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ICSE Class 10 Mathematics • Unit 1 Commercial Mathematics

Chapter 3: Shares and Dividend

To establish a large business corporation, a massive capital amount is divided into equal small units called shares. Individuals who purchase these shares become shareholders and joint owners of the enterprise. The profit declared by the company and distributed among shareholders is known as dividend. This master note covers all definitions, quotation conditions, core formulas, return percentage equations, portfolio switching mechanics, solved textbook master examples, and past ICSE board examination problems covering all 8 question types.

Official ICSE Class 10 Question Types Master Checklist

1. Fundamental Terms & Definitions

Commercial Definitions
  1. Share: The total capital required by a company is divided into small equal parts called shares (usually face value ₹10, ₹25, ₹50, or ₹100).
  2. Shareholder: An individual or institution that purchases one or more shares of a company.
  3. Nominal Value (N.V.) / Face Value (F.V.) / Register Value: The original printed value fixed for a share when issued by the company. The Nominal Value NEVER changes over time.
  4. Market Value (M.V.) / Cash Value: The price at which a share is currently bought or sold in the stock market. The Market Value changes continuously based on company performance and demand.
  5. Dividend: The share of company profit distributed annually or half-yearly to shareholders.
    • CRITICAL RULE: Dividend is ALWAYS calculated as a percentage of the Nominal Value (N.V.) of the share!
    • Dividend rate does NOT depend on the Market Value ($\text{M.V.}$).
  6. Return Rate / Income Rate / Yield Rate: The percentage profit earned by an investor on their actual Sum Invested (Investment): $$\text{Return Rate} = \frac{\text{Annual Income (Total Dividend)}}{\text{Sum Invested}} \times 100\%$$

Comparison: Nominal Value vs. Market Value

Property Nominal Value (N.V. / Face Value) Market Value (M.V. / Cash Value)
Definition Printed / Register value fixed by company Trading price of share in stock market
Time Variation Fixed (Never changes over time) Fluctuates (Changes daily based on market)
Dividend Basis Used to calculate Dividend Amount NOT used to calculate Dividend Amount
Investment Basis NOT used to calculate Sum Invested Used to calculate total Sum Invested

2. Quotation Conditions & Core Formula Derivations

A share's Market Value ($\text{M.V.}$) relative to its Nominal Value ($\text{N.V.}$) determines its price quotation in the stock market:

The 3 Share Quotation Conditions
  1. At Par: Market Value equals Nominal Value. $$\text{M.V.} = \text{N.V.}$$
  2. Above Par / At a Premium: Market Value is greater than Nominal Value. $$\text{M.V.} = \text{N.V.} + \text{Premium Amount}$$
  3. Below Par / At a Discount: Market Value is less than Nominal Value. $$\text{M.V.} = \text{N.V.} - \text{Discount Amount}$$
Derivation of Return % Relationship

For an investor purchasing $n$ shares of face value $\text{N.V.}$ at market price $\text{M.V.}$ with dividend rate $d\%$ and return rate $r\%$, the annual income can be expressed in two ways:

  1. Income from Company Dividend: $\text{Income} = n \times \frac{d}{100} \times \text{N.V.}$
  2. Income from Return on Investment: $\text{Income} = \frac{r}{100} \times \text{Sum Invested} = \frac{r}{100} \times (n \times \text{M.V.})$

Equating both expressions:

$$n \times \frac{r}{100} \times \text{M.V.} = n \times \frac{d}{100} \times \text{N.V.}$$
$$\text{Return Rate } (r\%) \times \text{Market Value (M.V.)} = \text{Dividend Rate } (d\%) \times \text{Nominal Value (N.V.)}$$

