[4 marks sum]

Question 14 Marks

Divide Rs. 20304 into two parts such that if one part is invested in 9% Rs. 50 shares at 8% premium and the other part is invested in 8% Rs. 25 shares at 8% discount, then the annual incomes from both the investment are equal

Answer

Total amount = Rs 20304
Let amount invested in $9 \%$ Rs 50 at $8 \%$
premium $=x$
Then amount invested in $8 \%$ Rs 25 at $8 \%$
Discount $=20304- x$
Income from both investments are equal Now income from first type of shares
$
=\frac{x \times 9}{100+8}=\frac{9 x}{108}=\frac{x}{12}
$
Income from second type of shares
$
\begin{aligned}
& =\frac{(20304-x) \times 8}{100-8} \\
& =\frac{(20304-x) \times 8}{92} \\
& =\frac{2(20304-x)}{23}
\end{aligned}
$
$\because$ In both cases, annual income is same
$
\begin{aligned}
& \therefore \frac{x}{12}=\frac{2(20304-x)}{23} \\
& \Rightarrow 23 x =24(20304- x ) \quad \ldots(\text {By cross multiplication) } \\
& \Rightarrow 23 x =24 \times 20304-24 x \\
& \Rightarrow 23 x +24 x =24 \times 20304 \\
& \Rightarrow 47 x =24 \times 20304 \\
& \therefore x =\frac{24 \times 20304}{47}=10368
\end{aligned}
$
$\therefore$ Amount invested in first kind of shares
$
=\text { ₹ }10368
$
and in second kind of shares
$
\begin{aligned}
& =\text { ₹ } 20304 \text { - \text { ₹ }10368 } \\
& =\text { ₹ } 9936 .
\end{aligned}
$

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Question 24 Marks

A man bought 360 ten-rupee shares paying $12 \%$ per annum. He sold them when the price rose to Rs. 21 and invested the proceeds in five-rupee shares paying $4 \frac{1}{2} \%$ per annum at Rs. 3.5 per share. Find the annual change in his income.

Answer

No. of shares bought $=360$
Face value of each share $=$ Rs. 10
Rate of dividend $=12 \%$
Total face value of 360 shares
$
\begin{aligned}
& =\text { Rs. } 10 \times 360 \\
& =\text { Rs. } 3600
\end{aligned}
$
$\therefore$ Yearly dividend $=$ Rs. $\frac{3600 \times 12}{100}=$ Rs. 432
On selling the share at Rs. 21 , the amount received = Rs. $21 \times 360=$ Rs. 7560
Face value of new shares $=$ Rs. 5.00
and market value $=$ Rs. 3.5
Rate of dividend $=4 \frac{1}{2} \%=\frac{9}{2} \%$
No. of share purchased
$
=\frac{7560}{3.5} \times \frac{7560 \times 10}{35}=2160
$Face value of 2160 shares $=$ Rs. $5 \times 2160$
$
\text { = Rs. } 10800
$
$
\therefore \text { Dividend }=\frac{10800 \times 9}{100 \times 2}=\text { Rs. } 486
$Change in income = Rs. 486 - Rs. 432
$
=\text { Rs. } 54 \text { gain. }
$

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Question 34 Marks

Divide Rs. 101520 into two parts such that if one part is invested in 8% Rs. 100 shares at 8% discount and the other in 9% Rs. 50 shares at 8% premium, the annual incomes are equal.

