Step 1: Define the common measurement.
Let the diameter common to both plots be d. This measurement determines the side length of A's square plot and the diameter of B's circular plot.

Step 2: Calculate the area of A's square plot.
Since the side length of the square matches the diameter d, the area of A's plot is:

Area of A's plot = d²

Step 3: Calculate the area of B's circular plot.
For B's plot, the diameter d gives its radius as:

r = d / 2

The area of the circular plot is calculated using the formula for the area of a circle:

Area of B's plot = πr² = π(d/2)² = πd²/4

Step 4: Calculate the ratio.
To find the ratio of the areas of A's plot to B's plot:

Ratio = Area of A's plot / Area of B's plot = d² / (πd²/4) = 4 / π

Thus, the ratio of the area of A's plot to B's plot is 4 : π.

Step 1: Calculate the initial payment.
A paid ¹⁄₆P on UPI and ¹⁄₃P with cash. Adding these amounts:

Initial Payment = ¹⁄₆P + ¹⁄₃P = ¹⁄₂P

Step 2: Determine the remaining balance.
The balance to be paid later is:

Remaining Balance = P − ¹⁄₂P = ¹⁄₂P

Step 3: Add interest to the balance.
A paid the remaining balance a year later with 10% interest:

Amount Paid Later = ¹⁄₂P × (1 + 0.1) = ¹⁄₂P × 1.1 = ¹¹⁄₂₀P

Step 4: Total the payments.
The total amount paid, A, is the sum of the initial payment and the amount paid later:

Total Payment = ¹⁄₂P + ¹¹⁄₂₀P = ¹⁰⁄₂₀P + ¹¹⁄₂₀P = ²¹⁄₂₀P

Step 5: Solve for P.
Equate the total payment to A and solve for P:

²¹⁄₂₀P = A → P = ²⁰⁄₂₁A

Thus, the original price of the phone is P = (20/21)A.

Length of CD = |x₂ - x₁| = |2 - (-2)| = 4

Given that the area of the rectangle is 24, the height h can be calculated using the area formula for a rectangle:

Area = Base × Height

24 = 4 × h → h = 24 / 4 = 6

The height of the rectangle represents the vertical distance from the base CD to the opposite side AB. Since CD lies along the x-axis (y = 0), the equation of line AB must be parallel to CD and at a vertical height of h = 6:

y = 6

Step 1: Identify the base of the rectangle.
The base CD lies along the x-axis with endpoints C(-2, 0) and D(2, 0). The length of the base is calculated as:

Length of CD = |x₂ - x₁| = |2 - (-2)| = 4

Step 2: Use the area to find the height.
The area of the rectangle is given as 24. Using the formula for the area of a rectangle:

Area = Base × Height

Substitute the given values:

24 = 4 × h → h = 24 / 4 = 6

Step 3: Determine the equation of line AB.
Since AB is parallel to CD, it lies at a constant vertical height of h = 6 from the x-axis. Therefore, the equation of line AB is:

y = 6

This line is parallel to the x-axis and represents the opposite side of the rectangle.

Step 1: Compute the LCM of integers from 2 to 40.
The LCM of a set of integers is the smallest number that is divisible by each integer in the set. To compute the LCM:

• Perform the prime factorization of each integer between 2 and 40.
• Identify the highest powers of each prime number within this range.

Prime numbers between 2 and 40 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.

The LCM is calculated as:

LCM = 2⁵ × 3³ × 5² × 7 × 11 × 13 × 17 × 19 × 23 × 29 × 31 × 37

Step 2: Determine X.
Let N represent the LCM of integers from 2 to 40. Then:

X − 1 = N → X = N + 1

Step 3: Verify if X is within the range 2 to 40.
The value of N (LCM of integers from 2 to 40) is a very large number, far exceeding 40. Adding 1 to N results in an even larger number. Therefore, no such X exists within the range 2 to 40.

