Ramita, Suman, Neetu Work Rates

This problem involves distributing a total payment (Rs. $6000$) among three individuals – Ramita, Suman, and Neetu – based on the amount of work each contributes. The fundamental idea is that the payment received by each person is directly proportional to their individual work rate. First, we determine the rate at which each person completes the work.

We are given the time each person takes to complete the entire work alone:

• Ramita can do the work in 100 days. Her work rate is $\frac{1}{100}$ of the work per day.
• Suman can do the same work in 120 days. Suman's work rate is $\frac{1}{120}$ of the work per day.
• Neetu can do the same work in 150 days. Neetu's work rate is $\frac{1}{150}$ of the work per day.

Work Ratio Calculation

To distribute the payment of Rs. $6000$ fairly, we need to find the ratio of the work done by Ramita, Suman, and Neetu. This ratio is equivalent to the ratio of their individual work rates:

Ramita : Suman : Neetu = $\frac{1}{100} : \frac{1}{120} : \frac{1}{150}$

To simplify this ratio into whole numbers, we calculate the Least Common Multiple (LCM) of the denominators: 100, 120, and 150.

Prime factorizations:

• $100 = 2^2 \times 5^2$
• $120 = 2^3 \times 3 \times 5$
• $150 = 2 \times 3 \times 5^2$

The LCM is calculated by taking the highest power of each prime factor present:

LCM(100, 120, 150) = $2^3 \times 3^1 \times 5^2 = 8 \times 3 \times 25 = 600$.

Multiplying each part of the ratio by the LCM (600):

Ratio = $(600 \times \frac{1}{100}) : (600 \times \frac{1}{120}) : (600 \times \frac{1}{150})$

Ratio = $6 : 5 : 4$

This ratio ($6:5:4$) represents how the total payment should be divided among Ramita, Suman, and Neetu, respectively.

Suman's Share of Payment

The total number of parts in the ratio is the sum of the individual ratio parts:

Total parts = $6 + 5 + 4 = 15$ parts.

The total payment to be distributed is Rs. $6000$. Suman's share corresponds to 5 parts out of the total 15 parts.

Suman's share = $\left( \frac{\text{Suman's ratio part}}{\text{Total ratio parts}} \right) \times \text{Total Payment}$

Suman's share = $\left( \frac{5}{15} \right) \times 6000$

Simplifying the fraction $\frac{5}{15}$ to $\frac{1}{3}$:

Suman's share = $\left( \frac{1}{3} \right) \times 6000$

Suman's share = Rs. $2000$.

Therefore, Suman receives Rs. $2000$ from the total payment of Rs. $6000$.