Understanding the Work Rates of X and Y

This problem involves calculating the share of payment for individuals working together based on their individual work capacities. We need to determine how much X earns when working with Y for a total payment of Rs. 1800.

Calculating Individual Work Rates

First, let's find out how much work X and Y can do individually per day.

• X's Work Rate:

  X completes ¹⁄₄ of the work in 3 days.

  This means X completes the entire work in $3 \times 4 = 12$ days.

  Therefore, X's work rate is ¹⁄₁₂ of the work per day.

• Y's Work Rate:

  Y completes ¹⁄₆ of the work in 4 days.

  This means Y completes the entire work in $4 \times 6 = 24$ days.

  Therefore, Y's work rate is ¹⁄₂₄ of the work per day.

Calculating Combined Work Rate

When X and Y work together, their combined work rate is the sum of their individual rates.

Combined Rate = X's Rate + Y's Rate

Combined Rate = ¹⁄₁₂ + ¹⁄₂₄

To add these fractions, we find a common denominator, which is 24:

Combined Rate = ²⁄₂₄ + ¹⁄₂₄ = ³⁄₂₄

Simplifying the fraction, the combined rate is ¹⁄₈ of the work per day.

Determining the Ratio of Work Done

The total payment is distributed based on the ratio of work done by each person. The ratio of work done is equivalent to the ratio of their work rates.

Ratio of Work (X : Y) = X's Rate : Y's Rate

Ratio = ¹⁄₁₂ : ¹⁄₂₄

To simplify this ratio, we can multiply both sides by the least common multiple of the denominators (12 and 24), which is 24:

Ratio = (¹⁄₁₂ $\times$ 24) : (¹⁄₂₄ $\times$ 24)

Ratio = 2 : 1

This means that for every 3 units of work done together, X contributes 2 units and Y contributes 1 unit.

Distributing the Total Payment

The total payment is Rs. 1800. This amount needs to be divided between X and Y in the ratio 2:1.

Total parts in the ratio = 2 (for X) + 1 (for Y) = 3 parts.

Calculating X's Share

X's share is calculated based on his proportion of the work (2 parts out of 3).

X's Share = (X's Ratio Part / Total Ratio Parts) $\times$ Total Payment

X's Share = (²⁄₃) $\times$ Rs. 1800

X's Share = 2 $\times$ (Rs. 1800 / 3)

X's Share = 2 $\times$ Rs. 600

X's Share = Rs. 1200

Calculating Y's Share (for verification)

Y's Share = (Y's Ratio Part / Total Ratio Parts) $\times$ Total Payment

Y's Share = (¹⁄₃) $\times$ Rs. 1800

Y's Share = Rs. 600

Total Payment = X's Share + Y's Share = Rs. 1200 + Rs. 600 = Rs. 1800. This matches the total payment given.