Earnings in joint work: Ram can finish a job alone in 9 days, while Shyam can finish it alone in 15 days. If they work together and share the payment in proportion to their actual work done, in what ratio should their total earnings be divided (Ram : Shyam)?

Introduction / Context:
Payment in joint work is divided according to the work contributed by each person. When individual times to complete a job are known, their rates (work per day) are inverses of these times. The ratio of rates gives the ratio of earnings for any common working period.

Given Data / Assumptions:

• Time(Ram) = 9 days ⇒ Rate(Ram) = 1/9 job per day
• Time(Shyam) = 15 days ⇒ Rate(Shyam) = 1/15 job per day
• They work together for the same duration

Concept / Approach:
Earnings ratio equals work ratio equals rate ratio for equal time worked. Therefore, divide 1/9 by 1/15 to get the ratio Ram : Shyam.

Step-by-Step Solution:
Rate ratio = (1/9) : (1/15)= (1/9) * (15/1) = 15/9 = 5/3Hence, Ram : Shyam = 5 : 3

Verification / Alternative check:
Consider an LCM of days (45). In 45 days, Ram does 45*(1/9) = 5 jobs; Shyam does 45*(1/15) = 3 jobs. Therefore 5 : 3.

Why Other Options Are Wrong:
3 : 5 or 2 : 3 invert or distort the correct rate comparison.'None of these' is unnecessary since 5 : 3 fits perfectly.

Common Pitfalls:
Confusing time ratios with work ratios. Remember: smaller time means higher rate and larger share.

Final Answer:
5 : 3