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) = $ ₹ $8700$