Commercial Mathematics - Shares and Dividends

A person invests ₹ $9000$ in shares of a company which is paying $8 \%$ dividend. If ₹ $100$ shares are available at a discount of $10 \%$, find the number of shares purchased and the annual income.

Money invested $= $ ₹ $9000$

Face value (FV) of each share $= $ ₹ $100$

Market value (MV) of each share $= $ ₹ $100 - 10 \%$ of ₹ $100$

$ = $ ₹ $\left(100 - \dfrac{10}{100} \times 100\right) = $ ₹ $90$

$\therefore \;$ Number of shares bought $= \dfrac{\text{money invested}}{\text{MV of each share}} = \dfrac{9000}{90} = 100$

Dividend (income) on one share $= 8 \%$ of FV $= \dfrac{8}{100} \times $ ₹ $100 = $ ₹ $8$

$\therefore \;$ Total income from the shares $= 100 \times $ ₹ $8 = $ ₹ $800$

Commercial Mathematics - Banking

A person deposits a certain sum of money each month in a recurring deposit account of a bank. If the rate of interest is $8 \%$ per annum and the person gets ₹ $8088$ from the bank after $3$ years, find the value of the monthly installment.

Let money deposited per month $= P = $ ₹ $x$

Time for which money is deposited $= n = 3$ years $= 36$ months

Rate of interest $= r = 8\%$ per annum

Interest $= I = P \times \dfrac{n \left(n + 1\right)}{2 \times 12} \times \dfrac{r}{100}$

i.e. $\;$ $I =$ ₹ $\left[\dfrac{x \times 36 \times 37}{2 \times 12} \times \dfrac{8}{100}\right]$

i.e. $\;$ $I =$ ₹ $\left[\dfrac{111 x}{25}\right] = $ ₹ $4.44 x$

Total money deposited in $36$ months $= $ ₹ $36 x$

Maturity value (MV) $=$ Total money deposited $+ $ Interest

i.e. $\;$ $MV = $ ₹ $\left(36 x + 4.44 x\right) = $ ₹ $40.44 x$

Given: $\;$ $MV =$ ₹ $8088$

$\implies$ $40.44x = 8088$ $\implies$ $x = 200$

i.e. $\;$ money deposited per month $= $ ₹ $200$

Commercial Mathematics - Goods and Services Tax

A wholesaler buys a TV from a manufacturer for ₹ $25,000$. The wholesaler marks the price of the TV $20 \%$ above the cost price and sells it to a retailer at a discount of $10 \%$ on the marked price. If the rate of GST is $28 \%$, find

1. the marked price;
2. retailer's cost price inclusive of tax;
3. GST paid by the wholesaler.

1. Cost Price of TV for wholesaler $= $ ₹ $25,000$
   
   Marked price of TV $= $ ₹ $\left(25,000 + 20 \% \text{ of } 25,000\right)$
   
   i.e $\;$ Marked price $= $ ₹ $\left(25,000 + \dfrac{20}{100} \times 25,000\right) = $ ₹ $\left(25,000 + 5,000\right) = $ ₹ $30,000$



2. Discount given by wholesaler $= 10 \% \text{ of }$ ₹ $30,000$
   
   i.e. $\;$ Discount $= $ ₹ $\dfrac{10}{100} \times 30,000 = $ ₹ $3,000$
   
   $\therefore \;$ Amount paid by retailer without GST $= $ ₹ $\left(30,000 - 3,000\right) = $ ₹ $27,000$
   
   Rate of GST $= 28\%$
   
   $\therefore \;$ Amount of GST paid by retailer $= 28 \% \text{ of }$ ₹ $27,000$
   
   i.e. $\;$ Amount of GST paid by retailer $= $ ₹ $\left(\dfrac{28}{100} \times 27,000\right) =$ ₹ $7,560$
   
   $\therefore \;$ Retailer's cost price (inclusive of tax) $= $ ₹ $\left(27,000 + 7,560\right) = $ ₹ $34,560$



