12.2 Pie Charts & Paragraph-Based Caselet DI

12.2 Pie Charts & Paragraph-Based Caselet DI

Circular graphical representations and paragraph-based passages represent two of the most frequent DI configurations in the SBI Clerk Examination. A Pie Chart divides a circular area into proportional sectors representing categorical shares of a unified whole. A Caselet DI presents quantitative data embedded within descriptive, paragraph-length text without pre-existing tabular or graphical structures.

Mastering these formats requires fluency in angular-percentage conversions, double-pie base alignments, and set-theory Venn diagram breakdowns.

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1. Single Pie Chart Mathematical Principles

A full circle encompasses an angle of $360^\circ$ at its center and represents $100\%$ of the evaluated universe ($T$).

Fundamental Conversion Identity

$$100\% = 360^\circ \iff 1\% = 3.6^\circ \iff 1^\circ = \frac{100}{360}\% = \frac{5}{18}\% \approx 0.2778\%$$

Sector Value Formulas

• When sector share is given as a percentage ($P\%$): $$\text{Sector Value} = \frac{P}{100} \times T, \qquad \text{Central Angle } (\theta) = P \times 3.6^\circ$$
• When sector share is given as a central angle ($\theta^\circ$): $$\text{Sector Value} = \frac{\theta}{360} \times T, \qquad \text{Percentage Share } (P) = \theta \times \frac{5}{18}\%$$

Degree-to-Percentage Benchmark Reference Table

Memorizing these common angular conversions saves valuable minutes during the exam:

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2. Double Pie Charts & Base-Disparity Rules

In SBI Clerk Mains, questions frequently present two related pie charts. For example:

• Pie Chart 1: Distribution of Total Employees across 5 Bank Departments ($T_1 = 4,000$).
• Pie Chart 2: Distribution of Female Employees across the same 5 Departments ($T_2 = 1,500$).

Finding Derived Metrics (e.g., Male Employees)

To find the number of male employees in Department $k$:

$$\text{Males}_k = \text{Total}_k - \text{Females}_k = \left(\frac{P_{1,k}}{100} \times T_1\right) - \left(\frac{P_{2,k}}{100} \times T_2\right)$$

3. Caselet DI: Unstructured Text to Structured Matrices

A Caselet presents quantitative data within continuous prose. Solving caselets requires a systematic 3-step extraction technique:

1. First Pass (Structural Scan): Scan the paragraph to identify the primary entities (e.g., branches, products, years) and variables (e.g., male/female, savings/current, pass/fail). Set up an empty 2D Grid Matrix or Venn Diagram on your scratchpad.
2. Second Pass (Anchor Population): Locate absolute numerical anchors (e.g., "Total students = 1,200", "Branch A disbursed Rs. 450 Crores") and enter them directly into the grid.
3. Third Pass (Relational Deduction): Use relative statements (ratios, percentages more/less, differences) to formulate simple linear equations and solve for empty cells.

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4. Set Theory & 3-Circle Venn Diagram Architecture

When a caselet describes overlapping preferences, memberships, or product adoptions (e.g., customers utilizing SBI YONO, Net Banking, and Credit Cards), use a Venn Diagram.

The 3-Set Venn Diagram Breakdown

Let the three categories be $A$, $B$, and $C$ within a universal set $U$:

• $a$: Elements in Only A
• $b$: Elements in Only B
• $c$: Elements in Only C
• $d$: Elements in Only A and B (excluding C)
• $e$: Elements in Only B and C (excluding A)
• $f$: Elements in Only A and C (excluding B)
• $g$: Elements in All Three ($A \cap B \cap C$)
• $N$: Elements in Neither

Key Algebraic Formulations

1. Total Population ($U$): $$U = a + b + c + d + e + f + g + N$$
2. Set Totals: $$n(A) = a + d + f + g$$ $$n(B) = b + d + e + g$$ $$n(C) = c + e + f + g$$
3. Exactly One Category: $$\text{Count}_{\text{exact 1}} = a + b + c$$
4. Exactly Two Categories: $$\text{Count}_{\text{exact 2}} = d + e + f = [n(A \cap B) - g] + [n(B \cap C) - g] + [n(A \cap C) - g]$$
5. At Least Two Categories: $$\text{Count}_{\ge 2} = d + e + f + g$$
6. The Master Inclusion-Exclusion Identity: $$n(A \cup B \cup C) = n(A) + n(B) + n(C) - [n(A \cap B) + n(B \cap C) + n(A \cap C)] + n(A \cap B \cap C)$$

5. Fully Worked Caselet Passage: Digital Banking Customer Adoption

Passage: An SBI branch conducts a customer survey among 1,200 active account holders regarding their adoption of three primary digital banking platforms: SBI YONO ($Y$), Internet Banking ($I$), and Mobile UPI ($U$). Every surveyed customer uses at least one of these three platforms (meaning $N = 0$).

• $45\%$ of total surveyed customers use SBI YONO.
• $50\%$ of total surveyed customers use Mobile UPI.
• $40\%$ of total surveyed customers use Internet Banking.
• $15\%$ of total customers use both YONO and Mobile UPI.
• $12\%$ of total customers use both Mobile UPI and Internet Banking.
• $10\%$ of total customers use both YONO and Internet Banking.

Step-by-Step Mathematical Resolution

1. Convert Given Percentages to Absolute Numbers ($Total = 1,200$):

   • $n(Y) = 45\% \times 1,200 = 540$
   • $n(U) = 50\% \times 1,200 = 600$
   • $n(I) = 40\% \times 1,200 = 480$
   • $n(Y \cap U) = 15\% \times 1,200 = 180$
   • $n(U \cap I) = 12\% \times 1,200 = 144$
   • $n(Y \cap I) = 10\% \times 1,200 = 120$
   • $n(Y \cup U \cup I) = 1,200$

2. Solve for the Central Core ($g = n(Y \cap U \cap I)$) using Inclusion-Exclusion:

   $$1,200 = 540 + 600 + 480 - (180 + 144 + 120) + g$$ $$1,200 = 1,620 - 444 + g$$ $$1,200 = 1,176 + g \implies g = 1,200 - 1,176 = \mathbf{24}$$

   Exactly 24 customers use all three digital banking platforms.

3. Deduce Disjoint 2-Platform Regions:

   • Only YONO and UPI ($d$): $n(Y \cap U) - g = 180 - 24 = \mathbf{156}$
   • Only UPI and Net Banking ($e$): $n(U \cap I) - g = 144 - 24 = \mathbf{120}$
   • Only YONO and Net Banking ($f$): $n(Y \cap I) - g = 120 - 24 = \mathbf{96}$

4. Deduce Exclusive Single-Platform Regions:

   • Only YONO ($a$): $n(Y) - (d + f + g) = 540 - (156 + 96 + 24) = 540 - 276 = \mathbf{264}$
   • Only UPI ($b$): $n(U) - (d + e + g) = 600 - (156 + 120 + 24) = 600 - 300 = \mathbf{300}$
   • Only Net Banking ($c$): $n(I) - (e + f + g) = 480 - (120 + 96 + 24) = 480 - 240 = \mathbf{240}$

Final Extracted Data Summary Table

Analytical Application Questions

• Customers using exactly two platforms: $d + e + f = 156 + 120 + 96 = \mathbf{372}$.
• Customers using only one platform: $a + b + c = 264 + 300 + 240 = \mathbf{804}$.
• Ratio of 'Only YONO' to 'Only UPI': $264 : 300 = \frac{264}{12} : \frac{300}{12} = \mathbf{22 : 25}$.