Explanation : 

Problem: Find (PQ) × 3 = RQQ, where P, Q, and R are distinct digits and R ≠ 0

Step 1: Express the Equation
(10P + Q) × 3 = 100R + 10Q + Q
30P + 3Q = 100R + 11Q
30P = 100R + 8Q
15P = 50R + 4Q

Step 2: Analyze the Equation
15P – 50R = 4Q
15P ≡ 50R (mod 4)

Step 3: Find the Values
Case 1: P = 8, R = 2
15 × 8 = 120 = 50 × 2 + 4Q
120 = 100 + 4Q
4Q = 20
Q = 5

Verification:
85 × 3 = 255
This confirms P = 8, R = 2, Q = 5 are distinct digits and R ≠ 0

Step 4: Calculate (P + R)/Q
(8 + 2)/5 = 10/5 = 2

Final Answer: 2

Explanation : 

3P + 4P + PP + PP = RQ2

Breaking it down:
(30 + P) + (40 + P) + (10P + P) + (10P + P) = 100R + 10Q + 2

Simplifying:
24P + 70 = 100R + 10Q + 2

Further Breaking Down:
20P + 70 + 4P = 100R + 10Q + 2

Finding Value of P:
– Unit digit is 2
– Must come from 4 × P
– Therefore, P must be 3 or 8

Case 1 (P = 3):
24P + 70 = (24 × 3) + 70
= 72 + 70
= 142

Case 2 (P = 8):
24P + 70 = (24 × 8) + 70
= 192 + 70
= 262

Arithmetic Mean:
(142 + 262) ÷ 2 = 202

Answer: 202

Explanation : 

Objective type test of 90 questions

Given Information:
Total questions = 90
Marks for correct answer = +5
Marks for incorrect answer = -2
Total marks obtained = 387

Let’s solve:
x = number of correct answers
y = number of incorrect answers

Equation 1 (Total Questions):
x + y = 90

Equation 2 (Total Marks):
5x – 2y = 387

Solving the System of Equations:
From Equation 1: x + y = 90
x = 90 – y
Substitute in Equation 2:
5(90 – y) – 2y = 387
450 – 5y – 2y = 387
450 – 7y = 387
-7y = -63
y = 9

Therefore: Incorrect answers = 9
Correct answers = 81 (90 – 9)

Explanation : 

A person p asks one of his three friends

Initial Conditions:
X has x rupees
Y has y rupees
Z has z rupees

Statement 1 – If Y gives X Rs. 40:
Y’s new amount = Half of Z’s amount
Equation: y – 40 = (1/2)z

Statement 2 – If Z gives X Rs. 40:
All three will have equal amounts
Equation: x + 40 = y = z – 40

Solving the Equations:
1. From first equation: y – 40 = (1/2)z
Therefore: 2y – 80 = z

2. From second equation: x + 40 = y = z – 40
Substituting z: x + 40 = y = (2y – 80) – 40
x + 40 = y = 2y – 120
Therefore: y = 120

Final Values:
X has Rs. 80
Y has Rs. 120
Z has Rs. 160

Explanation : 

Jay and Vijay spent an equal amount of money

Let’s say the price of a pen is x and the price of a pencil is y.

Since Jay and Vijay spent an equal amount of money, we can write an equation to represent this: 3x + 5y = 2x + 7y.

Solving this equation for x and y, we get: 3x + 5y = 2x + 7y
3x – 2x = 7y – 5y ; x = 2y

This means that the price of a pen is two times the price of a pencil. So the correct answer is © The price of a pen is two times the price of a pencil.

Explanation : 

Amount of money was distributed among a b and c

Ratio of distribution of money among A, B, and C is p : q : r.

Considering statement 1:
If p > (q + r), then p is definitely the largest number.
So, A must have got the maximum share. Hence, statement 1 is correct.

Considering statement 2:
If r < (p + q), then r may or may not be the smallest number.
For example, 5 < (2 + 4)
So, C may or may not have got the minimum share. Hence, statement 2 is incorrect.

Explanation : 

6 Person in a row handshake combinations

Let’s consider the 6 persons in a row as A, B, C, D, E, and F. The person can shake hands with 3 of them in the following distinct combinations:

1. A C E
2. A C F
3. A D F
4. B D F

In each of these combinations, the person does not shake hands with two consecutive persons. So there are a total of 4 distinct possible combinations for the handshakes to take place, and the correct answer is (b) 4.

Explanation : 

Cubical vessel of side 1 m

Calculating Volume of a Cube
Side length = 1 meter
Volume formula = side length³
Volume = 1 × 1 × 1 = 1 cubic meter

Converting to Milliliters:
1 cubic meter = 1000 liters
1 liter = 1000 milliliters

Final Calculation:
Total milliliters = 1000 × 1000
= 1,000,000 milliliters

Answer: (d) 1,000,000 milliliters

Explanation : 

Sum & Product of Consecutive Integers

Problem Part 1 – Five Consecutive Integers
Let’s say the first integer is x.
The next four numbers are: x+1, x+2, x+3, and x+4
The sum of these five numbers is 5x + 10
If this sum equals 100, then:
5x + 10 = 100
Solving for x: x = 18
Therefore, the five consecutive integers are: 18, 19, 20, 21, 22

Problem Part 2 – Numbers 1, 2, and 3
Sum: 1 + 2 + 3 = 6
Product: 1 × 2 × 3 = 6

Conclusion:
Both statements are correct since:
1. We found five consecutive integers that sum to 100
2. For numbers 1, 2, and 3, their sum equals their product

Explanation : 

Numbers greater than 30000, using 22333 as digits

Given:
– Digits available: 2,2,3,3,3 (two 2’s and three 3’s)
– Number must be > 30000
– First digit must be 3

Step 1: Organize possible cases
Case 1: First two digits are 33
Remaining digits: 2,2,3
Possible arrangements:
1. 33223
2. 33232
3. 33322

Case 2: First digit 3, second digit 2
Remaining digits: 2,3,3
Possible arrangements:
4. 32233
5. 32323
6. 32332

Step 2: List all numbers in ascending order
1. 32233
2. 32323
3. 32332
4. 33223
5. 33232
6. 33322

Answer: 6 distinct numbers can be formed (Option b is correct)