3. Cost price for wholesaler $= $ ₹ $25,000$
   
   $\therefore \;$ GST paid by wholesaler for purchase $= $ ₹ $28 \% \text{ of } 25,000$
   
   i.e. $\;$ GST paid by wholesaler $= $ ₹ $\left(\dfrac{28}{100} \times 25,000\right) = $ ₹ $7,000$
   
   Sale price for wholesaler $= $ ₹ $27,000$
   
   GST charged by wholesaler on selling of TV $= $ ₹ $28 \% \text{ of } 27,000 = $ ₹ $7,560$
   
   $\therefore \;$ GST paid by wholesaler $= $ GST charged on selling price $- $ GST paid against purchase price
   
   i.e. $\;$ GST paid by wholesaler $= $ ₹ $\left(7,560 - 7,000\right) = $ ₹ $560$

Commercial Mathematics - Shares and Dividends

A person invested ₹ $10,000$ in $8 \%$, ₹ $25$ shares at ₹ $40$. After a year, the shares were sold at ₹ $42$ each and the proceeds (including the dividend) were invested in $9 \%$, $\;$ ₹ $10$ shares at ₹ $11$. Find:

1. the dividend for the first year;
2. the new number of shares bought;
3. the percentage increase in the return on the original investment.

Investment $= $ ₹ $10,000$

Dividend $\% = 8 \%$

Nominal Value (N.V) of each share $= $ ₹ $25$

Market Value (M.V) of each share $= $ ₹ $40$

1. Number of shares bought $= \dfrac{\text{Investment}}{\text{M.V of each share}} = \dfrac{10,000}{40} = 250$
   
   Dividend on $1$ share $= 8 \% $ of ₹ $25 = \dfrac{8}{100} \times 25 = $ ₹ $2$
   
   $\therefore \;$ Dividend for first year $= $ ₹ $\left(2 \times 250\right) = $ ₹ $500$



2. Since each share is sold for ₹ $42$,
   
   $\therefore \;$ Proceeds (including dividend) $= 250 \times 42 + 500 = $ ₹ $11,000$
   
   $\therefore \;$ Sum invested $= $ proceeds $= $ ₹ $11,000$
   
   Nominal value (N.V) of each new share $= $ ₹ $10$
   
   Market value (M.V) of each new share $= $ ₹ $11$
   
   $\therefore \;$ Number of new shares bought $= \dfrac{\text{Investment}}{\text{M.V of each share}} = \dfrac{11,000}{11} = 1000$



3. New dividend $= 9 \%$
   
   Dividend on $1$ share $= 9 \% $ of ₹ $10 = \dfrac{9}{100} \times 10 = $ ₹ $0.90$
   
   $\therefore \;$ Annual dividend (income) in the second year $= $ ₹ $\left(0.90 \times 1000\right) = $ ₹ $900$
   
   $\therefore \;$ Increase in return $= $ ₹ $\left(900 - 500\right) = $ ₹ $400$
   
   $\therefore \;$ $\%$ Increase in return (on the original investment)
   
   $= \dfrac{\text{Increase}}{\text{Original investment}} \times 100 \% = \dfrac{400}{10,000} \times 100 = 4 \%$

Commercial Mathematics - Banking

A person has a recurring deposit account of ₹ $400$ per month at $10 \%$ per annum. If the person gets ₹ $260$ as interest at the time of maturity, find the total time for which the account was held.

Let the account be held for $n$ months.

