​X and Y together can finish work in 12 days, Y and Z can finish In 20 days and Z and X can finish in 15 days. Find in how many days

X,Y and Z will finish the work?​

Correct option is D

Given:

X and Y together can finish the work in 12 days.

Y and Z together can finish the work in 20 days.

Z and X together can finish the work in 15 days.

Concept Used:

Work done together is the sum of individual work rates.

If A and B can finish a job in tt days, then their combined work rate is \frac{1}{t} per day.

Solution:

Work rate of X + Y is $$\frac{1}{12}$$​(since they finish the work in 12 days).

Work rate of Y + Z is $$\frac{1}{20}$$​ (since they finish the work in 20 days).

Work rate of Z + X is $$\frac{1}{15}$$​(since they finish the work in 15 days).

Let the individual work rates of X, Y, and Z be x, y, and z, respectively.

We have the following system of equations:

​$$x + y = \frac{1}{12} \quad \text{(1)}\\ \ \\ y + z = \frac{1}{20} \quad \text{(2)} \\ \ \\ z + x = \frac{1}{15} \quad \text{(3)}$$​

Add all three equations:

​$$((x + y) + (y + z) + (z + x) = \frac{1}{12} + \frac{1}{20} + \frac{1}{15}$$​

​$$\frac{1}{12} + \frac{1}{20} + \frac{1}{15} = \frac{5}{60} + \frac{3}{60} + \frac{4}{60} = \frac{12}{60} = \frac{1}{5}$$​

​$$2(x + y + z) = \frac{1}{5}$$​

​$$x + y + z = \frac{1}{10}$$​

The combined work rate of X, Y, and Z is $$\frac{1}{10}$$​, meaning they can finish the work in 10 days.