If a person completes a work in 8 hours and another person completes the same work in 12 hours, how long will it take them to complete the work together?

Steps to Solve

1. Determine Individual Work Rates

Calculate the work rate for each individual. For the first person, who can complete the task in 8 hours, the work rate is:

$$ \text{Work rate of Person A} = \frac{1 \text{ task}}{8 \text{ hours}} = \frac{1}{8} \text{ tasks/hour} $$

For the second person, who can complete the task in 12 hours, the work rate is:

$$ \text{Work rate of Person B} = \frac{1 \text{ task}}{12 \text{ hours}} = \frac{1}{12} \text{ tasks/hour} $$

2. Combine the Work Rates

Add the work rates of both individuals to find their combined work rate:

$$ \text{Combined work rate} = \frac{1}{8} + \frac{1}{12} $$

To add these fractions, find a common denominator. The least common multiple of 8 and 12 is 24:

$$ \frac{1}{8} = \frac{3}{24} \quad \text{and} \quad \frac{1}{12} = \frac{2}{24} $$

So,

$$ \text{Combined work rate} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \text{ tasks/hour} $$

3. Calculate the Time to Complete the Task Together

To find the total time it takes for both individuals to complete 1 task together, we use the formula:

$$ \text{Time} = \frac{1 \text{ task}}{\text{Combined work rate}} $$

Substituting in the combined work rate:

$$ \text{Time} = \frac{1}{\frac{5}{24}} = \frac{24}{5} \text{ hours} $$

4. Convert into Hours and Minutes

Convert the fractional hours into minutes:

$$ \frac{24}{5} = 4.8 \text{ hours} = 4 \text{ hours} + 0.8 \times 60 \text{ minutes} = 4 \text{ hours} + 48 \text{ minutes} $$