If 'a' number of males can do a work in (2a-8) days while (a-8) number of males can do that work in (2a + 12) days, then find in how many days `(3)/(2)`a number of males can do `33(1)/(3)%` of that work?

Options

A. 12 days
B. `8(8)/(9)` days
C. `6(2)/(3)` days
D. 8 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the total work done
We know that the total work can be calculated using the formula:
\[ \text{Total Work} = \text{Number of Workers} \times \text{Days Taken} \]

From the first condition, we have:
- Number of males = \( a \)
- Days = \( 2a - 8 \)

Thus, the total work from the first condition is:
\[ \text{Total Work} = a \times (2a - 8) = 2a^2 - 8a \]

From the second condition, we have:
- Number of males = \( a - 8 \)
- Days = \( 2a + 12 \)

Thus, the total work from the second condition is:
\[ \text{Total Work} = (a - 8) \times (2a + 12) = 2a^2 + 12a - 16a - 96 = 2a^2 - 4a - 96 \]

### Step 2: Equate the two expressions for total work
Since both expressions represent the same total work, we can set them equal to each other:
\[ 2a^2 - 8a = 2a^2 - 4a - 96 \]

### Step 3: Simplify the equation
Cancelling \( 2a^2 \) from both sides gives:
\[ -8a = -4a - 96 \]

Now, rearranging the equation:
\[ -8a + 4a = -96 \]
\[ -4a = -96 \]
\[ a = 24 \]

### Step 4: Calculate the total work
Substituting \( a = 24 \) back into the total work formula:
\[ \text{Total Work} = a \times (2a - 8) = 24 \times (2 \times 24 - 8) \]
\[ = 24 \times (48 - 8) = 24 \times 40 = 960 \]

### Step 5: Determine the work needed
We need to find out how much work \( \frac{1}{3} \) of \( 33 \frac{1}{3}\% \) of the total work is:
\[ 33 \frac{1}{3}\% = \frac{33.33}{100} = \frac{1}{3} \]
So, \( \frac{1}{3} \) of the total work is:
\[ \text{Work Needed} = \frac{1}{3} \times 960 = 320 \]

### Step 6: Calculate the efficiency of \( \frac{3}{2} a \) males
The number of males is:
\[ \frac{3}{2} a = \frac{3}{2} \times 24 = 36 \]

### Step 7: Calculate the time taken to complete the work
The efficiency of one male is considered as 1 unit of work per day. Therefore, the total efficiency of 36 males is:
\[ \text{Total Efficiency} = 36 \]

Now, the time taken to complete 320 units of work is given by:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} = \frac{320}{36} = \frac{80}{9} \]

### Step 8: Convert to mixed fraction
To express \( \frac{80}{9} \) as a mixed fraction:
\[ \frac{80}{9} = 8 \frac{8}{9} \]

### Final Answer
Thus, the time taken by \( \frac{3}{2} a \) males to do \( 33 \frac{1}{3}\% \) of the work is:
**8 \( \frac{8}{9} \) days**