Q.In a simple partnership, A and B receive profits of $6000$ rupees and $9000$ rupees respectively for capitals invested over the same period. If A's capital is $8000$ rupees, find B's capital.

Because time is equal, the ratio of the partners' profits equals the ratio of their capitals, which fixes B's capital.

Setting the capital ratio equal to the profit ratio and using A's known capital gives B's capital.

In a simple partnership the profit ratio equals the capital ratio, so B's capital can be found from the profits and A's capital.

Step 1 — profit ratio. $6000 : 9000 = 2 : 3 \quad (\text{dividing both by } 3000).$ This is therefore also the capital ratio: $\text{A's capital} : \text{B's capital} = 2 : 3$.

Step 2 — value of one part from A's capital. A corresponds to $2$ parts and A's capital is $8000$ rupees, so $\text{one part} = \frac{8000}{2} = 4000 \text{ rupees}.$

Step 3 — B's capital. B corresponds to $3$ parts: $\text{B's capital} = 3 \times 4000 = 12000 \text{ rupees}.$

Verification. With capitals $8000$ and $12000$, the capital ratio is $8000 : 12000 = 2 : 3$, which matches the given profit ratio $6000:9000 = 2:3$ exactly — confirming B's capital.

Using a direct proportion: $\frac{\text{A's capital}}{\text{B's capital}} = \frac{\text{A's profit}}{\text{B's profit}} \Rightarrow \frac{8000}{\text{B}} = \frac{6000}{9000}$, giving $\text{B} = \frac{8000\times 9000}{6000} = 12000$ rupees.

Students sometimes set the capital ratio equal to the inverse of the profit ratio — but in a simple partnership the two ratios are directly equal, so the partner with the larger profit is the one with the larger capital.