MASTER FORMULAS FOR SHARES AND DIVIDEND

$$\text{Sum Invested} = \text{No. of Shares } (n) \times \text{M.V. of 1 share}$$ $$n = \frac{\text{Sum Invested}}{\text{M.V. of 1 share}} = \frac{\text{Total Dividend}}{\text{Dividend on 1 share}}$$ $$\text{Dividend on 1 Share} = \frac{\text{Dividend Rate}}{100} \times \text{N.V. of 1 share}$$ $$\text{Total Annual Dividend (Income)} = n \times \frac{\text{Dividend Rate}}{100} \times \text{N.V.}$$ $$\text{Return Rate } (r\%) = \frac{\text{Total Annual Income}}{\text{Sum Invested}} \times 100\%$$

3. Formula Quick Reference Grid

Sum Invested
$$\text{Investment} = n \times \text{M.V.}$$
Number of Shares
$$n = \frac{\text{Investment}}{\text{M.V.}}$$
Annual Dividend
$$I = n \times \frac{d}{100} \times \text{N.V.}$$
Return Equation
$$r \times \text{M.V.} = d \times \text{N.V.}$$
Sale Proceeds
$$\text{Proceeds} = n \times \text{Selling M.V.}$$
Net Income (TDS)
$$\text{Net } I = I \times \left(1 - \frac{t}{100}\right)$$

4. Solved Master Examples Covering All 8 Question Types

Type 1 — Basic Share Purchase & Investment Calculation

Problem: Calculate the money required to buy: (i) 350, ₹20 shares at a premium of ₹7, (ii) 275, ₹60 shares at a discount of ₹10, (iii) 50, ₹40 shares quoted at ₹38.50.

Solution:

  1. (i) $\text{N.V.} = \text{₹}20$, $\text{Premium} = \text{₹}7 \implies \text{M.V.} = 20 + 7 =$ ₹27.
    $\text{Money required for 350 shares} = 350 \times 27 =$ ₹9,450.
  2. (ii) $\text{N.V.} = \text{₹}60$, $\text{Discount} = \text{₹}10 \implies \text{M.V.} = 60 - 10 =$ ₹50.
    $\text{Money required for 275 shares} = 275 \times 50 =$ ₹13,750.
  3. (iii) Quoted price means Market Value = ₹38.50.
    $\text{Money required for 50 shares} = 50 \times 38.50 =$ ₹1,925.
Type 2 — Annual Income Calculation (ICSE 2008)

Problem: Rakhee invested ₹12,500 in shares of a company paying 6% dividend per annum. If she bought ₹50 shares for ₹62.50 each, find her annual income from the investment.

Solution:

  1. $\text{Sum Invested} = \text{₹}12,500$, $\text{M.V. of 1 share} = \text{₹}62.50$, $\text{N.V. of 1 share} = \text{₹}50$.
  2. Number of shares bought ($n$) = $\frac{\text{Sum Invested}}{\text{M.V.}} = \frac{12500}{62.50} = 200$ shares.
  3. Dividend on 1 share = 6% of ₹50 = $\frac{6}{100} \times 50 =$ ₹3.
  4. Total Annual Income = $200 \times 3 =$ ₹600.
  5. Direct Formula Method: $\text{Income} = n \times d\% \times \text{N.V.} = 200 \times 6\% \times 50 =$ ₹600.
Type 3 — Finding Return % / Yield % on Investment

Problem: Ramesh buys ₹100 shares at ₹20 premium in a company paying 15% dividend. Find: (i) the market value of 600 shares, (ii) his annual income, (iii) his percentage income (return %).

Solution:

  1. $\text{N.V.} = \text{₹}100$, $\text{Premium} = \text{₹}20 \implies \text{M.V.} = 100 + 20 =$ ₹120.
  2. (i) Market value of 600 shares = $600 \times 120 =$ ₹72,000.
  3. (ii) Annual Income = $600 \times 15\% \text{ of } 100 = 600 \times 15 =$ ₹9,000.
  4. (iii) Percentage Income (Return Rate) = $\frac{\text{Annual Income}}{\text{Sum Invested}} \times 100\% = \frac{9000}{72000} \times 100\% =$ 12.5%.
  5. Alternative Shortcut Equation: $\text{Return Rate} \times \text{M.V.} = \text{Dividend Rate} \times \text{N.V.} \implies r \times 120 = 15 \times 100 \implies r = \frac{1500}{120} =$ 12.5%.
Type 4 — Half-Yearly Dividend & Income Tax Deduction (TDS)

Problem: Find the dividend due at the end of a year on 250 shares of ₹50 each, if the half-yearly dividend is 4% of the value of the share. If the company deducts 20% income tax from dividend, find the net annual income.

