Question

A completes \( \frac{7}{10} \) of a work in 15 days and then he completes the remaining work with the help of B in 5 days. In how many days can A and B together complete the entire work?

• A. 13 \( \frac{1}{3} \)
• B. 15 \( \frac{3}{4} \)
• C. 14 \( \frac{1}{4} \)
• D. 16 \( \frac{2}{3} \)

Hint:

When working with rates, combine the rates of individuals working together to find the total work rate, and then calculate the time needed to complete the task.

Answer 1

The Correct Option is A

Step 1: A's Work Rate.
A completes \( \frac{7}{10} \) of the work in 15 days. A's daily work rate is calculated as:\[\text{Work rate of A} = \frac{7}{10} \div 15 = \frac{7}{150}\]

Step 2: B's Work Rate.
The remaining work is \( 1 - \frac{7}{10} = \frac{3}{10} \). This remaining work is completed by A and B together in 5 days. Their combined daily work rate is:\[\text{Work rate of A and B} = \frac{3}{10} \div 5 = \frac{3}{50}\]To find B's work rate, subtract A's work rate from their combined rate:\[\text{Work rate of B} = \frac{3}{50} - \frac{7}{150} = \frac{9}{150} - \frac{7}{150} = \frac{2}{150} = \frac{1}{75}\]

Step 3: Total Time to Complete the Work.
The combined work rate of A and B is the sum of their individual rates:\[\text{Work rate of A and B together} = \frac{7}{150} + \frac{1}{75} = \frac{7}{150} + \frac{2}{150} = \frac{9}{150} = \frac{3}{50}\]The total time required to complete the entire work is the reciprocal of their combined work rate:\[\text{Time} = \frac{1}{\text{Work rate of A and B together}} = \frac{50}{3} = 13 \frac{1}{3} \, \text{days}\]

Final Answer: \[ \boxed{13 \frac{1}{3} \, \text{days}} \]