Money deposited per month $= P =$ ₹ $400$

$\therefore \;$ Total money deposited $= $ ₹ $\left(400 \times n\right)$

Rate of interest $= r = 10\%$

Interest $= I = P \times \dfrac{n \left(n + 1\right)}{2 \times 12} \times \dfrac{r}{100}$

i.e. $\;$ $I = $ ₹ $\left[400 \times \dfrac{n \left(n + 1\right)}{2 \times 12} \times \dfrac{10}{100}\right]$

i.e. $\;$ $I = $ ₹ $\left[\dfrac{5 n \left(n + 1\right)}{3}\right]$

But, interest earned $= $ ₹ $260$

$\therefore \;$ $\dfrac{5n \left(n + 1\right)}{3} = 260$

i.e. $\;$ $5n^2 + 5n = 780$

i.e. $\;$ $n^2 + n = 156$

i.e. $\;$ $n^2 + n - 156 = 0$

i.e. $\;$ $n^2 + 13 n - 12n - 156 = 0$

i.e. $\;$ $n \left(n + 13\right) - 12 \left(n + 13\right) = 0$

i.e. $\;$ $\left(n + 13\right) \left(n - 12\right) = 0$

i.e. $\;$ $n + 13 = 0$ $\;$ or $\;$ $n - 12 = 0$

i.e. $\;$ $n = -13$ $\;$ or $\;$ $n = 12$

Since time cannot be negative, the time for which the account was held $= n = 12$ months $= 1$ year

Commercial Mathematics - Banking

A person has a recurring deposit account in a bank for $2$ years at $9\%$ per annum. If the person gets ₹ $7837.50$ at the time of maturity, find the monthly installment.

Let money deposited per month $= P =$ ₹ $100$

Number of months $= n = 24$ $\;\;\;$ [$2$ years]

Rate of interest $= r = 9 \%$

Interest $= I = P \times \dfrac{n \left(n + 1\right)}{12} \times \dfrac{r}{100}$

i.e. $\;$ $I =$ ₹ $\left(100 \times \dfrac{24 \times 25}{12} \times \dfrac{9}{100}\right) = $ ₹ $450$

Money deposited in $24$ months $= 24 \times $ ₹ $100 = $ ₹ $2400$

Maturity value (M.V) $= $ ₹ $\left(2400 + 450\right) = $ ₹ $2850$

When M.V is ₹ $2850$, monthly installment $= $ ₹ $100$

When M.V is ₹ $7837.50$, monthly installment $= $ ₹ $\dfrac{100 \times 7837.50}{2850} = $ ₹ $275$

Commercial Mathematics - Shares and Dividends

A person invests ₹ $9900$ on buying shares of face value of ₹ $100$ each at a premium of $10 \%$ in a company. If the person earns ₹ $1350$ at the end of the year, find

1. the number of shares the person has in the company and
2. the dividend percent per share.

Amount invested $= $ ₹ $9900$

Face value (F.V) of each share $= $ ₹ $100$

Market value (M.V) of each share $= $ ₹ $\left(100 + \dfrac{10}{100} \times 100\right) = $ ₹ $110$

$\text{Number of shares bought} = \dfrac{\text{Amount invested}}{\text{M.V of each share}} = \dfrac{9900}{110} = 90$

Let dividend percent per share $= d \%$

Dividend on $1$ share $= d \% \text{ of F.V} = d \% \text{ of }$ ₹ $100 = $ ₹ $d$

$\therefore \;$ Income from $90$ shares $= $ ₹ $90 d$

Given: Annual income $= $ ₹ $1350$

i.e. $\;$ $90 d = 1350$ $\implies$ $d = 15$

$\therefore \;$ Dividend percent per share $= 15 \%$

Commercial Mathematics - Goods and Services Tax

Find the amount of bill for the following intra state transaction of goods / services.

MRP (₹)	$12,000$	$15,000$	$9,500$	$18,000$
Discount $\left(\%\right)$	$30$	$20$	$30$	$40$
CGST $\left(\%\right)$	$6$	$9$	$14$	$2.5$

MRP (₹)	Discount $\left(\%\right)$	Discounted Price (₹)	Selling Price (₹)	CGST $\left(\%\right)$	CGST (₹)	SGST (₹)
$12,000$	$30$	$3,600$	$8,400$	$6$	$504$	$504$
$15,000$	$20$	$3,000$	$12,000$	$9$	$1,080$	$1,080$
$9,500$	$30$	$2,850$	$6,650$	$14$	$931$	$931$
$18,000$	$40$	$7,200$	$10,800$	$2.5$	$270$	$270$

