A alone can do a work in 15 days. A is 20% less efficient than B. In how many days can B alone do the work?

Options

A. a. 12
B. b. 18
C. c. 20
D. d. 24

Answer

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

### Step 1: Determine A's efficiency
A can complete the work in 15 days. Therefore, A's efficiency can be calculated as:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{15 \text{ days}} = \frac{1}{15} \text{ work per day} \]

**Hint:** Efficiency is the amount of work done in one day.

### Step 2: Relate A's efficiency to B's efficiency
We know that A is 20% less efficient than B. To express this in terms of fractions:
\[ 20\% = \frac{20}{100} = \frac{1}{5} \]
If we denote B's efficiency as \( E_B \), then A's efficiency \( E_A \) can be expressed as:
\[ E_A = E_B - \frac{1}{5}E_B = \frac{4}{5}E_B \]

**Hint:** To find the relationship between two efficiencies, you can express one in terms of the other.

### Step 3: Set up the equation for efficiency
From Step 1, we know:
\[ E_A = \frac{1}{15} \]
Substituting the expression for \( E_A \) from Step 2:
\[ \frac{4}{5}E_B = \frac{1}{15} \]

**Hint:** Substitute known values into equations to find unknowns.

### Step 4: Solve for B's efficiency
To find \( E_B \), we can rearrange the equation:
\[ E_B = \frac{1}{15} \cdot \frac{5}{4} = \frac{5}{60} = \frac{1}{12} \text{ work per day} \]

**Hint:** When solving for a variable, isolate it on one side of the equation.

### Step 5: Calculate the total work
Total work can be calculated using A's efficiency and the number of days A takes:
\[ \text{Total Work} = E_A \times \text{Days} = \frac{1}{15} \times 15 = 1 \text{ work} \]

**Hint:** Total work remains constant regardless of who is doing it.

### Step 6: Find the time taken by B to complete the work
Now, we can find out how many days B will take to complete the same work:
\[ \text{Time taken by B} = \frac{\text{Total Work}}{E_B} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

**Hint:** To find time taken, divide total work by efficiency.

### Final Answer
B can complete the work alone in **12 days**.

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