Efficiency of Asha is `25%` more than Usha and Usha takes 25 days to complete a piece of work. Asha started a work alone and then usha joined her 5 days before actual completion of the work. For how many days Asha worked alone?

Options

A. 9
B. 11
C. 10
D. 15

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Determine Usha's Efficiency
Usha takes 25 days to complete the work. Therefore, her efficiency can be calculated as:
\[ \text{Efficiency of Usha} = \frac{1 \text{ work}}{25 \text{ days}} = \frac{1}{25} \text{ work/day} \]

**Hint:** Efficiency is the amount of work done in a unit of time.

### Step 2: Calculate Asha's Efficiency
Asha's efficiency is 25% more than Usha's. To find Asha's efficiency:
\[ \text{Efficiency of Asha} = \text{Efficiency of Usha} + 25\% \text{ of Efficiency of Usha} \]
\[ = \frac{1}{25} + 0.25 \times \frac{1}{25} \]
\[ = \frac{1}{25} + \frac{1}{100} \]
To add these fractions, we need a common denominator (100):
\[ = \frac{4}{100} + \frac{1}{100} = \frac{5}{100} = \frac{1}{20} \text{ work/day} \]

**Hint:** To find a percentage increase, multiply the original value by the percentage and add it to the original value.

### Step 3: Calculate Total Work
The total work can be calculated using Usha's efficiency:
\[ \text{Total Work} = \text{Efficiency of Usha} \times \text{Time taken by Usha} \]
\[ = \frac{1}{25} \times 25 = 1 \text{ work unit} \]

**Hint:** Total work is the product of efficiency and time.

### Step 4: Set Up the Equation for Asha's Work
Let \( x \) be the number of days Asha worked alone. Usha joins Asha 5 days before the work is completed. Therefore, Asha and Usha worked together for 5 days.

The total work done can be expressed as:
\[ \text{Work done by Asha alone} + \text{Work done by Asha and Usha together} = \text{Total Work} \]
\[ 5k \cdot x + (5k + 4k) \cdot 5 = 100k \]

**Hint:** When calculating total work, consider the work done separately and together.

### Step 5: Substitute and Solve for \( x \)
Substituting the values:
\[ 5x + 9 \cdot 5 = 100 \]
\[ 5x + 45 = 100 \]
Subtract 45 from both sides:
\[ 5x = 55 \]
Now, divide by 5:
\[ x = 11 \]

**Hint:** To isolate \( x \), perform inverse operations (subtract, then divide).

### Conclusion
Asha worked alone for **11 days**.

**Final Answer:** Asha worked alone for **11 days**.