Rahul started a business by investing $Rs. 45,000$. $6$ months later,Sanjay joined him with $Rs. 30,000$. In what ratio should the earned profit be distributed at the end of the year?

$A$ and $B$ are partners in a business. $A$ contributes $\frac{1}{4}$ of the capital for $15$ months and $B$ received $\frac{2}{3}$ of the profit. For how long was $B$'s money used (in months)?

Anju and Brijesh enter into a partnership with their capitals in the ratio of $5:9$. At the end of $8$ months,Anju withdraws her capital. If they receive their share of profit in the ratio of $4:9$,find out how long Brijesh's capital was invested in the business (in months).

$A$ and $B$ started a partnership business by investing in the ratio of $3:8$. $C$ joined them after $4$ months with an amount equal to $\frac{3}{4}$ of $B$'s investment. What was the total profit (in ₹) at the end of the year if $C$ received ₹ $24000$ as his share?

$A$ and $B$ entered into a partnership investing ₹ $16000$ and ₹ $12000$ respectively. After $3$ months,$A$ withdrew ₹ $5000$ while $B$ invested ₹ $5000$ more. After another $3$ months,$C$ joined the business with a capital of ₹ $21000$. If the total profit after $1$ year is ₹ $26400$,by how much does the share of $B$ exceed that of $C$?

$A$ and $B$ enter into a partnership with capitals in the ratio $5: 6$. At the end of $8$ months,$A$ withdraws. If they receive profits in the ratio $5: 9$,find how long $B$'s capital was used (in months)?