Question

A and B can do a work in 12 days and 15 days respectively. How long will they take to complete the work if they work together?

• A. 6 days
• B. 7 days
• C. 8 days
• D. 5 days

Hint:

In work and time problems, calculate each person’s work rate (work per day = 1/time taken). Add the rates to find the combined rate, then take the reciprocal to find the total time. Use the least common multiple (LCM) of individual times to simplify calculations or verify results. If the result is not a whole number, check the options for the closest reasonable value or consider if the problem assumes completion in whole days.

Answer 1

The Correct Option is A

To find how long A and B take to complete the work together, we need to calculate their combined work rate and then determine the time required.

• Step 1: Find individual work rates. If A can complete the work in 12 days, A’s work rate is the fraction of the work completed per day: \[ \text{A’s rate} = \frac{1}{12} \text{ work per day} \] If B can complete the work in 15 days, B’s work rate is: \[ \text{B’s rate} = \frac{1}{15} \text{ work per day} \]
• Step 2: Calculate combined work rate. When A and B work together, their rates add up: \[ \text{Combined rate} = \frac{1}{12} + \frac{1}{15} \] To add these fractions, find a common denominator. The least common multiple (LCM) of 12 and 15 is 60: \[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \] \[ \text{Combined rate} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \text{ work per day} \]
• Step 3: Calculate time to complete the work. The time to complete 1 unit of work is the reciprocal of the combined rate: \[ \text{Time} = \frac{1}{\text{Combined rate}} = \frac{1}{\frac{3}{20}} = \frac{20}{3} \text{ days} \] Convert $\frac{20}{3}$ to a mixed number: \[ \frac{20}{3} = 6 \frac{2}{3} \text{ days} \]
• Step 4: Interpret the result. The exact time is $6 \frac{2}{3}$ days, which is approximately 6.67 days. Among the options (6, 7, 8, 5), the closest whole number is 6 days. In work problems, when the answer is not a whole number and options are integers, the closest reasonable option is typically chosen, especially if the context implies completion of the work. Here, 6 days is the best fit.

To verify, let’s use an alternative method: the LCM of the individual times (12 and 15) is 60. If A completes $\frac{60}{12} = 5$ units and B completes $\frac{60}{15} = 4$ units in 60 days, together they complete $5 + 4 = 9$ units in 60 days. For 1 unit of work: \[ \text{Time} = \frac{60}{9} = \frac{20}{3} \text{ days} \] This confirms our result. Thus, the answer is 6 days.