Reasoning

Arithmetic Reasoning

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HARD

Suppose a variation of the chocolate problem: If Ayushi, Sadhana, and Tanya each took 1/2 of the remaining chocolates instead of 1/3. If 27 chocolates were left, what would be the initial total?

108

216

162

Explanation

🌟 Quick Verification

Direct Answer: The initial total would be 216.

🧠 Concept Blueprint

Modified Unit Method:
- Three operations of 1/2 implies $2^3 = 8$ units.
- Ayushi takes 1/2 (4 units), remaining 4.
- Sadhana takes 1/2 of 4 (2 units), remaining 2.
- Tanya takes 1/2 of 2 (1 unit), remaining 1.
- We know 1 unit = 27 chocolates.
- Total = 8 units × 27 = 216.

📊 Fact Spotlight

Divisor	Stages	Total Units ( $D^n$ )
3	3	27
2	3	8
4	2	16

MEDIUM

Consider the following statements regarding student ratios and bus travel metrics:

If the number of 10th-grade students is 1/3 more than 12th-grade students, the ratio of 10th to 12th grade is 4:3.
In a bus where 90 students are present with a 2:1 boy-to-girl ratio, changing the ratio to 3:2 results in a change of 6 boys.
The total population in a Venn diagram problem is the sum of those participating in only one event, exactly two events, and all three events.

Which of the statements given above are correct?

1 and 2 only

2 and 3 only

1 and 3 only

1, 2, and 3

Explanation

🌟 Quick Verification

Direct Answer: All three statements are correct.
Core Concept: Ratio application, population shifting, and set theory logic (Venn diagrams).

🧠 Concept Blueprint

Statement 1 Verification: A ratio of '1/3 more' implies $1 + 1/3 = 4/3$ , hence the ratio is indeed 4:3. For 112 students, 7 units equal 112, 1 unit = 16, so 10th grade (4 units) = 64.
Statement 2 Verification: Day 1: 90 students, 2:1 ratio (60 boys, 30 girls). Day 2: 90 students, 3:2 ratio (5 units total = 18/unit). 3 units * 18 = 54 boys. The difference is 60 - 54 = 6.
Statement 3 Verification: In Venn logic, the Universal set equals the sum of mutually exclusive partitions: (Only One) + (Exactly Two) + (All Three).

📊 Fact Spotlight

Scenario	Method/Formula	Exam Significance
1/N more ratio	(N+1)/N	Standard ratio conversion trap
Population Shift	(Total * Ratio1) vs (Total * Ratio2)	Often tests changes in specific subsets
Venn Logic	Total = Sum of disjoint regions	Essential for group participation problems

EASY

Match the following problem types with their corresponding solution methodologies:

Type	Methodology
A. Taxi Fare	1. Linear Equation system (Total = Fixed + Per km × Dist)
B. Salary Calculation	2. Multiplication of monthly rate - Deduction for leaves
C. Work/Time	3. Total work = Rate × Time
D. Probability	4. Favorable outcomes / Total outcomes

Select the correct code from the options below:

A-1, B-2, C-3, D-4

A-2, B-1, C-4, D-3

A-1, B-3, C-2, D-4

A-4, B-2, C-3, D-1

Explanation

🌟 Quick Verification

Direct Answer: A-1, B-2, C-3, D-4.
Core Concept: Mathematical problem-solving frameworks.

🧠 Concept Blueprint

Taxi Fare: Uses a linear model where a fixed base cost is added to a variable cost based on distance traveled.
Salary Calculation: Annual income is determined by fixed monthly revenue minus penalty deductions for absent days.
Work/Time: Fundamental physics/math principle: Work is the product of capacity/rate and duration.
Probability: Basic statistical definition of occurrence likelihood.

💡 Exam Strategy & Trivia

Tip

Common Trap: In salary/leave problems, always convert monthly salary to annual salary before subtracting total deductions to avoid scaling errors.

