Solving the Work and Wages Problem for X, Y, and Z

This problem involves calculating the share of payment each person receives based on the amount of work they complete. The principle is that the payment for completing work should be distributed in proportion to the amount of work done by each individual.

Understanding Individual Work Rates

First, let's determine the rate at which X and Y can do the work individually per day. We consider the total work to be 1 unit.

• X can do the work in 18 days. So, X's work rate per day is $\frac{1}{18}$ of the total work.
• Y can do the work in 24 days. So, Y's work rate per day is $\frac{1}{24}$ of the total work.

Calculating the Combined Work Rate of X, Y, and Z

With the assistance of Z, X and Y finished the work in 8 days. This means the combined work rate of X, Y, and Z per day is $\frac{1}{8}$ of the total work.

Finding Z's Individual Work Rate

Z's work rate per day can be found by subtracting the combined work rate of X and Y from the combined work rate of X, Y, and Z.

Combined work rate of X and Y = X's rate + Y's rate

Combined work rate of X and Y = $\frac{1}{18} + \frac{1}{24}$

To add these fractions, we find a common denominator. The least common multiple (LCM) of 18 and 24 is 72.

$\frac{1}{18} = \frac{1 \times 4}{18 \times 4} = \frac{4}{72}$

$\frac{1}{24} = \frac{1 \times 3}{24 \times 3} = \frac{3}{72}$

Combined work rate of X and Y = $\frac{4}{72} + \frac{3}{72} = \frac{7}{72}$

Now, we can find Z's work rate:

Z's rate = (X+Y+Z)'s rate - (X+Y)'s rate

Z's rate = $\frac{1}{8} - \frac{7}{72}$

To subtract, we convert $\frac{1}{8}$ to a fraction with a denominator of 72:

$\frac{1}{8} = \frac{1 \times 9}{8 \times 9} = \frac{9}{72}$

Z's rate = $\frac{9}{72} - \frac{7}{72} = \frac{2}{72} = \frac{1}{36}$

So, Z's work rate per day is $\frac{1}{36}$.

Calculating the Amount of Work Done by Each Person in 8 Days

The work was completed in 8 days. The amount of work done by each person is their daily rate multiplied by the number of days they worked (which is 8 days for everyone in this case, as they worked together to finish it).

• Work done by X in 8 days = X's rate $\times$ 8 = $\frac{1}{18} \times 8 = \frac{8}{18} = \frac{4}{9}$
• Work done by Y in 8 days = Y's rate $\times$ 8 = $\frac{1}{24} \times 8 = \frac{8}{24} = \frac{1}{3}$
• Work done by Z in 8 days = Z's rate $\times$ 8 = $\frac{1}{36} \times 8 = \frac{8}{36} = \frac{2}{9}$

Let's quickly check if the total work done is 1 unit: $\frac{4}{9} + \frac{1}{3} + \frac{2}{9} = \frac{4}{9} + \frac{3}{9} + \frac{2}{9} = \frac{4+3+2}{9} = \frac{9}{9} = 1$. This confirms the amounts are correct.

Determining the Ratio of Work Done and Payment Share

The payment of Rs 5400 is distributed among X, Y, and Z in the ratio of the work they have done.

Ratio of work done by X : Y : Z = $\frac{4}{9} : \frac{1}{3} : \frac{2}{9}$

To simplify the ratio, we multiply by the common denominator, which is 9:

Ratio = $(\frac{4}{9} \times 9) : (\frac{1}{3} \times 9) : (\frac{2}{9} \times 9)$

Ratio = $4 : 3 : 2$

The total ratio parts are $4 + 3 + 2 = 9$.

The total amount to be distributed is Rs 5400.

The value of one ratio part = $\frac{\text{Total Amount}}{\text{Total Ratio Parts}} = \frac{5400}{9} = 600$

Calculating Z's Share of the Payment

Z's share corresponds to 2 parts of the ratio.

Z's share = Z's ratio parts $\times$ Value per ratio part

Z's share = $2 \times 600 = 1200$

So, the share of Z is Rs 1200.

We can also calculate the shares of X and Y for completeness:

• X's share = 4 parts $\times$ 600 = Rs 2400
• Y's share = 3 parts $\times$ 600 = Rs 1800
• Total share = Rs 2400 + Rs 1800 + Rs 1200 = Rs 5400, which matches the total contract amount.

The share of Z is Rs 1200.

Revision Table: Work and Wages Concepts

Additional Information: Efficiency and Payment

In Work and Wages problems, efficiency is directly related to the work rate. A person with a higher work rate is more efficient and completes more work in the same amount of time, or finishes the same work in less time.

When multiple people work together for the same duration to complete a task, the ratio of the payment they receive is directly equal to the ratio of their individual work rates because the work done by each is proportional to their rate (Work = Rate $\times$ Time, and Time is constant).

If they work for different durations, the payment ratio is based on the actual amount of work done by each person over the time they worked (as calculated in this problem).

The assistance of Z means Z joins X and Y to speed up the work, thereby reducing the total time taken to finish the project compared to X and Y working alone.