Two typists of varying skills can do a job in 6 minutes if they work together. If the first typist typed alone for 4 minutes and then the second typist typed alone for 6 minutos, they would be left with 1/5 of the whole work. How many minutes would it take the slower typist to complete the typing job working alone?

Options

A. 10 minutes
B. 15 minutes
C. 12 minutes
D. 20 minutes

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the combined work rate of both typists.
If both typists can complete the job together in 6 minutes, their combined work rate is:
\[ \text{Work rate} = \frac{1 \text{ job}}{6 \text{ minutes}} = \frac{1}{6} \text{ jobs per minute} \]

### Step 2: Express the work rates of the individual typists.
Let the work rate of the first typist (Typist A) be \( x \) jobs per minute. Then, the work rate of the second typist (Typist B) will be:
\[ \text{Work rate of B} = \frac{1}{6} - x \]

### Step 3: Calculate the work done by each typist.
Typist A works alone for 4 minutes, so the work done by A is:
\[ \text{Work done by A} = 4x \]
Typist B works alone for 6 minutes, so the work done by B is:
\[ \text{Work done by B} = 6\left(\frac{1}{6} - x\right) = 1 - 6x \]

### Step 4: Set up the equation based on the remaining work.
According to the problem, after both typists have worked, they are left with \( \frac{1}{5} \) of the job. Therefore, the total work done by both typists is:
\[ 4x + (1 - 6x) = 1 - \frac{1}{5} \]
This simplifies to:
\[ 4x + 1 - 6x = \frac{4}{5} \]
Combining like terms gives:
\[ -2x + 1 = \frac{4}{5} \]

### Step 5: Solve for \( x \).
Subtract 1 from both sides:
\[ -2x = \frac{4}{5} - 1 = \frac{4}{5} - \frac{5}{5} = -\frac{1}{5} \]
Now, divide by -2:
\[ x = \frac{1}{10} \]

### Step 6: Determine the work rate of Typist B.
Now we can find the work rate of Typist B:
\[ \text{Work rate of B} = \frac{1}{6} - \frac{1}{10} \]
Finding a common denominator (30):
\[ \text{Work rate of B} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15} \]

### Step 7: Calculate the time taken by each typist to complete the job alone.
The time taken by Typist A to complete the job alone is:
\[ \text{Time for A} = \frac{1 \text{ job}}{\frac{1}{10} \text{ jobs/min}} = 10 \text{ minutes} \]
The time taken by Typist B to complete the job alone is:
\[ \text{Time for B} = \frac{1 \text{ job}}{\frac{1}{15} \text{ jobs/min}} = 15 \text{ minutes} \]

### Step 8: Identify the slower typist.
Since Typist B takes longer (15 minutes) than Typist A (10 minutes), the slower typist is Typist B.

### Final Answer:
It would take the slower typist (Typist B) **15 minutes** to complete the typing job working alone.

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