Time & Work / Work Rates

This topic is designed to cover most patterns students encounter in aptitude tests, without the use of a calculator. The real difficulty in this topic is recognizing what stays constant, what changes, and whether rates should be added, subtracted, or split across time periods.

Almost every difficult question in this topic is a disguised application of the following three relationships:

However, there are still certain rules that apply to solve the work rates question successfully.

1. Convert Time Into Rate

If a person can complete a whole job in T units of time, then the amount of work they complete in one unit of time is their work rate:

So if A completes a job in 8 days:

This does not mean A does 8 units per day. It means A completes one-eighth of the entire job each day.

Lets’s do an example:

A can complete a project in 12 days. How much of the project does A complete in 5 days?

First get the daily rate:

Then the work in 5 days:

So if the remaining project rate is asked, one can simply:

Another example to get a deeper insight of this rule:

A can complete a job in 12 days. B takes 50% more time than A to complete the same job. If both work together for 4 days, what fraction of the job remains unfinished?

A takes 12 days, so A’s work rate is:

B takes 50% more time than A:

Therefore, B’s work rate is:

Their combined rate is:

Using 36 as the common denominator:

So together they complete \(\frac{5}{36}\) of the job each day.

In 4 days, they complete:

Therefore, the fraction remaining is:

Point to be noted here is: The 50% increase applies to B’s time, not B’s work rate. If someone takes 50% more time, you must first calculate the new completion time and then convert it into a rate.

2. Adding Work Rates

When two people work simultaneously toward the same objective, their rates are added.

If A takes $a$ days and B takes $b$ days:

Therefore:

Let’s do an example:

A can complete a job 6 days faster than B. Working together, they can complete the job in 4 days. How many days would A take to complete the job alone?

Suppose B takes x days to complete the job.

Since A is 6 days faster, A takes:

Their individual work rates are therefore:

Because they work simultaneously toward the same job, add their rates:

The denominators are:

A common denominator that contains all three is:

So we multiply every term on both sides by 4x(x-6):

Now cancel matching factors:

Now factor:

Since, \(x=2\) isn’t possible as otherwise A would then take \(2-6=-4 days\). Therefore:

B takes 12 days, so A takes:

Whena question says workers are working on the same job at the same time, don’t add their completion times, convert each person’s time into a rate first and then add the rates.

3. Finding One Worker’s Time From Combined Time

Let’s understand this with an example:

A and B together complete a job in 6 days. A alone can complete it in 10 days. How long would B take alone?

Combined rate as given:

A’s rate:

Therefore B’s rate:

B’s rate:

As a result the core rule is:

If

Then

4. Efficiency and Time Have an Inverse Relationship

For the same amount of work, efficiency and completion time move in opposite directions. A more efficient worker takes less time, while a less efficient worker takes more time.

Therefore:

So if the efficiency ratio is:

then the time ratio is reversed:

Let’s do an example:

A is 25% more efficient than B. If B takes 15 days to complete a job, how many days will A take?

If B’s efficiency is 100, A’s efficiency is:

So their efficiency ratio is:

Because time is inversely proportional to efficiency, reverse the ratio:

Since B takes 15 days:

Note that if A is 25% more efficient, it does not mean A takes 25% less time. Efficiency and time have an inverse, rather than a direct, relationship.

Similarly,

A is $50\%$ more efficient than B. B takes 18 days. How long does A take?

We can use assumptions to make things easier.

If B’s efficiency = 2 units,

A’s efficiency = 3 units.

So:

Therefore:

Thus,

An important point to remember: being 50% more efficient does not mean taking 50% less time. Since efficiency and time are inversely related, percentage changes in efficiency do not translate directly into equal percentage changes in completion time.

In the same, a percentage change in the efficiency also makes up an excellent aptitude test question. For example:

A worker normally takes 15 days. His efficiency increases by 25%. How long will he now take?

First calculate the new efficiency:

Therefore time becomes:

Another important aptitude-test patterns include:

5. When another worker joins

The strategy to solve such questions is to always break them into phases:

For example:

A can complete a job in 12 days. A works alone for 4 days, after which B joins. B alone could complete the job in 8 days. How much longer will they need?

A’s work in 4 days:

remaining:

combined rate:

required time:

So they need:

6. Alternate day working

Such questions become tricky because students calculate an average rate too early, slightly falling into the wrong choice among the options.

For example:

A can complete a job in 12 days and B in 18 days. They work on alternate days, starting with A. How much work is completed after 6 days?

A works 3 days:

B works 3 days:

Total:

7. Ratios and work rates

As mentioned earlier, some topics form the foundation for several other concepts and are closely interconnected with them. Ratios is one such fundamental topic. When combined with work rates, ratios can be used to determine how work, wages, and earnings should be divided among workers. (For more on ratios click here)

For example:

A is twice as efficient as B. A works for 6 days while B works for 9 days. They receive Rs. 7,000. How should it be divided?

Let B’s efficiency = 1.

Then A’s = 2.

Work contributions:

Ratio:

Total parts = 7.

Therefore:

Practice questions