MEDIUM

In a travel scenario, 2 tickets from Mohali to Delhi and 3 tickets from Mohali to Jammu cost ₹800. If 3 tickets from Mohali to Delhi and 2 tickets from Mohali to Jammu cost ₹700, what would be the cost of 5 tickets from Mohali to Delhi and 5 tickets from Mohali to Jammu?

₹1,500

₹1,200

₹1,600

₹1,400

Explanation

🌟 Quick Verification

Direct Answer: ₹1,500.
Core Concept: System of linear equations.

🧠 Concept Blueprint

Equation Setup:
- 2D + 3J = 800
- 3D + 2J = 700
Logical Shortcut: Add both equations:
- (2D + 3D) + (3J + 2J) = 800 + 700
- 5D + 5J = 1500.

📊 Fact Spotlight

Ticket Type	Individual Price
Mohali-Delhi (D)	₹100
Mohali-Jammu (J)	₹200
Combined (D+J)	₹300

HARD

Consider the following scenario for a training camp:

14% of students play Cricket.
22% of the remaining students play Hockey.
Half of the remaining students play Badminton.

What is the final percentage of students with no training?

30.50%

33.54%

36.25%

28.75%

Explanation

🌟 Quick Verification

Direct Answer: The percentage of students with no training is 33.54%.

🧠 Concept Blueprint

Step-by-Step Calculation:
- Start: 100%.
- After Cricket: 100% - 14% = 86% remaining.
- Hockey (22% of 86%): 0.22 * 86 = 18.92%. Remaining: 86 - 18.92 = 67.08%.
- Badminton (Half of 67.08%): 67.08 / 2 = 33.54%. Remaining: 67.08 - 33.54 = 33.54%.

💡 Exam Strategy & Trivia

Tip

Common Trap: Many students mistakenly add percentages (14% + 22% + 50%) and subtract from 100%. Always subtract sequentially from the remaining balance to correctly model the 'of the remaining' clause.

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MEDIUM

Consider the following statements regarding the 'Chocolate Distribution' problem provided in the text:

The initial total number of chocolates was assumed to be 27 units because the operation of taking 1/3 was performed three times (3^3).
After Ayushi, Sadhana, and Tanya finished taking their portions, exactly 24 chocolates remained on the table.
The actual number of chocolates each person should have received to ensure an equal distribution is 27.

Which of the statements given above are correct?

1 and 2 only

2 and 3 only

1 and 3 only

1, 2, and 3

Explanation

🌟 Quick Verification

Direct Answer: All statements provided are correct.
Core Concept: Algebraic distribution using units to avoid fractional complexity.

🧠 Concept Blueprint

Unit Method Efficiency:
- By assuming a total of 27 units ( $3^3$ ), we avoid dealing with fractions at every step.
- Ayushi takes 1/3 (9 units), leaving 18. Sadhana takes 1/3 of 18 (6 units), leaving 12. Tanya takes 1/3 of 12 (4 units), leaving 8 units.
- Since 8 units = 24 chocolates, 1 unit = 3 chocolates.
- Total initial chocolates = 27 units × 3 = 81.
- Equal share = 81 / 3 = 27 per person.

📊 Fact Spotlight

Participant	Units Taken	Real Value (Chocolates)
Ayushi	9	27
Sadhana	6	18
Tanya	4	12

💡 Exam Strategy & Trivia

Tip

Exam Trap: Always check if the question asks for the 'remaining' or the 'total' chocolates. Here, 24 is the final remainder, not the total.

Memorable Trivia: Using the power of the divisor (e.g., $3^n$ ) is a classic technique in competitive exams to solve cascading fractional reduction problems quickly.

EASY

A person has a monthly salary of ₹6,000. He takes 15 leaves in a year, with a deduction of ₹200 per leave. What is his total annual earning?

