Q.A and B start a business investing capitals of $30000$ rupees and $45000$ rupees respectively for the same period. At the end of the year the profit is $15000$ rupees. Find each partner's share.

Since both capitals are invested for the same time, profit is shared simply in the ratio of the capitals.

The ratio has $2+3=5$ parts; each part is worth $15000/5 = 3000$ rupees, giving A two parts and B three parts.

Because both partners invest for the same length of time, this is a simple partnership, and profit is divided in the ratio of the capitals.

Step 1 — form the capital ratio. $30000 : 45000 = 2 : 3 \quad (\text{dividing both by } 15000).$

Step 2 — find the value of one part. The ratio has $2 + 3 = 5$ parts in total, and these parts share the whole profit of $15000$ rupees, so $\text{one part} = \frac{15000}{5} = 3000 \text{ rupees}.$

Step 3 — find each share. $A = 2 \times 3000 = 6000 \text{ rupees}, \qquad B = 3 \times 3000 = 9000 \text{ rupees}.$

Verification. The two shares add to $6000 + 9000 = 15000$ rupees, which is exactly the total profit, and their ratio is $6000 : 9000 = 2 : 3$, matching the capital ratio — both checks confirm the division is correct.

The shares can also be found directly as fractions of the total profit: $A = \frac{2}{5}\times 15000 = 6000$ and $B = \frac{3}{5}\times 15000 = 9000$ rupees, without separately computing the value of one part.

A frequent error is to divide the profit equally between the two partners, ignoring that their capitals differ — profit follows the capital ratio, not a 50-50 split, whenever the capitals are unequal.