Shares and Dividend

A. The Core Definitions:

• Nominal Value (NV) / Face Value (FV) / Par Value: The original base price printed on the share certificate at the company's incorporation. Golden Rule: The Nominal Value remains completely constant and never changes, regardless of market fluctuations!
• Market Value (MV) / Cash Value: The actual price at which a share is bought or sold on the stock exchange. It fluctuates continuously based on supply, demand, and company profitability.
• Dividend: The portion of annual profit distributed to shareholders. Golden Rule: Dividend is ALWAYS calculated as a percentage of the Nominal Value (NV), NEVER on the Market Value!

B. The Three Trading Conditions:

The Master Equation Suite:

1. 1. Number of Shares Purchased ($n$): $$n = \frac{\text{Total Investment}}{\text{Market Value of 1 Share (MV)}} = \frac{\text{Total Annual Income}}{\text{Annual Income from 1 Share}} = \frac{\text{Total Face Value}}{\text{Face Value of 1 Share}}$$
2. 2. Total Investment ($I$): $$I = n \times \text{Market Value of 1 Share (MV)}$$
3. 3. Total Annual Income / Total Dividend ($D$): $$D = n \times \left( \frac{d}{100} \times \text{NV} \right) = \frac{d}{100} \times (n \times \text{NV}) = \frac{d}{100} \times \text{Total Face Value}$$
4. 4. Return Percentage / Yield Percentage ($R\%$): $$R\% = \frac{\text{Total Annual Income}}{\text{Total Investment}} \times 100$$

   Single-Share Yield Identity: For any individual share:
   $$\text{Return } \% \times \text{MV} = \text{Dividend } \% \times \text{NV} \implies R\% = \frac{d \times \text{NV}}{\text{MV}}$$

Stock Switching & Reinvestment Walkthrough:

Problem: A man invests ₹7,770 in 5% ₹100 shares at ₹111. When the market price rises to ₹120, he sells these shares and reinvests the proceeds in 8% ₹100 shares at ₹125. Calculate:

1. The number of shares originally purchased.
2. The total dividend received from the first company.
3. The sale proceeds obtained on selling the shares.
4. The change in his annual income after reinvestment.

Step-by-Step Solution:

Stage 1 (First Company):
NV = ₹100, MV = ₹111, Dividend $d_1 = 5\%$, Investment = ₹7,770.
(i) Number of shares $n_1 = \frac{7770}{111} = \mathbf{70 \text{ shares}}$.
(ii) Annual Income from 1st company $D_1 = 70 \times \frac{5}{100} \times 100 = \mathbf{₹350}$.

Stage 2 (Sale of Shares):
Selling price per share = ₹120.
(iii) Total Sale Proceeds = $70 \times 120 = \mathbf{₹8,400}$.

Stage 3 (Reinvestment in Second Company):
New Investment = ₹8,400; New NV = ₹100; New MV = ₹125; New Dividend $d_2 = 8\%$.
Number of new shares $n_2 = \frac{8400}{125} = 67.2$ shares (or 67 whole shares). Let us take fractional calculation: $n_2 = \frac{8400}{125} = \frac{336}{5} = 67.2$.
New Annual Income $D_2 = n_2 \times \frac{8}{100} \times 100 = 67.2 \times 8 = ₹537.60$.
(iv) Increase in Annual Income = $D_2 - D_1 = ₹537.60 - ₹350 = \mathbf{₹187.60}$.

A. Multi-Stage Portfolio Reallocation Walkthrough:

Problem: A person invested ₹20,000 in ₹100 shares of a company at a discount of 20%. The company declared a dividend of 15%. After receiving the first year's dividend, he sold these shares at a premium of 10% and invested the total proceeds (including the dividend received) in ₹50 shares of another company at ₹60 paying an 18% dividend. Calculate:

1. The number of shares purchased in the first company.
2. The annual dividend received from the first company.
3. The total capital reinvested in the second company.
4. The number of shares bought in the second company.
5. The percentage change in his annual income.

Step-by-Step Algebraic Solution:

Company 1 (First Investment):
Face Value $NV_1 = ₹100$. Market Value $MV_1 = 100 - 20 = ₹80$ (20% discount).
Total investment = ₹20,000.
1. Number of shares $n_1 = \frac{20000}{80} = \mathbf{250 \text{ shares}}$.
2. Dividend from Company 1: $D_1 = n_1 \times \frac{15}{100} \times 100 = 250 \times 15 = \mathbf{₹3,750}$.

Sale of Company 1 Shares:
Selling price per share = ₹100 + ₹10 = ₹110 (10% premium).
Sale Proceeds = $250 \times 110 = ₹27,500$.
3. Total capital reinvested = $\text{Sale proceeds} + \text{Dividend received} = ₹27,500 + ₹3,750 = \mathbf{₹31,250}$.

Company 2 (Second Investment):
Face Value $NV_2 = ₹50$. Market Value $MV_2 = ₹60$. Dividend rate $d_2 = 18\%$.
4. Number of shares in Company 2: $n_2 = \frac{31250}{60} = 520.83$ shares (taking whole shares: $520$ shares; remaining cash = ₹50).
Let us compute with exact fraction: $n_2 = \frac{3125}{6}$ shares.
Annual dividend from Company 2: $D_2 = \frac{3125}{6} \times \frac{18}{100} \times 50 = \frac{3125}{6} \times 9 = \mathbf{₹4,687.50}$.
5. Increase in annual income = $D_2 - D_1 = 4687.50 - 3750 = ₹937.50$.
$$\text{Percentage Increase} = \frac{937.50}{3750} \times 100 = \mathbf{25\%}.$$

Rigorous Step-by-Step Solutions for Top-Band Scores:

Below is a curated compendium of standard ICSE board-level examination questions designed to test mathematical derivation, algebraic accuracy, unit precision, and rigorous geometric justifications.

How ICSE Examiners Award Marks in this Topic:

In the official CISCE evaluation rubrics, marks are systematically distributed across three distinct cognitive dimensions:

• Method Marks (M): Awarded for writing the correct formula, establishing the proper geometric theorem, setting up the correct equation, or constructing the proper table columns. Even if a careless calculation error occurs later, method marks are fully preserved!
• Accuracy Marks (A): Awarded for correct intermediate arithmetic steps, accurate factorization, algebraic simplification, and correct radical reduction.
• Final Statement & Unit Marks (B/A): Awarded for stating the final numerical answer with correct units, proper rounding (e.g. two decimal places), and answering all sub-parts explicitly.

Top 5 Practical Recommendations to Maximize Scores:

1. Never skip writing the standard formula: Always write out the formula in algebraic terms before substituting numerical values.
2. Keep rough calculations neatly organized: Draw a dedicated 2-inch rough margin on the right side of your answer sheet for long divisions and square root extractions.
3. Check boundary conditions: Ensure your final solutions belong strictly to the specified domain (e.g., natural numbers, positive lengths, valid quadrants).
4. State geometric reasons in parentheses: In geometry proofs, every equality must be accompanied by its supporting theorem name (e.g., '[angles in the same segment are equal]').
5. Proofread before moving to the next question: Spend 30 seconds verifying signs, basic arithmetic, and decimal point placements.