₹69,000

₹72,000

₹75,000

₹68,500

Explanation

🌟 Quick Verification

Direct Answer: The total annual earning is ₹69,000.

🧠 Concept Blueprint

Calculation:
- Gross Annual Salary: 6,000 * 12 = 72,000.
- Total Deduction: 15 leaves * 200/leave = 3,000.
- Net Earnings: 72,000 - 3,000 = 69,000.

MEDIUM

A private taxi company has a fixed charge and a per-kilometer rate. For a 24 km journey, the cost is ₹368, and for a 32 km journey, the cost is ₹464. How much will a customer pay for a 15 km journey?

₹260

₹270

₹280

₹250

Explanation

🌟 Quick Verification

Direct Answer: ₹260.
Core Concept: Solving linear equations in two variables for cost estimation.

🧠 Concept Blueprint

Setup Equations:
- Let Fixed = F, Per-km = x.
- F + 24x = 368
- F + 32x = 464
Calculate Variable: Subtracting the two gives 8x = 96, so x = 12.
Calculate Fixed: Substitute x=12: F + 24(12) = 368 => F + 288 = 368 => F = 80.
Final Result: For 15 km, Cost = 80 + 15(12) = 80 + 180 = 260.

💡 Exam Strategy & Trivia

Tip

Time Saver: In such problems, the 'Per-kilometer' charge is simply the difference in cost divided by the difference in distance: (464 - 368) / (32 - 24) = 96 / 8 = 12.

MEDIUM

In the hotel problem, if the initial total number of players was 120, and the distribution logic remains unchanged, what was the initial count of players in Hotel X?

Explanation

🌟 Quick Verification

Direct Answer: The initial count in Hotel X was 40.

🧠 Concept Blueprint

Step-by-Step Logic:
- The problem assumes total players = 300 units, leading to a total of 120 players.
- Initially, Hotel X held 100 units out of 300.
- Proportion = 100/300 = 1/3.
- Count in X = (1/3) × 120 = 40.

💡 Exam Strategy & Trivia

Tip

Common Pitfall: Mixing up the units (300) with the actual player count (120). Always verify the ratio first.

EASY

An employee earns a monthly salary of ₹6,000. There are 15 leaves permitted per year, but a deduction of ₹200 is made for every leave. What is the final annual earning?

₹68,000

₹69,000

₹70,000

₹72,000

Explanation

🌟 Quick Verification

Direct Answer: ₹69,000.
Core Concept: Annualization of monthly data with deduction adjustments.

🧠 Concept Blueprint

Gross Salary: ₹6,000 × 12 months = ₹72,000.
Deductions: 15 leaves × ₹200/leave = ₹3,000.
Net Annual Salary: 72,000 - 3,000 = ₹69,000.

📊 Fact Spotlight

Item	Value
Monthly Base	6,000
Annual Gross	72,000
Total Penalty	3,000
Effective Pay	69,000

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MEDIUM

Four friends (Priya, Kavya, Gaurvi, Shalini) distribute money according to specific conditions:

P + 88 = G
S + K = 550
G + 25 = K
S + K + G = 840

What is the amount held by Priya?

202

290

265

240

Explanation

🌟 Quick Verification

Direct Answer: 202.
Core Concept: Systems of linear equations with multiple variables.

🧠 Concept Blueprint

Solving for Gaurvi: Given S+K+G = 840 and S+K = 550, we substitute (S+K): 550 + G = 840. Therefore, G = 290.
Solving for Priya: Use the first equation: P + 88 = G. Substituting G = 290: P + 88 = 290. P = 290 - 88 = 202.

💡 Exam Strategy & Trivia

Tip

Efficiency Note: In such problems, identify variables that can be isolated immediately. Here, 'G' (Gaurvi) was the bridge between the equation involving all and the specific equation for Priya.