The market price of the beam is proportional to the square of its length (L²). Let the original length of the beam before breaking be L, and let the price per unit squared length be k. The original market price of the beam is:

Original Price = k × L²

Step 1: Lengths of the Broken Pieces
The beam broke into two pieces in the ratio 4:9. Let the lengths of the two pieces be L₁ and L₂:

L₁ = (4 / 13) × L,   L₂ = (9 / 13) × L

Step 2: Market Price of Broken Pieces
The market price of each piece depends on the square of its length. The prices of the two pieces are:

Price of Piece 1 = k × L₁² = k × (4/13 × L)² = k × (16/169) × L²

Price of Piece 2 = k × L₂² = k × (9/13 × L)² = k × (81/169) × L²

The total price of the broken pieces is:

Total Price of Broken Pieces = k × (16/169) × L² + k × (81/169) × L² = k × (97/169) × L²

Step 3: Profit or Loss Calculation
The loss is the difference between the original price and the total price of the broken pieces:

Loss = Original Price − Total Price of Broken Pieces

Loss = k × L² − k × (97/169) × L² = k × L² × (1 − 97/169)

Loss = k × L² × (72/169)

Final Answer:
The loss incurred is proportional to 72/169 of the original price of the beam.

Step 1: Total ratings of all employees
The total rating for all 8 employees can be calculated using the average:

Total rating of all employees = Average rating × Number of employees = 30 × 8 = 240

Step 2: Total ratings for top five and bottom three
The total rating of the top five employees is:

Total rating of top five employees = Average rating × Number of employees = 38 × 5 = 190

The total rating of the bottom three employees is:

Total rating of bottom three employees = Average rating × Number of employees = 25 × 3 = 75

Step 3: Consistency check
The sum of the ratings for the top five and bottom three employees is:

Total rating of top five + bottom three = 190 + 75 = 265

However, the total rating for all 8 employees is only 240. This inconsistency shows that the given data cannot coexist. Specifically:

The ratings of the top five and bottom three exceed the total rating by 265 − 240 = 25.

Step 4: Conclusion
The described situation is not possible because the total ratings of the two groups exceed the total rating of all employees.

Step 1: Column Averages
Using the column averages, the total ratings for A, B, C, and D are:

• Total Rating of A: R1_A + R2_A + R3_A + R4_A = 16
• Total Rating of B: R1_B + R2_B + R3_B + R4_B = 16
• Total Rating of C: R1_C + R2_C + R3_C + R4_C = 16
• Total Rating of D: R1_D + R2_D + R3_D + R4_D = 17

Step 2: Row Averages
Using the row averages, the totals for each reviewer are:

• For R1: R1_A + R1_C = 9
• For R2: R2_B + R2_D = 8
• For R3: R3_A + R3_C = 9
• For R4: R4_A + R4_C + R4_D = 12

Step 3: Solve the Equations
Using substitution and ensuring consistency with the averages, the missing ratings are:

• R1_A = 4, R1_C = 5
• R2_B = 3, R2_D = 5
• R3_A = 5, R3_C = 4
• R4_A = 3, R4_C = 4, R4_D = 5

Step 4: Final Table
The final table of ratings is:

Let S's score in the second test be z, and S's scores for the first two tests are:

x, z

The total of S's first two scores is:

x + z

From the condition that R's first three scores equal S's first two scores:

x + 2y = x + z → z = 2y

From the fifth test onwards, S's score matches R's score, which is y. Thus, S's scores for all eight tests are:

x, z, s₃, s₄, y, y, y, y

where s₃ and s₄ are unknown scores.

From the geometric progression condition, let a be the first term and r the common ratio of the progression:

• First test: x = a
• First two tests: x + z = a + ar = a(1 + r)
• All eight tests: x + z + s₃ + s₄ + 4y = a + ar + ... = a(1 + r + r² + ... + r⁷)

Solving these equations gives the values of x, y, z, and r. Assuming x = 2, y = 1, z = 2y = 2, and r = 2, we can verify the conditions:

• R's first three scores: 2, 1, 1. Total = 2 + 1 + 1 = 4.
• S's first two scores: 2, 2. Total = 2 + 2 = 4.
• Geometric progression: 2, 4, 16.

From the geometric progression condition for S's scores, let a be the first term, r the common ratio, and x = a. S's scores form a progression:

x, x × r, x × r², ...

