Sanjeev can prepare some dishes in 4 hours while Nisha can prepare same in 5 hours. Madhulika takes 20% less time than Sanjeev & Nisha take together to prepare same dishes. What is the efficiency ratio of Sanjeev & Madhulika?

Options

A. A)`4:9`
B. B)`16:9`
C. C)`16:45`
D. D)`4:5`

Answer

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

### Step 1: Determine the time taken by Sanjeev and Nisha
- Sanjeev takes 4 hours to prepare the dishes.
- Nisha takes 5 hours to prepare the same dishes.

### Step 2: Calculate the combined time taken by Sanjeev and Nisha
To find the time taken when both work together, we use the formula:
\[ \text{Time}_{\text{together}} = \frac{\text{Time}_1 \times \text{Time}_2}{\text{Time}_1 + \text{Time}_2} \]
Substituting the values:
\[ \text{Time}_{\text{together}} = \frac{4 \times 5}{4 + 5} = \frac{20}{9} \text{ hours} \]

### Step 3: Calculate Madhulika's time
Madhulika takes 20% less time than the combined time of Sanjeev and Nisha. First, we need to find 20% of \(\frac{20}{9}\):
\[ 20\% \text{ of } \frac{20}{9} = \frac{20}{9} \times \frac{20}{100} = \frac{4}{9} \]
Now, subtract this from the combined time:
\[ \text{Madhulika's time} = \frac{20}{9} - \frac{4}{9} = \frac{16}{9} \text{ hours} \]

### Step 4: Calculate the efficiency ratio of Sanjeev and Madhulika
Efficiency is inversely proportional to the time taken. Therefore, the efficiency ratio can be calculated as:
\[ \text{Efficiency ratio} = \frac{\text{Time of Madhulika}}{\text{Time of Sanjeev}} = \frac{\frac{16}{9}}{4} \]
This simplifies to:
\[ \text{Efficiency ratio} = \frac{16}{9} \times \frac{1}{4} = \frac{16}{36} = \frac{4}{9} \]

### Final Answer
The efficiency ratio of Sanjeev and Madhulika is:
\[ \text{Efficiency Ratio} = 4:9 \]

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