Solution:

  1. Half-yearly dividend = 4% of ₹50 = ₹2 per share.
  2. Yearly dividend per share = $2 \times 2 =$ ₹4 per share (or 8% of ₹50 = ₹4).
  3. Total Gross Annual Dividend = $250 \times 4 =$ ₹1,000.
  4. Income tax deducted = 20% of ₹1,000 = ₹200.
  5. Net Annual Income after TDS = $1000 - 200 =$ ₹800.
Type 5 — Extra Shares for Target Income Increase (ICSE 2002)

Problem: A man wants to buy 62 shares available at ₹132 (par value being ₹100). (i) How much must he invest? (ii) If dividend is 7.5%, what will be his annual income? (iii) If he wants to increase his annual income by ₹150, how many extra shares should he buy?

Solution:

  1. $n = 62$, $\text{M.V.} = \text{₹}132$, $\text{N.V.} = \text{₹}100$, $\text{Dividend} = 7.5\%$.
  2. (i) Sum Invested = $62 \times 132 =$ ₹8,184.
  3. (ii) Dividend per share = 7.5% of ₹100 = ₹7.50.
    Annual Income = $62 \times 7.50 =$ ₹465.
  4. (iii) Target income increase = ₹150.
    Number of extra shares needed = $\frac{\text{Required Income Increase}}{\text{Dividend per share}} = \frac{150}{7.50} =$ 20 extra shares.
Type 6 — Investment Comparison ("Which is a Better Investment?")

Problem: Which is a better investment: 12%, ₹100 shares at 120 OR 8%, ₹100 shares at 90?

Solution:

  1. 1st Investment Option:
    $\text{Return Rate } (r_1) \times \text{M.V.} = \text{Dividend Rate} \times \text{N.V.} \implies r_1 \times 120 = 12 \times 100 \implies r_1 = \frac{1200}{120} = 10\%$.
  2. 2nd Investment Option:
    $\text{Return Rate } (r_2) \times \text{M.V.} = \text{Dividend Rate} \times \text{N.V.} \implies r_2 \times 90 = 8 \times 100 \implies r_2 = \frac{800}{90} = 8.89\%$.
  3. Since $10\% > 8.89\%$, the 1st investment (12%, ₹100 shares at 120) is BETTER.
Type 7 — Selling Shares & Re-investing (Portfolio Switching)

Problem: A man sells 60, ₹15 shares of a company paying 12% dividend, at ₹21 each and invests the proceeds in ₹6 shares of another company at ₹9 each. Find his change in income if the 2nd company pays an 8% dividend.

Solution:

  1. 1st Company (Original Portfolio):
    Number of shares = 60, $\text{N.V.} = \text{₹}15$, $\text{Dividend} = 12\%$.
    Income per share = 12% of ₹15 = ₹1.80.
    Original Total Income = $60 \times 1.80 =$ ₹108.
  2. Sale Proceeds:
    Selling M.V. = ₹21. Total proceeds = $60 \times 21 =$ ₹1,260.
  3. 2nd Company (New Portfolio):
    Sum invested = ₹1,260, $\text{N.V.} = \text{₹}6$, $\text{M.V.} = \text{₹}9$, $\text{Dividend} = 8\%$.
    Number of new shares bought = $\frac{1260}{9} = 140$ shares.
    Dividend per share = 8% of ₹6 = ₹0.48.
    New Total Income = $140 \times 0.48 =$ ₹67.20.
  4. Change (Decrease) in Income = $108 - 67.20 =$ ₹40.80 decrease.
Type 8 — Partitioning Total Capital into Two Investments

Problem: Divide ₹40,608 into two parts such that if one part is invested in 8%, ₹100 shares at 8% discount and the other part is invested in 9%, ₹100 shares at 8% premium, the annual incomes from both investments are equal.

