Two typists undertake to do a job. The second typist begins working one hour after the first. Three hours after the first typist has begun working, there is still 9/20 of the work to be done. When the assignment is completed, it turns out that each typist has done half the work. How many hours would it take each one to do the whole job individually?

Options

A. 12 hr and 8 hr
B. 8 hr and 5.6 hr
C. 10 hr and 8 hr
D. 5 hr and 4 hr

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = time taken by the first typist (A) to complete the job alone (in hours).
- \( y \) = time taken by the second typist (B) to complete the job alone (in hours).

### Step 2: Work Done by Each Typist
The work done by A in one hour is \( \frac{1}{x} \).
The work done by B in one hour is \( \frac{1}{y} \).

### Step 3: Work Done in 3 Hours
After 3 hours of work by A, the amount of work done by A is:
\[ \text{Work done by A} = 3 \times \frac{1}{x} = \frac{3}{x} \]
B starts working after 1 hour, so B works for 2 hours. The amount of work done by B is:
\[ \text{Work done by B} = 2 \times \frac{1}{y} = \frac{2}{y} \]

### Step 4: Total Work Done
According to the problem, after 3 hours, \( \frac{9}{20} \) of the work is still left. Therefore, the work done by both A and B together is:
\[ 1 - \frac{9}{20} = \frac{11}{20} \]
Thus, we can write the equation:
\[ \frac{3}{x} + \frac{2}{y} = \frac{11}{20} \]

### Step 5: Condition of Equal Work Done
When the job is completed, both typists have done half the work. This means:
\[ \text{Work done by A} = \text{Work done by B} = \frac{1}{2} \]
This gives us the equations:
\[ \frac{3}{x} + \frac{2}{y} = \frac{11}{20} \quad \text{(1)} \]
And since both have done half the work, we can express the total work done by A and B:
\[ \frac{1}{x} \cdot T = \frac{1}{2} \quad \text{and} \quad \frac{1}{y} \cdot T = \frac{1}{2} \]
Where \( T \) is the total time taken to complete the job.

### Step 6: Solve the Equations
From the above equations, we can express \( T \) in terms of \( x \) and \( y \):
\[ T = \frac{x}{2} = \frac{y}{2} \]
Thus, \( x = y \).

### Step 7: Substitute and Solve
Now, substituting \( y = x \) into equation (1):
\[ \frac{3}{x} + \frac{2}{x} = \frac{11}{20} \]
This simplifies to:
\[ \frac{5}{x} = \frac{11}{20} \]
Cross-multiplying gives:
\[ 5 \cdot 20 = 11 \cdot x \implies 100 = 11x \implies x = \frac{100}{11} \approx 9.09 \text{ hours} \]
And since \( y = x \), we have:
\[ y = \frac{100}{11} \approx 9.09 \text{ hours} \]

### Conclusion
Each typist would take approximately \( 9.09 \) hours to complete the job individually.

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