Understanding Work and Time Problems

Work and time problems are common in quantitative aptitude. The basic concept is that the amount of work done is proportional to the time taken and the rate at which the work is done. If a person can complete a work in 'n' days, their rate of work is $\frac{1}{n}$ of the work per day.

Calculating Individual Work Rates

We are given the time taken by the mother, daughter, and son to complete the work individually:

• Mother takes 3 days.
• Daughter takes 4 days.
• Son takes 5 days.

Using the concept of work rate, we can find the amount of work each person can do in one day:

• Mother's work rate per day = $\frac{1}{\text{Days taken by Mother}} = \frac{1}{3}$ of the work.
• Daughter's work rate per day = $\frac{1}{\text{Days taken by Daughter}} = \frac{1}{4}$ of the work.
• Son's work rate per day = $\frac{1}{\text{Days taken by Son}} = \frac{1}{5}$ of the work.

Calculating Combined Work Rate

When they work together, their individual work rates add up to form a combined work rate. The combined work rate per day is the sum of their individual work rates:

Combined work rate = (Mother's rate) + (Daughter's rate) + (Son's rate)

Combined work rate = $\frac{1}{3} + \frac{1}{4} + \frac{1}{5}$

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 3, 4, and 5 is 60.

Combined work rate = $\frac{1 \times 20}{3 \times 20} + \frac{1 \times 15}{4 \times 15} + \frac{1 \times 12}{5 \times 12}$

Combined work rate = $\frac{20}{60} + \frac{15}{60} + \frac{12}{60}$

Combined work rate = $\frac{20 + 15 + 12}{60} = \frac{47}{60}$ of the work per day.

Calculating Time Taken Together

The total time taken to complete the work when working together is the reciprocal of the combined work rate.

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

Time taken together = $\frac{1}{\frac{47}{60}}$ days

Time taken together = $\frac{60}{47}$ days.

Converting to Mixed Number

The time taken is $\frac{60}{47}$ days. We can convert this improper fraction into a mixed number. Divide 60 by 47:

$60 \div 47 = 1$ with a remainder of $60 - (47 \times 1) = 13$.

So, $\frac{60}{47}$ days is equal to $1 \frac{13}{47}$ days.

Thus, if they work together, they will take $1 \frac{13}{47}$ days to complete the work.

Revision Table: Work and Time Concepts

Additional Information on Work and Time Problems

Work and time problems often involve scenarios where individuals work at different speeds, work for different durations, or share tasks. Understanding the concept of work rate is crucial for solving these problems. The total work is often considered as '1 unit' or 'the whole work'. If a fraction of work is done, the remaining work can be calculated by subtracting the done portion from 1.

Sometimes, problems might involve negative work (like a pipe emptying a tank while others fill it). In such cases, the rate of negative work is subtracted from the positive work rates to find the net rate.

Always ensure that the units of time (days, hours, minutes) are consistent throughout the calculation.