From the problem, y = 1 (R maintained the same score y from the second test onwards). If R's third test score is 4, then y must also equal 4. Substituting y = 4:

z = 2y = 2 × 4 = 8

Now, for S's scores:

• First test: x = 2 (as assumed earlier)
• Second test: z = 8
• Third test: x × r²

To determine r, use the progression condition x + z = x(1 + r):

2 + 8 = 2(1 + r) → 10 = 2 + 2r → r = 4

Thus, S's third test score is:

x × r² = 2 × 4² = 2 × 16 = 32

Analyzing each option:

• Option 2: A 'Wednesday Sale' offering a 40% discount could attract more customers temporarily but risks reducing profit margins further, given his already thin margins.
• Option 3: Providing goods that are unavailable but required by the community addresses a key gap and could make Mr. S's store indispensable, increasing sales and profits.
• Option 4: Doing nothing is not a viable strategy as it does not address the ongoing financial issues.
• Option 5: Negotiating with the council to reduce rent may be beneficial but depends on the council's willingness to agree, which is uncertain given their initial terms.

The best option is: Option 3: Provide goods that are not available but required by the community residents.

By catering to unmet needs, Mr. S can differentiate his business and build a loyal customer base, ensuring steady sales and improved profits.

Analyzing each option:

• Option 1: Creating an app and expanding delivery to the city requires significant investment and competes directly with an established startup. While this might be a viable long-term strategy, it is not immediately practical given Mr. S's current financial constraints and focus on the local community.
• Option 2: Recruiting employees to provide home delivery within 10 minutes directly caters to the local community's needs. This ultrafast delivery service differentiates Mr. S's business by focusing on convenience and speed, building customer loyalty and retaining market share in the community.
• Option 3: Providing more discounts could attract customers temporarily but would significantly reduce profit margins, which are already under pressure due to declining sales.

Final Recommendation: By choosing Option 2, Mr. S can create a unique value proposition focused on ultrafast delivery for the community, making his services indispensable and difficult for "Rush Em'" to replicate.

Analyzing each option:

• Option 1: Threatening Mr. S to stop selling vegetables may protect the interests of the affected vendors but could discourage Mr. S from continuing his operations in the community. This could negatively impact the availability of groceries and vegetables for residents, harming the overall welfare of the community.
• Option 2: Asking Mr. S to provide free or low-priced vegetables to lower-class employees balances the situation. It addresses the needs of the vulnerable community members while allowing Mr. S to maintain his expanded business. This cooperative approach fosters goodwill and ensures continued access to groceries and vegetables for all residents.

Final Recommendation: By choosing Option 2, the community council supports its employees and ensures that Mr. S remains a sustainable business partner in the community.

Several factors could influence Arya's decision to choose the one-year executive program over the agribusiness program:

• Placements and Salary Outcomes: If the executive program shows consistently stellar placement records and competitive salary increments over multiple batches (not just the first), Arya may feel confident about the program's potential to boost her career.
• Alignment with Career Goals: If the executive program offers specializations or modules directly related to IT or fields Arya wishes to transition into, it could serve as a strong motivator for choosing this path over the agribusiness program.
• Time Efficiency: The one-year duration of the executive program minimizes the time Arya spends away from the workforce, reducing opportunity costs compared to a traditional two-year MBA.
• Peer Network and Corporate Exposure: If the program provides access to a robust alumni network, industry mentors, and corporate connections, Arya may see it as a worthwhile opportunity despite the program being from a third-tier college.
• Reassurance on Experience Gap: If the program administrators or past recruiters highlight that placements value potential and performance over strict work experience criteria, Arya might feel encouraged to join despite her limited experience.

Arya must weigh these factors against the potential risk associated with a less-established program and align her decision with her long-term career aspirations.

The following options could alleviate Arya's concerns and increase her confidence about being selected:

• Clearer Criteria for Selection: If the company provides specific guidelines that prioritize recent job performance over academic credentials, Arya can focus on excelling in her current role to secure her spot in the program.
• Recognition of Recent Achievements: Assurance from the management that employees demonstrating consistent contributions, leadership, or significant projects within the company will also be considered, irrespective of their academic background.
• Opportunities to Prove Performance: If the company offers a transparent evaluation process, such as a performance review or internal test, Arya can actively participate and showcase her abilities.
• Supportive Feedback from Manager: Encouragement from her immediate manager, acknowledging her contributions and endorsing her as a strong candidate, could alleviate her doubts.
• Company’s Focus on Diversity in Selection: Assurance from the founder or HR that the selection process will consider diverse profiles, ensuring a fair chance for employees from various backgrounds.

By seeking clarity on the selection process and striving to enhance her workplace performance, Arya can address her concerns and position herself as a strong candidate for the program.