Jai Prakash wants to invest $₹ 3600$ . He invests $₹ 750$ in $₹ 100$ shares of $3.5 %$ at $₹ 75$ , $₹ 1050$ in $₹ 100$ shares of $3 %$ at $₹ 70$ and the remaining in $₹ 100$ shares of $6 %$ . If his total yield is $5 \frac{5}{9} %$ of his investment, at what price did he buy the shares of $6 %$ ?

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Solution

Step 1: Given data:
Total investment, $I = ₹ 3600$
Yield%, $Y % = 5 \frac{5}{9} %$
Type of data $1^{s t}$ part $2^{n d}$ part $3^{r d}$ part
Investment $I_{1}$ is $₹ 750$ $I_{2}$ is $₹ 1050$ $I_{3}$ is $(I - I_{1} - I_{2}) = (3600 - 750 - 1050) = ₹ 1800$
Nominal value $F_{1}$ is $₹ 100$ $F_{2}$ is $₹ 100$ $F_{3}$ is $₹ 100$
Market value $M_{1}$ is $₹ 75$ $M_{2}$ is $₹ 70$ Not given
Rate of dividend $D_{1}$ is $3.5 %$ $D_{2}$ is $3 %$ $D_{3}$ is $6 %$
Step 2: Find the annual income from $1^{s t}$ part of the investment:
As for the $1^{s t}$ part, he invests $₹ 750$ in $₹ 100$ shares of $3.5 %$ at $₹ 75$ . To calculate annual income, the number of shares is required.
∴ $N u m b e r o f s h a r e s = \frac{I n v e s t m e n t}{M a r k e t v a l u e}$
$N_{1} = \frac{I_{1}}{M_{1}} = \frac{750}{75} = 10 shares$
Now,
$A n n u a l i n c o m e = N u m b e r o f s h a r e s \times N o m i n a l v a l u e \times \frac{R a t e o f d i v i d e n d}{100}$
$A_{1} = N_{1} \times F_{1} \times \frac{D_{1}}{100} = 10 \times 100 \times \frac{3.5}{100} = ₹ 35$
Step 3: Find the annual income from $2^{n d}$ part of the investment:
As for the $2^{n d}$ part, he invests $₹ 1050$ in $₹ 100$ shares of $3 %$ at $₹ 70$ . To calculate annual income, the number of shares is required.
∴ $N u m b e r o f s h a r e s = \frac{I n v e s t m e n t}{M a r k e t v a l u e}$
$N_{2} = \frac{I_{2}}{M_{2}} = \frac{1050}{70} = 15 shares$
Now,
$A n n u a l i n c o m e = N u m b e r o f s h a r e s \times N o m i n a l v a l u e \times \frac{R a t e o f d i v i d e n d}{100}$
$A_{2} = N_{2} \times F_{2} \times \frac{D_{2}}{100} = 15 \times 100 \times \frac{3}{100} = ₹ 45$
Step 4: Find the annual income from $3^{r d}$ part of the investment:
As for the $3^{r d}$ part, he invests $₹ 1800$ in $₹ 100$ shares of $6 %$ .
Also, Jai Prakash's total yield is $5 \frac{5}{9} %$ of his investment.
∴ $Y i e l d % = \frac{T o t a l A n n u a l I n c o m e}{T o t a l I n v e s t m e n t} \times 100$
$Y % = \frac{T o t a l A n n u a l I n c o m e}{3600} \times 100 5 \frac{5}{9} = \frac{T o t a l A n n u a l I n c o m e}{36} \frac{50}{9} = \frac{T o t a l A n n u a l I n c o m e}{36} T o t a l A n n u a l I n c o m e = \frac{50 \times 36}{9} T o t a l A n n u a l I n c o m e = ₹ 200$
Let us assume that the annual income from $3^{r d}$ part be $A_{3}$ .
∴ $T o t a l A n n u a l i n c o m e = A_{1} + A_{2} + A_{3}$
$200 = 35 + 45 + A_{3} 200 = 80 + A_{3} A_{3} = 200 - 80 A_{3} = ₹ 120$
Step 5: Find the number of shares in $3^{r d}$ part of the investment:
Let us assume that the number of shares in $3^{r d}$ part be $N_{3}$ .
∴ $A n n u a l i n c o m e = N u m b e r o f s h a r e s \times N o m i n a l v a l u e \times \frac{R a t e o f d i v i d e n d}{100}$
$A_{3} = \frac{I_{3}}{M_{3}} \times F_{3} \times \frac{D_{3}}{100} [∵ N u m b e r o f s h a r e s = \frac{I n v e s t m e n t}{M a r k e t v a l u e}] 120 = N_{3} \times 100 \times \frac{6}{100} N_{3} = \frac{120}{6} N_{3} = 20 shares$
Step 6: Find the market value of the shares of $3^{r d}$ part of the investment:
$M a r k e t v a l u e = \frac{I n v e s t m e n t}{N u m b e r o f s h a r e s}$
$M_{3} = \frac{I_{3}}{N_{3}} = \frac{1800}{20} = ₹ 90$
Final Answer: Hence, Jai Prakash bought the shares of $6 %$ at $₹ 90$ .

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Type of data	$1^{s t}$ part	$2^{n d}$ part	$3^{r d}$ part
Investment	$I_{1}$ is $₹ 750$	$I_{2}$ is $₹ 1050$	$I_{3}$ is $(I - I_{1} - I_{2}) = (3600 - 750 - 1050) = ₹ 1800$
Nominal value	$F_{1}$ is $₹ 100$	$F_{2}$ is $₹ 100$	$F_{3}$ is $₹ 100$
Market value	$M_{1}$ is $₹ 75$	$M_{2}$ is $₹ 70$	Not given
Rate of dividend	$D_{1}$ is $3.5 %$	$D_{2}$ is $3 %$	$D_{3}$ is $6 %$

Jai Prakash wants to invest₹3600. He invests₹750 in ₹100 shares of3.5% at ₹75, ₹1050 in ₹100 shares of 3%at ₹70 and the remaining in ₹100 shares of6%. If his total yield is 559%of his investment, at what price did he buy the shares of6%?