Dhiru can dig `(1)/(a)` of a field in 20 hours. What fraction of the same field can Kaku dig in 20 hours If the two of them can dig the field in 60 hours, working together at their respective rates ?

Options

A. `((a-3))/(a)`
B. `(1)/(3a)`
C. `((a-3))/(a)`
D. `((a-3))/(3a)`

Answer

To solve the problem, we need to determine how much of the field Kaku can dig in 20 hours, given the rates of both Dhiru and Kaku.

### Step 1: Determine Dhiru's Rate
Dhiru can dig \( \frac{1}{a} \) of the field in 20 hours. Therefore, in 1 hour, Dhiru can dig:
\[ \text{Dhiru's rate} = \frac{1/a}{20} = \frac{1}{20a} \]

### Step 2: Determine the Combined Rate of Dhiru and Kaku
When Dhiru and Kaku work together, they can dig the entire field in 60 hours. Thus, their combined rate is:
\[ \text{Combined rate} = \frac{1}{60} \]

### Step 3: Set Up the Equation for Kaku's Rate
Let Kaku's rate be \( k \). Therefore, we can write the equation for their combined rates as:
\[ \text{Dhiru's rate} + \text{Kaku's rate} = \text{Combined rate} \]
Substituting the known values:
\[ \frac{1}{20a} + k = \frac{1}{60} \]

### Step 4: Solve for Kaku's Rate
To find Kaku's rate \( k \), we rearrange the equation:
\[ k = \frac{1}{60} - \frac{1}{20a} \]

Now, we need a common denominator to subtract these fractions. The least common multiple of 60 and \( 20a \) is \( 60a \). Thus, we can rewrite the fractions:
\[ k = \frac{a}{60a} - \frac{3}{60a} = \frac{a - 3}{60a} \]

### Step 5: Determine How Much Kaku Can Dig in 20 Hours
Now that we have Kaku's rate, we can find out how much of the field Kaku can dig in 20 hours:
\[ \text{Amount Kaku can dig in 20 hours} = k \times 20 = \left(\frac{a - 3}{60a}\right) \times 20 = \frac{20(a - 3)}{60a} = \frac{a - 3}{3a} \]

### Final Answer
Thus, the fraction of the field that Kaku can dig in 20 hours is:
\[ \frac{a - 3}{3a} \]