A can do a piece of work in 10 days and B in 15 days. How long will they take together to finish it?

A can do a piece of work in 10 days and B in 15 days. How long will they take together to finish it?

To determine how long A and B will take to finish a piece of work together, we start by understanding their individual work rates.In this scenario, A can complete the work in 10 days, and B can complete the same work in 15 days. These individual times can be converted into work rates, which tell us how much of the work each person can complete in one day.

First, we calculate A's work rate. Since A can finish the work in 10 days, A's work rate is 1/10 of the work per day. Similarly, B's work rate is 1/15 of the work per day because B can complete the task in 15 days. These work rates are the fractions of the total work that each person completes in one day.

If Person A can complete the task in 10 days, then A's work rate is 1/10 of the task per day.

If Person B can complete the task in 15 days, then B's work rate is 1/15 of the task per day.

Next, to determine how long A and B will take to finish the work together, we need to find their combined work rate. This involves adding their individual work rates. The combined work rate is the sum of A's work rate (1/10) and B's work rate (1/15). To add these fractions, we find a common denominator. The common denominator for 10 and 15 is 30, so we convert the fractions: 1/10 becomes 3/30 and 1/15 becomes 2/30.

After converting, we add the fractions:

3/30 + 2/30 = 5/30

Simplifying this fraction gives us 1/6, which means that together, A and B can complete 1/6 of the work in one day.

Finally, to find the total time taken to complete the work together, we take the reciprocal of their combined work rate.

The reciprocal of 1/6 is 6, meaning that A and B, working together, will take 6 days to complete the task.

How to Calculate Combined Work Rates?

In order to determine the time two people will take to complete a task together, follow the basic principles below.

Individual Work Rates:

• Each person's ability to complete a task is measured by their individual work rate.
• The work rate is the fraction of the task a person can complete in one day.

Express Work Rates as Fractions:

Work rates are often expressed as fractions, with the numerator representing the part of the task completed and the denominator representing the total time.

Combine Work Rates:

• To find the combined work rate of two people working together, add their individual work rates.
• Addition of fractions involves finding a common denominator.
• Simplify the resulting fraction to find the combined work rate in its simplest form.

Calculate the Reciprocal:

• To find the total time taken to complete the task together, calculate the reciprocal of the combined work rate.
• The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

Determine Total Completion Time:

The reciprocal gives the total number of days required for both individuals to complete the task together.

Mathematical Approach:

• This process involves basic arithmetic operations such as addition and finding reciprocals.
• It provides a systematic way to determine how collaboration can reduce the total completion time.

Importance in Problem-Solving:

• Understanding how to combine work rates is essential for solving problems related to shared tasks.
• This concept is widely used in various real-life and academic applications.

Applications of Time and Work Concept

These are the major applications of time and work methodologies used in many aspects.

Project Planning:

• Estimating time required for collaborative tasks.
• Allocating resources effectively based on individual capabilities.

Workforce Management:

• Assessing combined productivity of multiple workers.
• Determining the most efficient team combinations.

Construction Projects:

• Scheduling work phases to ensure timely project completion.
• Distributing tasks among workers with varying speeds.

Algebra:

• Solving linear equations involving rates of work.
• Applying the concept of least common multiples (LCM) to find combined work rates.

Number Theory:

• Employing the concept of fractions and ratios in solving work-related problems.
• Understanding the relationship between different rates and their common multiples.

Graph Theory:

• Representing work flow as graphs to find the shortest path to task completion.
• Analyzing network efficiency in collaborative work environments.

Operations Research:

• Developing algorithms to optimize task assignments and scheduling.
• Using linear programming techniques to maximize efficiency in multi-worker tasks.