Question

Ramesh and Ganesh can together complete a work in 16 days. After seven days of working together, Ramesh got sick and his efficiency fell by 30%. As a result, they completed the work in 17 days instead of 16 days. If Ganesh had worked alone after Ramesh got sick, in how many days would he have completed the remaining work?

• A. 13.5
• B. 11
• C. 12
• D. 14.5

Answer 1

The Correct Option is A

Step 1: Define Work Rates 

Let \( r \) be the rate of work of Ramesh, and \( g \) be the rate of work of Ganesh. Together, they complete the entire work in 16 days.

\[ (r + g) \times 16 = 1 \quad \Rightarrow \quad r + g = \frac{1}{16} \tag{1} \]

Step 2: Work Done in 7 Days

Together in 7 days:

\[ (r + g) \times 7 = \frac{7}{16} \]

Remaining work:

\[ 1 - \frac{7}{16} = \frac{9}{16} \tag{2} \]

Step 3: Next 10 Days with Reduced Ramesh Rate

In the next 10 days, Ramesh works at 70% efficiency, i.e. \( 0.7r \), and Ganesh works at full rate \( g \). Total work in 10 days:

\[ (0.7r + g) \times 10 = \frac{9}{16} \tag{3} \]

Step 4: Solve Equations

Use equation (1): \( g = \frac{1}{16} - r \)

Substitute into equation (3):

\[ 10(0.7r + g) = \frac{9}{16} \Rightarrow 7r + 10g = \frac{9}{16} \tag{4} \]

From equation (1): \( r + g = \frac{1}{16} \Rightarrow g = \frac{1}{16} - r \)

Substitute into (4):

\[ 7r + 10\left( \frac{1}{16} - r \right) = \frac{9}{16} \Rightarrow 7r + \frac{10}{16} - 10r = \frac{9}{16} \Rightarrow -3r = \frac{9 - 10}{16} = -\frac{1}{16} \Rightarrow r = \frac{1}{48} \]

Then, from equation (1):

\[ g = \frac{1}{16} - \frac{1}{48} = \frac{3 - 1}{48} = \frac{2}{48} = \frac{1}{24} \]

Step 5: Time for Ganesh to Complete Remaining Work Alone

From equation (2), remaining work is \( \frac{9}{16} \), and Ganesh's rate is \( \frac{1}{24} \):

\[ \text{Time} = \frac{\frac{9}{16}}{\frac{1}{24}} = \frac{9}{16} \times 24 = \frac{216}{16} = \boxed{13.5 \text{ days}} \]