of trees would be x+ (x+ 10) + (x+ 20) + (x+ 30) + (x+ 45) = 5x+ 100 + (5 × 1). Thus, if a person had planted 15 more trees than the previous month in month, the total number of trees planted by that person = 5x+ 100 + (5 × No. of months to go from month of planting to end of tenure (p)).

Total number of trees planted by Suresh = 36 × 5 = 180

∴(5 × 12) + 100 + 5p= 180 ⇒p= 4

This simply means that Suresh planted 15 more trees than the previous month in month 4.

Using this approach, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 4, month 2, month 3, month 4 and month 3 respectively.

Thus, Suresh planted 47 trees in month 4.

If Ajinkya did not plant 55 trees in month 5, the total number of trees planted by him would drop by 55 and his average would change by 55/5 = 11.

Note:Even if Ajinkya does not plant any trees in a month, his average would still be calculated for all the five months, and not just for the number of months in which he planted trees.

Question 32

Find the difference in the average number of trees planted by Virat and Mahendra in the first four months.___

SOLUTION

Solution :
$\begin{matrix} N a m e & M o n t h 1 & M o n t h 2 & M o n t h 4 & M o n t h 4 & M o n t h 5 & A v e r a g e S u r e s h & 12 & 27 & 37 & 47 & 57 & 36 V i r a t & 11 & 21 & 31 & 46 & 56 & 33 R o h i t & 9 & 24 & 34 & 44 & 54 & 33 A j i n k y a & 10 & 20 & 35 & 45 & 55 & 33 M a h e n d r a & 15 & 25 & 35 & 50 & 60 & 37 S h i k a r & 21 & 31 & 46 & 56 & 66 & 44 \end{matrix}$

Had each of the six planted 10 trees more in a month than in the previous month for all the five months, their averages would be:

Suresh: 32, Virat: 31, Rohit: 29, Ajinkya: 30, Mahendra: 35, Shikhar: 41.

But the actual averages are: Suresh: 36, Virat: 33, Rohit: 33, Ajinkya: 33, Mahendra: 37, Shikhar: 44.

Since there is a difference of 4 between the expected average and actual average, Suresh planted 4 × 5 = 20 more trees than expected. Similarly, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 10, 20, 15, 10 and 15 more trees respectively than expected.

Now If any of these people had planted 15 trees more than the previous month in month 5, then the number of excess trees planted would only have been 5. Similarly, had the person planted 15 trees more than the previous month in month 4, the number of excess trees planted would be 5 + 5 = 10. Thus, if a person planted 15 trees more than the previous month in month 2, month 3, month 4 or month 5, the corresponding number of total excess trees would be 20, 15, 10 and 5 respectively. Thus, Suresh, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 2, month 4, month 2, month 3, month 4 and month 3 respectively.

Thus, the number of trees planted by them in each month can be calculated as shown in the table above.

Thus, Suresh planted 47 trees in month 4.

Alternatively,

From month 2 onwards, each person plants 10 more trees in every subsequent month and 15 more trees than the previous month in exactly one month. If the person plants x trees in month 1 and had planted exactly 10 more trees in each subsequent month, the total number of trees planted by that person would be x + (x+ 10) + (x+ 20) + (x+ 30) + (x+ 40) = 5x+ 100.

Now, if the person had planted 15 trees more than the previous month in month 2, the total number of trees would be x + (x+ 15) + (x+ 25) + (x+ 35) + (x+ 45) = 5x+ 100 + (5 × 4). On the other hand, if the person had planted 15 trees more than the previous month in month 5, the total number of trees would be x+ (x+ 10) + (x+ 20) + (x+ 30) + (x+ 45) = 5x+ 100 + (5 × 1). Thus, if a person had planted 15 more trees than the previous month in month, the total number of trees planted by that person = 5x+ 100 + (5 × No. of months to go from month of planting to end of tenure (p)).

Total number of trees planted by Suresh = 36 × 5 = 180

∴(5 × 12) + 100 + 5p= 180 ⇒p= 4

This simply means that Suresh planted 15 more trees than the previous month in month 4.

Using this approach, Virat, Rohit, Ajinkya, Mahendra and Shikhar planted 15 more trees than the previous month in month 4, month 2, month 3, month 4 and month 3 respectively.

Thus, Suresh planted 47 trees in month 4.

The average number of trees planted by Virat in the first four months = (11 + 21 + 31 + 46)/4 = 27.25 and the average number of trees planted by Mahendra in the first four months = (15 + 25 + 35 + 50)/4 = 31.25.
Thus, the required difference is 4.

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