MEDIUM

In a group of 72 workers, all participate in at least one of three activities (Quiz, Mono-acting, Mimicry). 11 participate in Quiz only, 15 in Mono-acting only, and 14 in Mimicry only. 5 participate in all three activities. How many participate in exactly two activities?

Explanation

🌟 Quick Verification

Direct Answer: 27.
Core Concept: Set inclusion-exclusion principle.

🧠 Concept Blueprint

Step-by-Step Logic:
- Total Workers = 72.
- Only One Event = 11 + 15 + 14 = 40.
- All Three Events = 5.
- Let 'X' be those in exactly two events.
- Equation: Total = (Only One) + (Exactly Two) + (All Three).
- 72 = 40 + X + 5.
- 72 = 45 + X => X = 27.

HARD

Abhay and Rajiv consume mangoes at rates of 'x' and 'y' respectively. They can finish a stockpile of 60x mangoes in 36 days together. How many days would it take for Rajiv to finish the stockpile alone?

60 days

75 days

90 days

120 days

Explanation

🌟 Quick Verification

Direct Answer: 90 days.
Core Concept: Rate and work equivalence.

🧠 Concept Blueprint

Equation Formation: Total Work = 60x. Combined rate = (x+y). Time = 36 days.
Equation: 60x = 36(x + y).
Simplification: 60x = 36x + 36y => 24x = 36y => y = (24/36)x = 2/3x.
Rajiv's Time: Time = Total Work / Rajiv's Rate = 60x / (2/3)x = 60 * 3/2 = 90 days.

MEDIUM

Which of the following processes correctly describes the steps to find the percentage of students with NO training in a camp where 14% play Cricket, 22% of the remaining play Hockey, and half of the remaining play Badminton?

Subtract 14+22+50 from 100

Sequentially subtract percentages from the remaining balance starting from 100

Calculate 14% of 100, 22% of 100, 50% of 100, and sum them

Divide 100 by the sum of 14, 22, and 50

Explanation

🌟 Quick Verification

Direct Answer: Sequentially subtract percentages from the remaining balance starting from 100.

🧠 Concept Blueprint

Reasoning: The phrase 'of the remaining' implies that each subsequent percentage is calculated on a shrinking base. You must track the 'remaining' pool size after each group is removed to find the final percentage of untrained individuals (which results in 33.54%).

EASY

In the chocolate distribution scenario, how many additional chocolates must Renuka give to Sadhana to ensure everyone has an equal share?

Explanation

🌟 Quick Verification

Direct Answer: Renuka must give 9 additional chocolates.

🧠 Concept Blueprint

Goal: Everyone must end up with 27 chocolates (Total 81 / 3 daughters).
Calculation:
- Sadhana currently holds 18 chocolates (based on the calculation: 1/3 of 54 remaining chocolates).
- Required share for equality is 27.
- Difference = 27 - 18 = 9.

MEDIUM

Consider the following statements regarding mathematical work-rate, service charges, and ratios:

In a work problem where two individuals complete a task in 36 days with rates 'x' and 'y', if the second person's rate (y) is 2/3 of the first (x), the second person would take 90 days to finish the task alone.
In a taxi service with a fixed charge and a variable rate, if 24 km costs ₹368 and 32 km costs ₹464, the fixed charge is ₹80 and the per-kilometer charge is ₹12.
A ratio defined as '1/3 more' between two quantities implies a ratio of 4:3.

Which of the statements given above are correct?

1 and 2 only

2 and 3 only

1 and 3 only

1, 2, and 3

Explanation

🌟 Quick Verification

Direct Answer: All statements (1, 2, and 3) are correct.

🧠 Concept Blueprint

Statement 1 (Work Rates): If the first person works at rate 'x' and the second at 'y = 2/3x', their combined rate is 5/3x. The total work finished in 36 days is 36 * (5/3x) = 60x. Time for the second person (y) alone is 60x / (2/3x) = 90 days. The calculation holds.
Statement 2 (Taxi Fare): The difference in cost for 8 km (32 - 24) is ₹96 (464 - 368). The rate is 96/8 = ₹12/km. The fixed charge is 368 - (24 * 12) = 368 - 288 = ₹80. The calculation holds.
Statement 3 (Ratio): '1/3 more' means 1 + 1/3 = 4/3, which corresponds to the ratio 4:3. The definition holds.

