Question

Two men and seven boys can do a work in 14 days. Three men and eight boys can do the same work in 11 days. Further eight men and six boys can do three times the amount of this work in:

• A. 24 days
• B. 21 days
• C. 18 days
• D. 18 days

Hint:

Always convert each scenario into equations using daily work rates and solve for individual rates before finding combined rates.

Answer 1

The Correct Option is B

Step 1: Represent the work equation. Let 1 day's work of a man be \( m \) and of a boy be \( b \).
Two men and seven boys can do the work in 14 days, so: \((2m + 7b) \times 14 = 1 \ \text{(whole work)}\)
This simplifies to: \( 2m + 7b = \frac{1}{14} \).
Step 2: Second equation from second scenario. Three men and eight boys can do the work in 11 days: \((3m + 8b) \times 11 = 1\).
This gives: \( 3m + 8b = \frac{1}{11} \).
Step 3: Solve for \( m \) and \( b \). Multiply the first equation by 3: \( 6m + 21b = \frac{3}{14} \).
Multiply the second equation by 2: \( 6m + 16b = \frac{2}{11} \).
Subtract: \( (6m + 21b) - (6m + 16b) = \frac{3}{14} - \frac{2}{11} \).
This gives \( 5b = \frac{33 - 28}{154} = \frac{5}{154} \).
So \( b = \frac{1}{154} \).
Step 4: Find \( m \). From \( 2m + 7b = \frac{1}{14} \), substitute \( b \): \( 2m + 7 \times \frac{1}{154} = \frac{1}{14} \).
\( 2m + \frac{7}{154} = \frac{11}{154} \).
\( 2m = \frac{4}{154} $\Rightarrow$ m = \frac{2}{154} = \frac{1}{77} \).
Step 5: Work rate of eight men and six boys. One day's work = \( 8 \times \frac{1}{77} + 6 \times \frac{1}{154} \).
= \( \frac{8}{77} + \frac{6}{154} = \frac{16}{154} + \frac{6}{154} = \frac{22}{154} = \frac{11}{77} \).
Step 6: Time for three times the work. Three times the work means work = \( 3 \).
Time = \( \frac{\text{Total Work}}{\text{One day's work}} = \frac{3}{\frac{11}{77}} = 3 \times \frac{77}{11} = 21 \ \text{days} \).
Hence, the answer is \(\boxed{21 \ \text{days}}\).