Abhishek starts to paint a fence on one day. On the second day, two more friend of Abhishek join him. On the third day 3 more friends of him join him and so on. If the fence is completely painted this way in exactly 20 days, then find the number of days in which 10 girls painting together can paint the fence completely, given that every girl can paint twice as fast as Abhishek and his friends(Boys)7(Assume that the friends of Abhishek are all boys).

Options

A. 20
B. 40
C. 45
D. 77

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the Painting Pattern
Abhishek starts painting on the first day. On each subsequent day, the number of friends joining him increases by one. Thus:
- Day 1: 1 person (Abhishek)
- Day 2: 1 + 2 = 3 people (Abhishek + 2 friends)
- Day 3: 1 + 2 + 3 = 6 people (Abhishek + 3 friends)
- ...
- Day n: 1 + 2 + 3 + ... + n = n(n + 1)/2 people

### Step 2: Calculate Total Work Done in 20 Days
The total number of people painting on the nth day is given by the formula for the sum of the first n natural numbers:
\[ S_n = \frac{n(n + 1)}{2} \]

For 20 days, the total work done (in terms of the number of people working) is:
\[ \text{Total Work} = S_{20} = \frac{20(20 + 1)}{2} = \frac{20 \times 21}{2} = 210 \text{ units of work} \]

### Step 3: Determine Work Rate of Abhishek and Friends
Let’s assume the work done by Abhishek and each of his friends in one day is 1 unit of work. Thus, the total work done by them over 20 days is 210 units.

### Step 4: Calculate the Work Rate of Girls
According to the problem, every girl can paint twice as fast as Abhishek and his friends. Therefore, if one boy can paint 1 unit of work in a day, then one girl can paint 2 units of work in a day.

### Step 5: Calculate Total Work Done by 10 Girls
If 1 girl can do 2 units of work in a day, then 10 girls can do:
\[ 10 \text{ girls} \times 2 \text{ units/girl} = 20 \text{ units of work per day} \]

### Step 6: Calculate the Number of Days for 10 Girls to Complete the Work
To find out how many days it will take for 10 girls to complete the 210 units of work:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Work done by 10 girls per day}} = \frac{210 \text{ units}}{20 \text{ units/day}} = 10.5 \text{ days} \]

### Final Answer
Thus, it will take 10 girls a total of **10.5 days** to paint the fence completely.

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