EASY

Match the problem scenarios with the primary variable setup technique:

Problem Scenario	Primary Variable/Method
A. Taxi Fare Calculation	1. Linear Equation System with Fixed Cost
B. Boys/Girls Ratio Shift	2. Algebraic Substitution of Ratio Units
C. Work/Time Problems	3. Rate × Time = Total Capacity
D. Average Speed	4. Arithmetic Mean of Progression

Select the correct code from the options below:

A-1, B-2, C-3, D-4

A-2, B-1, C-4, D-3

A-1, B-3, C-2, D-4

A-3, B-1, C-2, D-4

Explanation

🌟 Quick Verification

Direct Answer: A-1, B-2, C-3, D-4.
Core Concept: Problem-solving frameworks.

MEDIUM

Consider the scenario where 90 students have a boy-girl ratio of 2:1 on Day 1 and 3:2 on Day 2. How many more boys were present on Day 2 than Day 1?

Explanation

🌟 Quick Verification

Direct Answer: There were 6 more boys on Day 2 (Wait, check calculation: Day 1 boys = 2/3 * 90 = 60. Day 2 boys = 3/5 * 90 = 54. Actually, the change is 6, let's re-verify: 60 - 54 = 6).

🧠 Concept Blueprint

Calculation:
- Total = 90.
- Day 1 (2:1): Boys = 2/3 * 90 = 60.
- Day 2 (3:2): Boys = 3/5 * 90 = 54.
- Difference = 60 - 54 = 6.

EASY

Which of the following describes the correct procedure for calculating annual earnings after accounting for leaves?

Annual Salary = Monthly Salary × 12; Final Earnings = Annual Salary + (Leaves × Deduction).

Annual Salary = Monthly Salary × 12; Final Earnings = Annual Salary - (Leaves × Deduction).

Annual Salary = Monthly Salary × 10; Final Earnings = Annual Salary - (Leaves × Deduction).

Annual Salary = Monthly Salary × 12; Final Earnings = (Annual Salary - Leaves) × Deduction.

Explanation

🌟 Quick Verification

Direct Answer: Annual Salary = Monthly Salary × 12; Final Earnings = Annual Salary - (Leaves × Deduction).
Core Concept: Annualization and penalty deduction logic.

🧠 Concept Blueprint

Logic:
- First, define the total annual capacity (12 months).
- Second, apply penalties (leaves × cost) as subtractions from the total.

MEDIUM

Assertion (A): Linear equations in two variables are highly effective for solving money distribution and ticket pricing problems.
Reason (R): These equations allow for the simultaneous representation of multiple constraints where distinct variables (e.g., ticket types, individuals) are interdependent.

Choose the correct option:

Both A and R are true and R is the correct explanation of A

Both A and R are true but R is NOT the correct explanation of A

A is true but R is false

A is false but R is true

Explanation

🌟 Quick Verification

Direct Answer: Both Assertion (A) and Reason (R) are true, and R is the correct explanation.

🧠 Concept Blueprint

The Utility of Linear Equations: In scenarios like ticket pricing (Delhi vs. Jammu tickets) or money distribution (Priya vs. Kavya), we are dealing with unknowns that satisfy specific, rigid conditions. By setting up equations like 2x + 3y = 800, we can solve for unique values of 'x' and 'y' because the system captures the dependency between variables perfectly.
Exam Strategy: When faced with multi-entity allocation problems, always define your unknowns first (e.g., Let Ticket A = x, Ticket B = y) to simplify the narrative text into solvable algebraic expressions.

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