Solution:

  1. Let the 1st part = ₹$x$ and 2nd part = ₹$(40608 - x)$.
  2. 1st Investment: $\text{N.V.} = \text{₹}100$, $\text{Discount} = 8\% \implies \text{M.V.} = 100 - 8 =$ ₹92.
    Number of shares = $\frac{x}{92}$. Dividend per share = 8% of ₹100 = ₹8.
    $\text{Income}_1 = 8 \times \frac{x}{92} = \frac{2x}{23}$.
  3. 2nd Investment: $\text{N.V.} = \text{₹}100$, $\text{Premium} = 8\% \implies \text{M.V.} = 100 + 8 =$ ₹108.
    Number of shares = $\frac{40608 - x}{108}$. Dividend per share = 9% of ₹100 = ₹9.
    $\text{Income}_2 = 9 \times \frac{40608 - x}{108} = \frac{40608 - x}{12}$.
  4. Since incomes are equal: $$\frac{2x}{23} = \frac{40608 - x}{12} \implies 24x = 23(40608 - x) = 933984 - 23x$$ $$47x = 933984 \implies x = \frac{933984}{47} = 19872$$
  5. 1st Part = ₹19,872   and   2nd Part = $40608 - 19872 =$ ₹20,736.

5. Comprehensive ICSE Board Practice Problems

Textbook Exercises & Past Board Questions
Q1 (ICSE 2000). A dividend of 9% was declared on ₹100 share selling at a certain price. If the rate of return is 7.5%, calculate: (i) the market value of the share, (ii) the amount to be invested to obtain an annual dividend of ₹630.
Solution:
(i) $\text{Return Rate} \times \text{M.V.} = \text{Dividend Rate} \times \text{N.V.} \implies 7.5\% \times \text{M.V.} = 9\% \times 100 \implies \text{M.V.} = \frac{900}{7.5} =$ ₹120.
(ii) Dividend on 1 share = 9% of ₹100 = ₹9.
Number of shares to get ₹630 dividend = $\frac{630}{9} = 70$ shares.
Sum to be invested = $70 \times 120 =$ ₹8,400.
Q2 (ICSE 2001). A man invests ₹1,680 in buying shares of nominal value ₹24 and selling at 12% premium. The dividend on the shares is 15% per annum. Calculate: (i) the number of shares he buys, (ii) the dividend he receives annually.
Solution:
$\text{N.V.} = \text{₹}24$, $\text{Premium} = 12\% \text{ of } 24 = \text{₹}2.88 \implies \text{M.V.} = 24 + 2.88 =$ ₹26.88.
(i) Number of shares bought = $\frac{1680}{26.88} =$ 60 shares.
(ii) Dividend per share = 15% of ₹24 = ₹3.60.
Total annual dividend = $60 \times 3.60 =$ ₹216.
Q3 (ICSE 2003). A man invests ₹20,020 in buying shares of N.V. ₹26 at 10% premium. The dividend on the shares is 15% per annum. Calculate: (i) the number of shares he buys, (ii) the dividend he receives annually, (iii) the rate of interest he gets on his money.
Solution:
$\text{N.V.} = \text{₹}26$, $\text{Premium} = 10\% \text{ of } 26 = \text{₹}2.60 \implies \text{M.V.} = 26 + 2.60 =$ ₹28.60.
(i) Number of shares = $\frac{20020}{28.60} =$ 700 shares.
(ii) Annual dividend = $700 \times 15\% \text{ of } 26 = 700 \times 3.90 =$ ₹2,730.
(iii) Rate of return = $\frac{2730}{20020} \times 100\% =$ 13.64% p.a.
Q4 (ICSE 2004). A man invested ₹45,000 in 15%, ₹100 shares quoted at ₹125. When the M.V. of these shares rose to ₹140, he sold some shares, just enough to raise ₹8,400. Calculate: (i) the number of shares he still holds, (ii) the dividend due to him on these remaining shares.
Solution:
Original shares bought = $\frac{45000}{125} = 360$ shares.