Total selling price $= S.P = $ ₹ $37,850$

Total CGST $= $ ₹ $2785$

Total SGST $= $ ₹ $2785$

Amount of bill $= S.P + CGST + SGST$

i.e. $\;$ Amount of bill $= $ ₹ $\left(37,850 + 2785 + 2785\right) = $ ₹ $43,420$

Commercial Mathematics - Shares and Dividends

A person receives an annual income of ₹ $900$ in buying ₹ $50$ shares selling at ₹ $80$. If the dividend declared is $20 \%$, find the amount invested and the percentage return on investment.

Income $= $ ₹ $900$

Face value (F.V) of each share $= $ ₹ $50$

Market value (M.V) of each share $= $ ₹ $80$

Rat of dividend $= 20 \%$

Let the number of shares bought $= n$

Income $= $ Number of shares $\times$ rate of dividend $\times$ F.V

i.e. $\;$ $900 = n \times \dfrac{20}{100} \times 50$

$\implies$ $n = 90$

i.e. $\;$ Number of shares bought $= n = 90$

$\text{Number of shares} = \dfrac{\text{Sum invested}}{\text{M.V of 1 share}}$

$\therefore \;$ Sum invested $= $ Number of shares $\times$ M.V of 1 share

i.e. Sum invested $= 90 \times 80 =$ ₹ $7200$

$\% \text{ Return on investment} = \dfrac{\text{Income}}{\text{Investment}} \times 100 \%$

i.e. $\;$ $\% \text{ Return on investment} = \dfrac{900}{7200} \times 100 = 12.5 \%$

Commercial Mathematics - Shares and Dividends

A person invests ₹ $16500$ partly in $10\% \;$ ₹ $100$ shares at ₹ $130$ and partly in $8\% \;$ ₹ $100$ shares at ₹ $120$. If the total annual income from these shares is ₹$1180$, find the investment in each kind of shares.

Let the money invested in $10\% \;$ ₹ $100$ shares at ₹ $130$ $= $ ₹ $x$

Let the money invested in $8\% \;$ ₹ $100$ shares at ₹ $120$ $= $ ₹ $\left(16500 - x\right)$

For ₹ $100$ shares at ₹ $130$:

Nominal Value (N.V) of each share $= $ ₹ $100$

Market Value (M.V) of each share $= $ ₹ $130$

$\therefore \;$ Number of shares bought $= \dfrac{\text{money invested}}{M.V} = \dfrac{x}{130}$

Dividend on each share $= 10\% \text{ of N.V } = \dfrac{10}{100} \times 100 = $ ₹ $10$

$\therefore \;$ Dividend (income) from $\dfrac{x}{130}$ shares $= \dfrac{x}{130} \times 10 = $ ₹ $\dfrac{x}{13}$

For ₹ $100$ shares at ₹ $120$:

Nominal Value (N.V) of each share $= $ ₹ $100$

Market Value (M.V) of each share $= $ ₹ $120$

$\therefore \;$ Number of shares bought $= \dfrac{\text{money invested}}{M.V} = \dfrac{16500 - x}{120}$

Dividend on each share $= 8\% \text{ of N.V } = \dfrac{8}{100} \times 100 = $ ₹ $8$

$\therefore \;$ Dividend (income) from $\dfrac{16500 - x}{120}$ shares

$=$ ₹ $ \dfrac{16500 - x}{120} \times 8 = $ ₹ $\left(\dfrac{16500 - x}{15}\right)$

$\therefore \;$ Total dividend $= $ ₹ $\left(\dfrac{x}{13} + \dfrac{16500 - x}{15}\right)$

Given: $\;$ Total dividend $= $ ₹ $1180$

i.e. $\;$ $\dfrac{x}{13} + \dfrac{16500 - x}{15} = 1180$

i.e. $\;$ $15 x + 214500 - 13x = 230100$

i.e. $\;$ $2x = 15600$ $\implies$ $x = 7800$

$\therefore \;$ Amount invested in $10\% \;$ ₹ $100$ shares at ₹ $130 = $ ₹ $7800$

Amount invested in $8\% \;$ ₹ $100$ shares at ₹ $120 = $ ₹ $\left(16500 - 7800\right) = $ ₹ $8700$

Commercial Mathematics - Banking

A person deposits a certain sum of money each month in a recurring deposit account of a bank. If the rate of interest is $8\%$ per annum and the person gets ₹ $8088$ from the bank after $3$ years, find the amount of money deposited each month.

