D can do a work in 18 days and E can do the same work in half that time .How many days will they take to finish the work ,doing it together ?

Options

A. A) 5
B. B) 4
C. C) 7
D. D) 6

Answer

To solve the problem of how many days D and E will take to finish the work together, we can follow these steps:

### Step 1: Determine the work rates of D and E
- D can complete the work in 18 days. Therefore, D's work rate is:
\[ \text{Work rate of D} = \frac{1 \text{ work}}{18 \text{ days}} = \frac{1}{18} \text{ work/day} \]

- E can complete the same work in half the time of D. Since D takes 18 days, E takes:
\[ \text{Time taken by E} = \frac{18}{2} = 9 \text{ days} \]
Thus, E's work rate is:
\[ \text{Work rate of E} = \frac{1 \text{ work}}{9 \text{ days}} = \frac{1}{9} \text{ work/day} \]

### Step 2: Combine the work rates
- To find the combined work rate when D and E work together, we add their individual work rates:
\[ \text{Combined work rate} = \text{Work rate of D} + \text{Work rate of E} = \frac{1}{18} + \frac{1}{9} \]

- To add these fractions, we need a common denominator. The least common multiple of 18 and 9 is 18:
\[ \frac{1}{9} = \frac{2}{18} \]
Therefore:
\[ \text{Combined work rate} = \frac{1}{18} + \frac{2}{18} = \frac{3}{18} = \frac{1}{6} \text{ work/day} \]

### Step 3: Calculate the total time to complete the work together
- If D and E together can complete \(\frac{1}{6}\) of the work in one day, then the total time taken to complete 1 whole work is:
\[ \text{Total time} = \frac{1 \text{ work}}{\text{Combined work rate}} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]

### Final Answer
Thus, D and E together will take **6 days** to finish the work.

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