Understanding Work and Time Problems

Work and time problems involve calculating how long it takes individuals or groups to complete a task, based on their work rates. A key concept is the 'work rate', which is the amount of work done per unit of time (e.g., per day).

If a person can complete a work in \(n\) days, their work rate is \( \frac{1}{n} \) of the work per day. When multiple people work together, their individual work rates are added to find their combined work rate.

Calculating Combined and Individual Work Rates

Let the total work be represented as 1 unit.

• A and B together can do the work in 10 days.
• This means their combined work rate is \( \frac{1}{10} \) of the work per day.

A alone can do the work in 15 days.

• This means A's individual work rate is \( \frac{1}{15} \) of the work per day.

Finding B's Work Rate

The combined work rate of A and B is the sum of A's work rate and B's work rate.

Combined Work Rate \( = \) A's Work Rate \( + \) B's Work Rate

We know the combined work rate and A's work rate. We can find B's work rate by subtracting A's work rate from the combined work rate:

B's Work Rate \( = \) Combined Work Rate \( - \) A's Work Rate

Let's calculate the numerical value:

\( \text{B's Work Rate} = \frac{1}{10} - \frac{1}{15} \)

To subtract these fractions, we find a common denominator, which is 30.

\( \frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30} \)

\( \frac{1}{15} = \frac{1 \times 2}{15 \times 2} = \frac{2}{30} \)

Now subtract:

\( \text{B's Work Rate} = \frac{3}{30} - \frac{2}{30} = \frac{3 - 2}{30} = \frac{1}{30} \)

So, B's work rate is \( \frac{1}{30} \) of the work per day.

Determining Time Taken by B Alone

If B's work rate is \( \frac{1}{30} \) of the work per day, it means B can complete \( \frac{1}{30} \) of the work in 1 day.

To find the total time B takes to complete the entire work (1 unit), we take the reciprocal of B's work rate:

\( \text{Time taken by B} = \frac{1}{\text{B's Work Rate}} \)

\( \text{Time taken by B} = \frac{1}{\frac{1}{30}} = 1 \times \frac{30}{1} = 30 \text{ days} \)

Therefore, B will take 30 days to do the same work alone.

Summary Steps to Solve Work and Time Problems

1. Identify the given times for individuals or groups working together.
2. Calculate the work rate for each given scenario (Work Rate = 1 / Time).
3. Use the relationship between combined and individual work rates (Combined Rate = Sum of Individual Rates) to find the unknown work rate.
4. Calculate the time taken by the unknown individual or group by taking the reciprocal of their work rate (Time = 1 / Work Rate).

Revision Table: Key Concepts in Work and Time

Additional Information: Work and Time Variations

Work and time problems can appear in many forms. Some variations include:

• Multiple Workers: Problems with more than two people working together.
• Changing Workers: Problems where some workers leave or join after a few days.
• Efficiency Ratios: Problems where the relative efficiency of workers is given instead of direct times.
• Work Done in Parts: Problems where a fraction of work is completed by different individuals or groups.

In all these variations, the fundamental principle remains: calculate the work rate and use it to find the time or vice versa.