Answer

Total investment = Rs. 101520
Let investment in first part $= x$
and in second part $=(101520-x)$
Market value of first kind of shares $=$ Rs. $100-$ Rs. 8
$
=\text { Rs. } 92
$
and rate of dividend $=8 \%$
$
\therefore \text { Dividend }=\frac{x \times 8}{92}=\text { Rs. } \frac{2 x}{23}
$
Market value of second kind $=(101520-x)$
Rate of dividend $=9 \%$
and market value $=$ Rs. $\frac{100+8}{100} \times 50$
$
\begin{aligned}
& =\frac{108}{100} \times 50 \\
& =\text { Rs. } 54
\end{aligned}
$
$\therefore$ Dividend $=(101520-x) \times \frac{9}{2 \times 54}=\frac{101520-x}{12}$
$\therefore$ According to the sum $\frac{2 x}{23}=\frac{101520-x}{12}$
$
\begin{aligned}
& \Rightarrow 24 x =101520 \times 23-23 x \\
& \Rightarrow 24 x +23 x =101520 \times 23 \\
& \Rightarrow 47 x =101520 \times 23 \\
& \therefore x =\frac{101520 \times 23}{47}=49680
\end{aligned}
$
$\therefore$ Investment of first part $=$ Rs. 49680
and in second part
$
\text { = Rs. } 101450 \text { - Rs. } 49680
$
$
=\text { Rs. } 51840 \text {. }
$

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Question 44 Marks

A man has some shares of Rs. 100 par value paying 6% dividend. He sells half of these at a discount of 10% and invests the proceeds in 7% Rs. 50 shares at a premium of Rs. 10. This transaction decreases his income from dividends by Rs. 120. Calculate:
(i) the number of shares before the transaction.
(ii) the number of shares he sold.
(iii) his initial annual income from shares.

Answer

Let no. of shares $=x$
Value of $x$ shares $=x \times 100=100 x$
and dividend $=\frac{100 x \times 6}{100}=$ Rs. $6 x$
and dividend on half-shares $=$ Rs. $\frac{6 x}{2}=$ Rs. $3 x$
Now, no of shares he sold out $=\frac{x}{2}$
Amount received at $10 \%$ discount
$
=\frac{x}{2} \times 90=\text { Rs. } 45 x
$
In investing Rs. $45 x$, no. of share he purchased $=\frac{45 x}{60}$
$\therefore$ Amount of shares $=\frac{45 x}{60} \times 50=$ Rs. $\frac{225 x}{6}$
Income at the rate of $7 \%=\frac{225}{6} x \times \frac{7}{100}=\frac{21 x}{8}$
Differece in income $=3 x-\frac{21 x}{8}=\frac{3 x}{8}$
According to the condition, $\frac{3 x}{8}=120$.
$
\Rightarrow x =\frac{120 \times 8}{3}=320
$
(i) $\therefore$ No. of share he hold initially $=320$
(ii) No. of share he hold later $=\frac{320}{2}=160$
(iii) Amount of income initially
$
\begin{aligned}
& =320 \times 6 \\
& =\text { Rs. } 1920 .
\end{aligned}
$

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Question 54 Marks

By selling at Rs. 92, some 2.5% Rs. 100 shares and investing the proceeds in 5% Rs. 100 shares at Rs. 115, a person increased his annual income by Rs. 90. Find:
(i) the number of shares sold.
(ii) the number of shares purchased.
(iii) the new income.
(iv) the rate percent which he earns on his investment.

Answer

Rate of dividend $=2.5 \%$ and market price $=$ Rs. 92
Let number of shares purchased $=x$.
Selling price of $x$ shares $=92 x$
Income from investing
$
\begin{aligned}
& \text { ₹ } x =\frac{92 x \times 2.5}{92} \\
& =\frac{92 \times x 25}{92 \times 10} \\
& =\frac{5}{2} x
\end{aligned}
$
Again by investing $92 x$ in $5 \%$ at $\text { ₹ }115$
the dividend $=\frac{92 x \times 5}{115}=4 x$
Difference $=4 x-\frac{115}{2} x=\frac{3}{2} x$
$\therefore \frac{3}{2} x=90$
$\Rightarrow x =\frac{90 \times 2}{3}=60$
(i) $\therefore$ No. of shares $=60$
(ii) No. of shares sold $=\frac{92 x}{115}$
$
=\frac{92 \times 60}{115}=48
$
(iii) New income $=4 x =4 \times 60=\text { ₹ } 240$
(iv) Rate percent interest on investment
$
\begin{aligned}
& =\frac{5 \times 100}{115} \\
& =\frac{100}{23} \\
& =4 \frac{8}{23} \%
\end{aligned}
$

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Mathematics [4 marks sum]

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