Question

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 1

Answer: Rs. 49680  needs to be invested in the Rs.100 shares while Rs. 51840  needs to be invested in the Rs.50 shares so that annual income from them are equal.

We follow these steps in order to arrive at the answer:

We first determine the price at which each of the shares can be purchased and the annual income from each share.

Let P₁₀₀ be the purchase price of the Rs.100 share.

Let I₁₀₀ be the income form one Rs.100 share.

Let P₅₀ be the purchase price of the Rs.50 share.

Let I₅₀ be the income form one Rs.50 share.

Since the Rs.100 share can be purchased at a discount of 8%, the purchase price will be:

$\mathbf{P_{100}= 100 - (100*0.08) = 92}$

The Rs.50 can be purchased at a premium of 8%, hence the purchase price is:

$\mathbf{P_{50}= 50 + (50*0.08) = 54}$

The income from the Rs.100 share will be:

$\mathbf{I_{100}= 100 * 0.08 = 8}$

The income from the Rs.50 share will be:

$\mathbf{I_{50}= 50 * 0.09 = 4.5}$

Since the question states that the income from both types of shares must be equal, we can determine the number of Rs.50 shares required to earn the same income from one Rs.100 share.

This is arrived at as:

$4.5x = 8$

$\mathbf{x= \frac{8}{4.5} = 1.777777778}$

Now, we can determine the purchase price of 1.777777778 Rs.50 shares as:

$\mathbf{1.777777778 *54 = 96}$

This means that 1.777778 shares with a Face Value of Rs.50 but bought at Rs.54 per share (worth Rs.96) will give the same annual income as one share with a Face Value of Rs.100 bought at Rs. 92.

Now, we can express this as mathematically in order to determine the amount to be invested in each share.

Let x be the number of  Rs.100 shares purchased.

$\mathbf{92x + 96x = 101520}$

Solving we get,

$\mathbf{188x = 101520}$

$\mathbf{x = 540}$

Since one Rs.100 share is equal to 1.1.777778 Rs.50 shares, the number of Rs.50 shares will be $\mathbf{540*1.777777778 = 960}$

Hence the amount to be invested in each type of share will be:

$\mathbf{92*540 = 49680}$ in Rs.100 shares and

$\mathbf{54*960= 51840}$ in Rs.50 shares.