Time and Work - Key Formulas and Examples - Part One

Question

Effectively managing time and calculating work done within a specific period is an invaluable skill in a variety of fields, from professional project management to competitive exam preparation. Understanding how to allocate time and estimate task completion can significantly enhance productivity. In this first installment of our "Time and Work" series, we will break down the fundamental formulas and walk you through practical examples to help you fully comprehend the key concepts.

Whether you're a student gearing up for exams or someone seeking to sharpen your time management abilities, this guide offers clear explanations and real-world applications to help you confidently solve time and work-related problems.

To solve these types of problems, the following principles are essential:

1. Work Done in a Day: If worker X can complete a task in n days, then in one day, X will finish $ \frac{1}{n} $ of the total work.

2. Time Taken to Complete Work: If X completes $ \frac{1}{n} $ of the task in a day, then X will take n days to finish the entire job.

3. Work Done in an Hour: If X completes $ \frac{1}{n} $ of the task per hour, then to complete the full task, X will take $ \frac{n}{m} $ hours, where m is the number of hours worked per day.

4. Work Efficiency and Ratio: If X works three times faster than Y, the ratio of work done by X and Y will be 3:2. The time taken by Y and Y will follow the inverse ratio, i.e., 1:3.

5. Multiple Workers Working Together: If X, Y, and Z can complete the task in T₁, T₂, and T₃ days respectively, and they have worked for D₁, D₂, and D₃ days, the work done by each is calculated as:  
   - Work done by X: $ \frac{D₁}{T₁} $
   - Work done by Y: $ \frac{D₂}{T₂} $
   - Work done by Z: $ \frac{D₃}{T₃} $

The total work done by X, Y, and Z together will be:  
   $ \frac{D₁}{T₁} + \frac{D₂}{T₂} + \frac{D₃}{T₃} $  
   If this sum equals 1, the task is fully completed.

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Useful Shortcuts

A. Shortcut for Two Workers Working Together

If A can complete a task in X days and B can finish the same task in Y days, then when both work together, they can complete the task in:

$
\frac{XY}{X + Y} \text{ days}
$

Explanation:

The formula comes from combining the work rates of A and B. The amount of work done per day by A is $ \frac{1}{X} $ and by B is $ \frac{1}{Y} $. Together, their combined work rate is:

$
\frac{1}{X} + \frac{1}{Y} = \frac{X + Y}{XY}
$

The reciprocal of this combined work rate gives the total time taken when they work together, which is:

$
\frac{XY}{X + Y}
$

Example:
A can complete a piece of work alone in 6 days.
B can complete the same work alone in 12 days.

Using the formula, the time taken by both working together is:

$
\frac{6 \times 12}{6 + 12} = \frac{72}{18} = 4 \text{ days}
$

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Thus, when A and B work together, they will complete the work in 4 days.

B. Shortcut for Three Workers Working Together (

If A, B, and C can each finish a task in X, Y, and Z days, respectively, then all three working together will complete the work in:

$
\frac{XYZ}{XY + YZ + ZX} \text{ days}
$

Explanation:
The total time for all three workers to finish the task together is calculated by summing their individual work rates and using the formula above.Example:
A can finish the work in 4 days.
B can finish the work in 6 days.
C can finish the work in 12 days.

The combined time taken when all three work together is:

$
\frac{4 \times 6 \times 12}{4 \times 6 + 6 \times 12 + 12 \times 4} = \frac{288}{24 + 72 + 48} = \frac{288}{144} = 2 \text{ days}
$

So, when A, B, and C work together, they will complete the task in exactly 2 days.

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C. Finding How Long Person B Takes to Complete a Task Alone

If A and B working together can complete a task in X days, and A working alone can complete the same task in Y days, then the time B would take to complete the task alone is:

$\frac{XY}{Y - X} \text{ days}$

Explanation:

When A and B work together, they share the work based on their individual work rates. The combined work rate for both is $ \frac{1}{X} $, meaning they complete $ \frac{1}{X} $ of the work per day together.

The individual rate of A working alone is $ \frac{1}{Y} $. Therefore, B's work rate can be found by subtracting A's work rate from their combined rate:

$\frac{1}{B} = \frac{1}{X} - \frac{1}{Y}$

Simplifying this equation gives the time taken by B alone to finish the task:

$B = \frac{XY}{Y - X}$

Example:

A and B together can complete a task in 6 days.
 A working alone can finish the task in 9 days.

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Using the formula, the time B takes to complete the task alone is:

$B = \frac{6 \times 9}{9 - 6} = \frac{54}{3} = 18 \text{ days}$

Thus, B alone can finish the task in 18 days.

