A, B and C enter into a partnership. Their capital contribution is in the ratio 21 : 18 : 14. At the end of the business term they share profits in the ratio 15 : 8 : 9. Find the ratio of time for which they invest their capitals.
Answer
Answer (as printed):
Explanation
Suppose A, B and C invest ₹$21x$ for $p$ months, ₹$18x$ for $q$ months and ₹$14x$ for $r$ months. Then, $21x\times p:18x\times q:14x\times r=15:8:9 \Rightarrow 21p:18q:14r=15:8:9$. Now, $\frac{21p}{18q}=\frac{15}{8} \Rightarrow p=\left(\frac{15}{8}\times\frac{18}{21}\right)q=\frac{45}{28}q$. And, $\frac{18q}{14r}=\frac{8}{9} \Rightarrow q=\left(\frac{8}{9}\times\frac{14}{18}\right)r=\frac{56}{81}r$. $\therefore p=\frac{45}{28}q=\left(\frac{45}{28}\times\frac{56}{81}\right)r=\frac{10}{9}r$. So, required ratio $=p:q:r=\frac{10}{9}r:\frac{56}{81}r:r=\frac{10}{9}:\frac{56}{81}:1=90:56:81$.
Explanation as extracted from the printed page; notation may be imperfect.