Number of shares sold at ₹140 to raise ₹8,400 = $\frac{8400}{140} = 60$ shares.
(i) Shares still held = $360 - 60 =$ 300 shares.
(ii) Dividend per share = 15% of ₹100 = ₹15.
Dividend on remaining shares = $300 \times 15 =$ ₹4,500.
Q5 (ICSE 2006). Mr. Ram Gopal invested ₹8,000 in 7%, ₹100 shares at ₹80. After a year, he sold these shares at ₹75 each and invested the proceeds (including his dividend) in 18%, ₹25 shares at ₹41. Find: (i) his dividend for 1st year, (ii) annual income in 2nd year, (iii) percentage increase in return on original investment.
Solution:
1st year shares = $\frac{8000}{80} = 100$ shares.
(i) 1st year dividend = $100 \times 7\% \text{ of } 100 =$ ₹700.
Proceeds from sale + dividend = $(100 \times 75) + 700 = 7500 + 700 =$ ₹8,200.
2nd year shares bought = $\frac{8200}{41} = 200$ shares.
(ii) 2nd year annual income = $200 \times 18\% \text{ of } 25 = 200 \times 4.50 =$ ₹900.
(iii) Increase in return = $900 - 700 =$ ₹200.
Percentage increase on original ₹8,000 = $\frac{200}{8000} \times 100\% =$ 2.5%.
Q6 (ICSE 2011). Mr. Parekh invested ₹52,000 in ₹100 shares at a discount of ₹20 paying 8% dividend. At the end of one year he sells the shares at a premium of ₹20. Find: (i) the annual dividend, (ii) the profit earned including his dividend.
Solution:
Purchase M.V. = $100 - 20 =$ ₹80. Shares bought = $\frac{52000}{80} = 650$ shares.
(i) Annual dividend = $650 \times 8\% \text{ of } 100 =$ ₹5,200.
Selling M.V. = $100 + 20 =$ ₹120.
Sale proceeds = $650 \times 120 =$ ₹78,000.
(ii) Total profit including dividend = $(\text{Sale Proceeds} - \text{Investment}) + \text{Dividend} = (78000 - 52000) + 5200 =$ ₹31,200.
Q7 (ICSE 2014). Salman invests a sum of money in ₹50 shares, paying 15% dividend quoted at 20% premium. If his annual dividend is ₹600, calculate: (i) the number of shares he bought, (ii) his total investment, (iii) the rate of return on his investment.
Solution:
$\text{N.V.} = \text{₹}50$, $\text{Premium} = 20\% \text{ of } 50 = \text{₹}10 \implies \text{M.V.} = 50 + 10 =$ ₹60.
Dividend on 1 share = 15% of ₹50 = ₹7.50.
(i) Number of shares bought = $\frac{600}{7.50} =$ 80 shares.
(ii) Total investment = $80 \times 60 =$ ₹4,800.
(iii) Rate of return = $\frac{600}{4800} \times 100\% =$ 12.5% p.a.
Q8 (ICSE 2015). Rohit invested ₹9,600 on ₹100 shares at ₹20 premium paying 8% dividend. Rohit sold the shares when the price rose to ₹160. He invested the proceeds (excluding dividend) in 10%, ₹50 shares at ₹40. Find: (i) original number of shares, (ii) sale proceeds, (iii) new number of shares, (iv) change in his dividend income.
Solution:
(i) Purchase M.V. = $100 + 20 =$ ₹120. Original shares = $\frac{9600}{120} =$ 80 shares.
Original annual dividend = $80 \times 8\% \text{ of } 100 =$ ₹640.
(ii) Sale proceeds = $80 \times 160 =$ ₹12,800.
(iii) New M.V. = ₹40. New number of shares = $\frac{12800}{40} =$ 320 shares.
New annual dividend = $320 \times 10\% \text{ of } 50 = 320 \times 5 =$ ₹1,600.
(iv) Change in dividend income = $1600 - 640 =$ ₹960 increase.
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