# PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning

**Answer the next 4 questions based on the information given below.**

In an examination, there are 100 questions. A student is awarded 6 marks for each correct answer. He loses 2 marks for each wrong answer and loses 1 mark for each un-attempted question.

The net score of Varun in that test is 321.

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

The number of ways in which Varun could have attempted the exam is _____.

- (a)
6

- (b)
5

- (c)
7

- (d)
None of these

Answer: Option B

**Explanation** :

Let

x = Number of correct answers

y = Number of wrong answers

z = Number of unattempted questions.

Given, x + y + z = 100 …(1)

6x – 2y – z = 321 …(2)

(1) + (2) :

⇒ 7x - y = 421

⇒ y = 7x - 421

We know y ≥ 0

Hence, 7x - 421 ≥ 0

⇒ x ≥ 421/7

⇒ x ≥ 60.14

∴ Least value of x can be 61, for which y = 6 and z = 33.

Next possible value of x = 62, for which y = 13 and z = 25.

Next possible value of x = 63, for which y = 20 and z = 17.

Next possible value of x = 64, for which y = 27 and z = 9.

Next possible value of x = 65, for which y = 34 and z = 1.

∴ There are 5 ways of attempting the exam.

Hence, option (b).

Workspace:

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

The maximum number of questions attempted by Varun could be .

Answer: 99

**Explanation** :

Consider the solution to first question of this set.

We need to maximize (x + y) and hence minimize z.

The least value of z is 1, hence maximum value of (x + y) = 100 – 1 = 99.

Hence, 99.

Workspace:

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

What is the number of questions that Varun got wrong if the number of correct questions that he got is a multiple of 8?

Answer: 27

**Explanation** :

Consider the solution to first question of this set.

The number of correct questions can only be 64 (since it has to be a multiple of 8).

∴ Number of wrong questions will be 27.

Hence, 27.

Workspace:

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

What can be the maximum number of questions Varun left un-attempted?

Answer: 33

**Explanation** :

Consider the solution to first question of this set.

We need to maximize z.

The highest value of z is 33.

Hence, 33.

Workspace:

**Directions for next 5 questions:**

Five friends, Achal, Jitesh, Karthik, Pratyush and Tapish, are trying to build a tower using blocks. Each person arranges certain number of blocks on top of existing blocks. Every day at least 4 persons play this game.

It is known that:

- On any day if both Achal and Jitesh arrange blocks, Jitesh puts 1 block less than Achal.
- On any day if both Jitesh and Karthik arrange blocks, Jitesh puts 2 blocks more than Karthik.
- On any day if both Pratyush and Karthik arrange blocks, Pratyush puts 3 blocks more than Karthik.
- On any day if both Pratyush and Tapish arrange blocks, Pratyush puts 6 blocks more than Tapish.
- On any day if both Tapish and Achal arrange blocks, Achal puts 5 blocks less than Tapish.

The following graph provides, the height of the tower (in terms of number of blocks) at the end of each day.

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

How many blocks were arranged by Tapish across all 5 days.

Answer: 46

**Explanation** :

Total number of blocks arranged on:

Day 1 = 31

Day 2 = 65 – 31 = 34

Day 3 = 98 – 65 = 33

Day 4 = 136 – 98 = 38

Day 5 = 169 – 136 = 33

Let number of blocks arranged on a day by Achal, Jitesh, Karthik, Pratyush and Tapish be A, J, K, P and T.

(1) ⇒ J = A - 1

(2) ⇒ J = K + 2

(3) ⇒ P = K + 3

(4) ⇒ P = T + 6

(5) ⇒ A = T – 5

On a day if all 5 persons arrange blocks and if K arranges x blocks,

(2) ⇒ J arranges x + 2 blocks

(1) ⇒ A arranges x + 3 blocks,

(3) ⇒ P arranges x + 3 blocks,

(4) ⇒ T arranges x – 3 blocks,

(5) ⇒ A arranges x – 8 blocks

Now, (2) and (5) are contradictory, hence all 5 cannot arrange blocks on any day.

Now considering cases where exactly 4 persons arrange blocks on a particular day:

On a day if all 4 persons except T arrange blocks ad if K arranges x blocks,

(2) ⇒ J arranges x + 2 blocks

(1) ⇒ A arranges x + 3 blocks,

(3) ⇒ P arranges x + 3 blocks,

Total blocks arranged on this day = 4x + 8

On a day if all 4 persons except P arrange blocks and if K arranges x blocks,

(2) ⇒ J arranges x + 2 blocks

(1) ⇒ A arranges x + 3 blocks,

(5) ⇒ T arranges x + 8 blocks

Total blocks arranged on this day = 4x + 13

On a day if all 4 persons except J arrange blocks if K arranges x blocks,

(3) ⇒ P arranges x + 3 blocks,

(4) ⇒ T arranges x – 3 blocks,

(5) ⇒ A arranges x – 8 blocks

Total blocks arranged on this day = 4x - 8

On a day if all 4 persons except A arrange blocks if K arranges x blocks,

(2) ⇒ J arranges x + 2 blocks

(3) ⇒ P arranges x + 3 blocks,

(4) ⇒ T arranges x – 3 blocks,

Total blocks arranged on this day = 4x + 2

On a day if all 4 persons except K arrange blocks and if J arranges x blocks,

(1) ⇒ A = x + 1

(5) ⇒ T = x + 6

(4) ⇒ P = x + 12

Total blocks arranged on this day = 4x + 19

Now, number of blocks arranged on day 1 = 31.

If T is not working ⇒ 4x + 8 = 31.

This is rejected as x should be an integer.

If P is not working ⇒ 4x + 13 = 31.

This is rejected as x should be an integer.

If J is not working ⇒ 4x - 8 = 31.

This is rejected as x should be an integer.

If A is not working ⇒ 4x + 2 = 31.

This is rejected as x should be an integer.

If K is not working ⇒ 4x + 19 = 31.

⇒ x = 3

∴ On day 1 number of blocks arranged by each of them is J = 3, A = 4, T = 9 and P = 15

Similarly, we can calculate blocks arranged for remaining 4 days as well.

Tapish arranges a total of 9 + 5 + 13 + 6 + 13 = 46 blocks,

Hence, 46.

Workspace:

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

What is the least number of blocks arranged by any of the five friends on any of the 5 days.

Answer: 3

**Explanation** :

Consider the solution to first question of this set.

Jitesh arranged the least number of blocks i.e., 3 on Day 1.

Hence, 3.

Workspace:

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

Who among the 5 friends arranged blocks on maximum number of days?

- (a)
Karthik

- (b)
Pratyush

- (c)
Tapish

- (d)
None of these

Answer: Option C

**Explanation** :

Consider the solution to first question of this set.

Jitesh and Tapish both arranged blocks on all 5 days.

Hence, (c).

Workspace:

**PE 2 - Mathematical Reasoning | LR - Mathematical Reasoning**

What is the difference between the highest and least number of total blocks arranged by these 5 friends?

Answer: 26

**Explanation** :

Consider the solution to first question of this set.

Highest total number of blocks arranged is by Tapish i.e., 46

Lowest total number of blocks arranged is by Achal i.e., 20.

Required difference = 46 – 20 = 26.

Hence, 26.

Workspace:

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