Let money deposited per month $= P = $ ₹ $x$

Rate of interest $= r = 8 \%$ per annum

Number of months $= n = 36$ $\;\;\;$ [$3$ years $= 36$ months]

Interest $= I = P \times \dfrac{n \left(n + 1\right)}{2 \times 12} \times \dfrac{r}{100}$

i.e. $\;$ $I = $ ₹ $x \times \dfrac{36 \times 37}{2 \times 12} \times \dfrac{8}{100} = $ ₹ $4.44 \; x$

Money deposited in $36$ months $= $ ₹ $36 \; x$

$\therefore \;$ Maturity value (M.V) $= $ ₹ $36 \; x + $ ₹ $4.44 \; x = $ ₹ $40.44 \; x$

Given: $\;$ M.V $= $ ₹ $8088$

$\implies$ $40.44 \; x = 8088$ $\implies$ $x = \dfrac{8088}{40.44} = 200$

$\therefore \;$ Money deposited per month $= $ ₹ $200$

Commercial Mathematics - Shares and Dividends

A person buys $500$, ₹ $20$ shares at a discount of $20\%$ and receives a return of $10\%$ on her money. Calculate the amount invested and the rate of dividend paid by the company.

Number of shares $= 500$

Nominal value of each share $= N.V = $ ₹ $20$

Market value of each share $= M.V = $ ₹ $20 - 20 \%$ of ₹ $20 = $ ₹ $20 \; - $ ₹ $4 = $ ₹ $16$

Rate of return $= 10\%$

Amount invested $=$ Number of shares $\times$ M.V of 1 share

i.e. $\;$ Amount invested $= 500 \times $ ₹ $16 = $ ₹ $8000$

Now, $\;$ $\text{Rate of return} \times M.V = \text{Rate of dividend} \times N.V$

$\therefore \;$ $\text{Rate of dividend} = \dfrac{\text{Rate of return} \times M.V}{N.V}$

i.e. $\;$ $\text{Rate of dividend} = \dfrac{10}{100} \times 16 \times \dfrac{1}{20} = \dfrac{8}{100} = 8\%$

Commercial Mathematics - Shares and Dividends

An investment of ₹ $8800$ is made on ₹ $100$ shares at $10\%$ premium paying $22\%$ dividend. Find the return percent on the investment and the dividend earned in one year.

Nominal value of $1$ share $= N.V = $ ₹ $100$

Market value of $1$ share $= M.V = $ ₹ $100 + $ ₹ $10 = $ ₹ $110$

Investment $=$ ₹ $8800$

$\therefore \;$ Number of shares bought $= \dfrac{\text{Investment}}{\text{M.V of 1 share}} = \dfrac{8800}{110}= 80$

Rate of dividend $= 22 \%$ (Given)

$\therefore \;$ Income (i.e. dividend) earned $=$ Number of shares $\times$ Rate of dividend $\times$ N.V

i.e. $\;$ Dividend earned $= 80 \times \dfrac{22}{100} \times 100 = $ ₹ $1760$

i.e. $\;$ ₹ $1760$ is the income obtained on investing ₹ $8800$

$\therefore \;$ Return $\%$ on investment $= \dfrac{1760}{8800} \times 100 = 20 \%$

Commercial Mathematics - Banking

If ₹ $\; 15,000$ is earned as interest on a monthly deposit of ₹ $\; 5,000$ for $2$ years. Find the rate of interest on the recurring deposit.

