Q.A man sells ₹5000, 12% stock at 156 and invests the proceeds partly in 8% stock at 90 and 9% stock at 108. He thereby increases his income by ₹70. How much of the proceeds were invested in each stock?

Concept: Stock Valuation – Income from stock = (Face Value × Rate%) / Market Price × Investment.

Step 1: Find total proceeds from sale.

Sale of ₹5000, 12% stock at 156 (market price per ₹100 face value).

Proceeds = $\frac{5000}{100} \times 156 = 50 \times 156 = ₹7800$.

Step 2: Original income.

Income from 12% stock = $5000 \times \frac{12}{100} = ₹600$.

Step 3: New income and investment split.

New income = $600 + 70 = ₹670$.

Let investment in 8% stock at 90 be $x$; then in 9% stock at 108 be $7800 - x$.

Income from 8% stock = $\frac{x}{90} \times 8 = \frac{8x}{90}$.

Income from 9% stock = $\frac{7800 - x}{108} \times 9 = \frac{9(7800 - x)}{108}$.

Total new income: …

Concept and Intuition

This is a classic stock valuation problem from Indian finance and mathematics exams. The core idea: when you buy a stock, you pay its market price (e.g., ₹156 per ₹100 face value), but your income (dividend) is calculated on the face value (usually ₹100 per share). So the yield (effective return) depends on both the dividend rate and the market price.

Here, the man sells one stock and buys two others. The total proceeds from the sale are fixed. He wants to allocate that money between the two new stocks so that his total annual income increases by ₹70 compared to what he was earning before. We need to find the exact split.

The trick: income from a stock = (Face Value × Dividend Rate) / 100. But the amount invested = (Market Price / 100) × Face Value. So we must convert between face value and investment amount carefully.

Step-by-Step Solution

1. Find the annual income from the original stock.

The man holds ₹5000 of 12% stock. This means the face value of his holding is ₹5000. The dividend is 12% per annum on the face value.

2. Calculate the sale proceeds.

He sells this stock at a market price of ₹156 per ₹100 face value. So for every ₹100 face value, he gets ₹156.

So he has ₹7800 to invest in the two new stocks.

3. Set up variables for the new investments.

Let the amount invested in the 8% stock at 90 be ₹$x$.

Then the amount invested in the 9% stock at 108 is ₹$(7800 - x)$.

4. Convert investment amounts into face values for each stock.

For the 8% stock at 90: Market price ₹90 buys ₹100 face value.

So if investment is ₹$x$, the face value bought is:

For the 9% stock at 108: Market price ₹108 buys ₹100 face value.

So if investment is ₹$(7800 - x)$, the face value bought is:

5. Compute the new total income.

Income from the 8% stock = 8% of its face value:

Income from the 9% stock = 9% of its face value:

$\text{Income}_2 = \frac{9}{100} \times \frac{25}{27} (7800 - x) = \frac{225}{2700} (7800 - x) = \frac{1}{12} (7800 - x)$ …