D. Three Workers Working Together or in Pairs

When A, B, and C work together or in pairs, we can calculate how long it takes for them to finish the work using the following formulas:

i) A, B, and C working together: If A and B together can complete the work in X days, B and C in Y days, and C and A in Z days, then the time taken for A, B, and C to finish the work together is:
$\frac{2XYZ}{XY + YZ + ZX} \text{ days}$

ii) A alone finishing the work:
The time taken by A to finish the task alone is: 
$\frac{2XYZ}{XY + YZ - ZX} \text{ days}$

iii) B alone finishing the work: Similarly, the time taken by B to finish the task alone is: 
$\frac{2XYZ}{YZ + ZX - XY} \text{ days}$

iv) C alone finishing the work: The time taken by C to finish the task alone is:
$\frac{2XYZ}{ZX + XY - YZ} \text{ days}$

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Explanation:

1. A, B, and C working together: 
The formula $ \frac{2XYZ}{XY + YZ + ZX} $ comes from combining their pairwise work rates. When two people work together, their combined work rate is calculated, and when three people work together, we account for the combined work of all pairs. The factor of 2 accounts for the double counting of each pair&rsquo;s contribution.

2. A working alone:
To find how long A will take to complete the work alone, we adjust the formula for pairwise work by excluding C's contribution. Hence, the subtraction in the formula $ \frac{2XYZ}{XY + YZ - ZX} $.

The same logic applies for calculating the time taken by B and C to complete the work alone, but we modify the terms based on their pairwise contributions.

Example:

Let&rsquo;s assume the following:
A and B can finish a job in 6 days.
B and C can finish the same job in 12 days.
C and A can finish the job in 8 days.

i) Time taken by A, B, and C working together:
Using the formula for three workers working together:
$\frac{2 \times 6 \times 12 \times 8}{6 \times 12 + 12 \times 8 + 8 \times 6} = \frac{1152}{72 + 96 + 48} = \frac{1152}{216} = 5.33 \text{ days}$

Thus, A, B, and C together will finish the work in approximately 5.33 days.

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ii) Time taken by A working alone:
Using the formula for A working alone:
$\frac{2 \times 6 \times 12 \times 8}{6 \times 12 + 12 \times 8 - 8 \times 6} = \frac{1152}{72 + 96 - 48} = \frac{1152}{120} = 9.6 \text{ days}$

So, A alone will complete the work in approximately 9.6 days.

iii) Time taken by B working alone:
Using the formula for B working alone:
$\frac{2 \times 6 \times 12 \times 8}{12 \times 8 + 8 \times 6 - 6 \times 12} = \frac{1152}{96 + 48 - 72} = \frac{1152}{72} = 16 \text{ days}$

Thus, B alone will finish the work in 16 days.

iv) Time taken by C working alone:
Using the formula for C working alone:
$\frac{2 \times 6 \times 12 \times 8}{8 \times 6 + 6 \times 12 - 12 \times 8} = \frac{1152}{48 + 72 - 96} = \frac{1152}{24} = 48 \text{ days}$

Thus, C alone will finish the work in 48 days.

E. Efficiency and Work Time Relationship

i) Time taken by A and B working together when B is k times more efficient than A
If A can finish a task in x days and B is k times more efficient than A, then the time taken for both A and B working together to finish the task is:
$\frac{x}{1 + k}$

Explanation:
If B is k times more efficient than A, then B's work rate is k times that of A. Therefore, their combined work rate is the sum of A's work rate and B's work rate. Using this, we calculate the time taken for both to complete the work together.

Example:

A can finish a task in 10 days.
B is 2 times more efficient than A.

The time taken by A and B together is:
$\frac{10}{1 + 2} = \frac{10}{3} \approx 3.33 \text{ days}$

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So, A and B together will finish the task in approximately 3.33 days.

ii) Time taken by A and B working alone if B is k times more efficient than A

a) Time taken by A alone:
If A and B working together can finish a task in x days, and B is k times more efficient than A, the time taken by A alone to finish the task is:
$A = x(1 + k)$

b) Time taken by B alone:
Since B is k times more efficient than A, the time taken by B alone to finish the task is:
$B = \frac{x(1 + k)}{k}$

Explanation:
When A and B work together, their combined rate completes the task in x days. A's rate contributes $ \frac{1}{A} $, and B's rate contributes $ \frac{k}{A} $. By rearranging these relationships, we can determine the individual times for A and B to finish the task alone.

Example:

A and B together can finish the task in 4 days.
 B is 3 times more efficient than A.

Time taken by A alone:
$A = 4(1 + 3) = 4 \times 4 = 16 \text{ days}$

So, A alone will finish the task in 16 days.

Time taken by B alone:
$B = \frac{4(1 + 3)}{3} = \frac{4 \times 4}{3} = \frac{16}{3} \approx 5.33 \text{ days}$

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So, B alone will finish the task in approximately 5.33 days.

When B is k times more efficient than A, we use the formulas above to calculate the time taken by A and B working together or individually, depending on the situation.

In this first part of our *Time and Work series, we've introduced key formulas and examples to help you understand how individuals and teams can collaborate to complete tasks efficiently. We've explored various scenarios, including workers with different efficiencies, time taken for groups to finish tasks, and methods to calculate individual contributions when working together.

However, this is just the beginning! In Part Two, we'll dive deeper into more complex scenarios, such as handling varying work rates, dealing with breaks, and understanding how to allocate resources effectively. Stay tuned for more advanced tips to master time and work problems!