Money deposited each month $= P =$ ₹ $5,000$

Time for which money deposited $= n = 2 $ years $= 24 $ months

Let, rate of interest $= r \%$

Interest received $= $ ₹ $15,000$

$\text{Interest} = P \times \dfrac{n \left(n + 1\right)}{2 \times 12} \times \dfrac{r}{100}$

i.e. $\;$ $15000 = 5000 \times \dfrac{24 \times 25}{2 \times 12} \times \dfrac{r}{100}$

i.e. $r = \dfrac{15000 \times 2 \times 12 \times 100}{5000 \times 24 \times 25} = 12 \%$

$\therefore \;$ Rate of interest on the recurring deposit $= 12 \%$

Commercial Mathematics - Banking

A person deposited ₹ $ 400$ at the beginning of every month in a recurring deposit account and received ₹ $ 16,398$ at the end of $3$ years. Find the rate of interest given by the bank.

Money deposited each month $= P =$ ₹ $ 400$

Time for which money deposited $= n = 3 $ years $= 36 $ months

Let, rate of interest $= r \%$

Amount received on maturity $= $ ₹ $ 16,398$

$\begin{aligned} \text{Interest} & = P \times \dfrac{n \left(n + 1\right)}{2 \times 12} \times \dfrac{r}{100} \\\\ & = 400 \times \dfrac{36 \times 37}{2 \times 12} \times \dfrac{r}{100} \\\\ & = 222 \; r \end{aligned}$

i.e. $\;$ Interest $= $ ₹ $ 222 \; r$

Total money deposited $= 400 \times 36 = $ ₹ $ 14,400$

$\text{Amount received on maturity} = \text{Money deposited} + \text{Interest}$

i.e. $\;$ $16398 = 14400 + 222 \; r$

i.e. $\;$ $222 \; r = 1998$

i.e. $\;$ $r = 9$

$\therefore \;$ Rate of interest given by the bank $= 9 \%$

Commercial Mathematics - Shares and Dividends

A person bought ₹ $\; 100$ shares of dividend $9 \%$ selling at a certain price. If the rate of return is $7.5 \%$, calculate:

1. the market value of each share;
2. the amount to be invested to obtain an annual income of ₹ $\; 1260$;
3. how many more shares should be bought to increase the income to ₹ $\; 1890$.

Nominal value (N.V) of each share $= $ ₹ $ 100$

Rate of dividend $= 9 \%$

Rate of return $= 7.5 \%$

1. Let market value (M.V) of each share $= $ ₹ $ x$
   
   Now, $\;$ $\text{Rate of return} \times M.V = \text{Rate of dividend} \times N.V$
   
   i.e. $\;$ $\dfrac{7.5}{100} \times x = \dfrac{9}{100} \times 100$
   
   $\implies$ $x = \dfrac{900}{7.5} = 120$
   
   i.e. $\;$ Market value of each share $= $ ₹ $ 120$
   
   
2. Total annual income $= $ ₹ $ 1260$
   
   Annual income on $1$ share $= 9 \%$ of ₹ $ 100 = $ ₹ $ 9$
   
   $\begin{aligned} \text{Number of shares bought} & = \dfrac{\text{Total annual income}}{\text{Annual income on 1 share}} \\\\ & = \dfrac{1260}{9} \\\\ & = 140 \end{aligned}$
   
   $\therefore \;$ Amount to be invested $=$ Number of shares bought $\times$ M.V of 1 share
   
   i.e. $\;$ Amount to be invested $= 140 \times $ ₹ $ 120 = $ ₹ $ 16,800$
   
   
3. New annual income $= $ ₹ $ 1890$
   
   $\begin{aligned} \therefore \; \text{Number of shares} & = \dfrac{\text{Total annual income}}{\text{Annual income on 1 share}} \\\\ & = \dfrac{1890}{9} \\\\ & = 210 \end{aligned}$
   
   $\therefore \;$ Number of extra shares $= 210 - 140 = 70$