A and B are partners in a business. They invest in the ratio of 4 : 9, at the end of 6 months A withdraws. If they receive profits in the ratio of 4 : 9, then find how long B’s investment was used.

Understanding the Problem

In a partnership business, A and B invest in the ratio of 4:9. A withdraws his investment after 6 months. Despite A’s withdrawal, they still receive profits in the same ratio as their initial investments (4:9). We need to find out for how long B’s investment was used.

Solving the Problem

Let’s denote the period for which B’s investment was used as \(x\) months.

The profit earned by a partner in a business is directly proportional to the product of their investment and the time period for which the investment is made.

Given that A’s investment was used for 6 months and B’s investment was used for \(x\) months, we can write the following equation based on the proportion of their profits:

\[ \frac{\text{A’s investment} \times \text{A’s time}}{\text{B’s investment} \times \text{B’s time}} = \frac{4}{9} \]

Plugging in the values:

\[ \frac{4 \times 6}{9 \times x} = \frac{4}{9} \]

Simplifying:

\[ \frac{24}{9x} = \frac{4}{9} \]

\[ 24 = 4x \]

\[ x = \frac{24}{4} = 6 \]

Conclusion

B’s investment was used for 6 months.