# Work and Time

**Work and Time**

In quantitative ability, work and time problems involve calculating the amount of work done or time taken to complete a task when multiple individuals or entities are working together or at different rates. These types of problems are commonly encountered in mathematics and are often found in competitive exams, aptitude tests, and real-life situations involving work distribution and efficiency.

There are three main types of work and time problems:

**Work Done Together:** In this type of problem, two or more entities work together to complete a task. The problem usually provides information about the individual work rates of each entity. The task is to determine the time taken to complete the task when they work together.

Example: If person A can complete a job in 6 days, and person B can complete it in 8 days, how long will it take if they work together to finish the job?

**Work Done Sequentially:** In this type of problem, two or more entities work on a task one after the other. The problem will provide information about the individual work rates of each entity and sometimes the amount of work already completed. The task is to find the time taken to complete the entire task.

Example: If person X can build a wall in 10 days and person Y can build the same wall in 15 days, and they start working together but Y joins after the first half of the work is done, how long will it take to complete the wall?

**Efficiency and Ratio:** These problems involve entities with different work rates or efficiencies, and the ratio of their work rates is given. The problem can ask to find the individual work rates or time taken for each entity to complete the task.

Example: If machine A can produce 500 widgets in 10 hours and machine B can produce 600 widgets in 12 hours, find the ratio of their efficiencies, and how long will it take for them to produce 3000 widgets if they work together?

To solve work and time problems, you need to understand the concept of "work" as the amount of work done per unit of time and use formulas that relate work, time, and rate (work rate = work done / time taken). You may also need to apply concepts of fractions, ratios, and simultaneous equations, depending on the complexity of the problem.

Examples:

Example 1: Work Done Sequentially

Tom can build a wall in 10 days. After the first half of the wall is built, Jerry joins Tom to complete the rest of the wall. Jerry can build the same wall in 15 days. How long will it take to complete the entire wall?

Solution:

Let's assume the total work to build the wall is represented by "W."

Tom's work rate = 1 wall / 10 days = 1/10 W/day.

Jerry's work rate = 1 wall / 15 days = 1/15 W/day.

Tom completes half of the work, which is (1/2) * W, in 5 days (half of his time).

Now, the remaining work to be done is (1/2) * W.

When Tom and Jerry work together, their combined work rate is the sum of their individual work rates.

Combined work rate = 1/10 W/day + 1/15 W/day = (3 + 2) / 30 W/day = 5/30 W/day = 1/6 W/day

To find the time taken to complete the remaining work when they work together, we use the formula:

Time = Remaining work / Combined work rate

Time = (1/2) * W / (1/6) = (1/2) * W * (6/1) = 3W

Therefore, it will take Tom and Jerry three days to complete the entire wall.

Example 2: Work Done Together (with Fractions)

A and B can build a wall in 12 days and 18 days, respectively. They start working together, but after 4 days, C joins them, and they complete the wall in a total of 8 days. How long would it take for C alone to build the wall?

Solution:

Let's assume the total work to build the wall is represented by "W."

A's work rate = 1 wall / 12 days = 1/12 W/day

B's work rate = 1 wall / 18 days = 1/18 W/day

Combined work rate of A and B = 1/12 W/day + 1/18 W/day = (3 + 2) / 36 W/day = 5/36 W/day

Let "x" represent the number of days it takes C to complete the wall on their own.

C's work rate = 1 wall / x days = 1/x W/day

In the 4 days when A and B worked together, they completed (4 days) * (Combined work rate) = 4 * (5/36) = 5/9 of the work.

So, the remaining work to be done after 4 days is (1 - 5/9) = 4/9 of the total work.

Now, when A, B, and C work together, their combined work rate is the sum of their individual work rates.

Combined work rates of A, B, and C = 5/36 W/day + 1/x W/day

According to the problem, they completed the remaining 4/9 of the work in 8 days, so:

(8 days) * (Combined Work Rate) = 4/9

8 * (5/36 + 1/x) = 4/9

Now, solve for "x":

40/36 + 8/x = 4/9

8/x = 4/9 - 40/36

8/x = (4 - 40)/36

8/x = -36/36

x = -8

The negative value of "x" is not meaningful in this context, so there seems to be an error in the problem statement or calculations. Please double-check the problem and the values provided.

Example 3: Work Done Sequentially (with Percentages)

A can complete a task in 5 days, and B can complete the same task in 8 days. A starts the task but gives up after working for 2 days. B takes over and completes the remaining tasks. By what percentage did B's efficiency change compared to his normal efficiency?

Solution:

Suppose that the total work to be done is represented by "W."

A's work rate = 1 task / 5 days = 1/5 W/day

B's work rate = 1 task / 8 days = 1/8 W/day

In 2 days, A completed 2 * (1/5) = 2/5 of the work. So, the remaining work for B is (1 - 2/5) = 3/5 of the total work.

Now, B completes the remaining 3/5 of the work in 6 days (8 days - 2 days).

B's efficiency to complete the remaining work

B's work rate = Remaining work / Time taken

B's work rate = (3/5) / 6 = 3/30 = 1/10 W/day

Now, we need to find the percentage change in B's efficiency compared to his normal efficiency (1/8 W/day).

Percentage change = [(New Value - Old Value) / Old Value] * 100

Percentage change = [(1/10 - 1/8) / (1/8)] * 100

Percentage change = [(4/40 - 5/40) / (5/40)] * 100

Percentage change = [-1/40] * 100

Percentage change = -2.5%

B's efficiency decreased by 2.5% when he took over from A to complete the task.

Practise is crucial to getting better at solving work and time problems, and with time, you will become more proficient in handling these types of quantitative ability questions.