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X and Y can do a piece of work in 10 day...

X and Y can do a piece of work in 10 days, Y and Z in 12 days and Z and X in 15 days. In how many days can Z alone do the work?

A

40

B

24

C

18

D

36

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will use the information provided about the work done by X, Y, and Z together. ### Step 1: Determine the total work done We know: - X and Y can complete the work in 10 days. - Y and Z can complete the work in 12 days. - Z and X can complete the work in 15 days. To find the total work, we can take the least common multiple (LCM) of the days taken by each pair to complete the work. The LCM of 10, 12, and 15 is 60. This means we can assume the total work is 60 units. ### Step 2: Calculate the efficiencies of each pair Now we can calculate the efficiencies of each pair: - Efficiency of X and Y: \[ \text{Efficiency of (X + Y)} = \frac{60 \text{ units}}{10 \text{ days}} = 6 \text{ units/day} \] - Efficiency of Y and Z: \[ \text{Efficiency of (Y + Z)} = \frac{60 \text{ units}}{12 \text{ days}} = 5 \text{ units/day} \] - Efficiency of Z and X: \[ \text{Efficiency of (Z + X)} = \frac{60 \text{ units}}{15 \text{ days}} = 4 \text{ units/day} \] ### Step 3: Set up equations for individual efficiencies Let the efficiencies of X, Y, and Z be represented as \( x \), \( y \), and \( z \) respectively. We can set up the following equations based on the efficiencies calculated: 1. \( x + y = 6 \) (from X and Y) 2. \( y + z = 5 \) (from Y and Z) 3. \( z + x = 4 \) (from Z and X) ### Step 4: Solve the equations Now we can solve these equations step by step. From equation (1): \[ y = 6 - x \tag{4} \] Substituting equation (4) into equation (2): \[ (6 - x) + z = 5 \] \[ z = 5 - 6 + x = x - 1 \tag{5} \] Now substituting equation (5) into equation (3): \[ (x - 1) + x = 4 \] \[ 2x - 1 = 4 \] \[ 2x = 5 \implies x = \frac{5}{2} = 2.5 \text{ units/day} \] Now substituting \( x \) back into equation (4) to find \( y \): \[ y = 6 - 2.5 = 3.5 \text{ units/day} \] Substituting \( x \) into equation (5) to find \( z \): \[ z = 2.5 - 1 = 1.5 \text{ units/day} \] ### Step 5: Calculate the time taken by Z alone Now that we have the efficiency of Z, we can calculate how long it will take Z to complete the total work of 60 units. Using the formula: \[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of Z}} = \frac{60}{1.5} = 40 \text{ days} \] ### Final Answer Z alone can complete the work in **40 days**. ---
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