Person A and Person B can do a piece of work in 70 and 60 days respectively. They began the work together, but Person A leaves after some days and Person B finished the remaining work.

Person A and Person B can do a piece of work in 70 and 60 days respectively. They began the work together, but Person A leaves after some days and Person B finished the remaining work in 47 days. After how many days did Person A leave?

Let's denote the number of days person A worked before leaving as "x".

Now, person A's work rate is 1/70 of the work per day, and person B's work rate is 1/60 of the work per day.

When they work together, their combined work rate is the sum of their individual work rates:

Combined work rate = 1/70 + 1/60 = (6 + 7)/420 = 13/420

So, in x days, they complete x * (13/420) of the work.

After person A leaves, person B works alone and completes the remaining work in 47 days, so his work rate is 1/60 of the work per day.

So, in 47 days, person B completes 47 * (1/60) of the work.

Given that person B completes the remaining work, we can set up an equation:

x * (13/420) + 47 * (1/60) = 1

Now, we can solve for x:

(13x/420) + (47/60) = 1

Multiply both sides by 420 * 60 to clear the denominators:

13x * 60 + 47 * 420 = 420 * 60

780x + 19740 = 25200

780x = 25200 - 19740

780x = 5450

x = 5450/780

x = 7

So, person A worked for 7 days before leaving.

Work and Time in Mathematics

Work and time problems are a common type of mathematical question that involve determining the amount of work done by individuals or machines working together at different rates. These problems typically involve finding how long it takes for a certain amount of work to be completed when individuals or machines work together or at different rates.

Here's a basic overview of how to approach work and time problems:

1. Understanding the Concept: In these problems, "work" refers to a task to be completed, which could be anything from painting a wall to building a structure. The rate at which work is done is usually given in terms of the amount of work done per unit time, such as "2 workers can paint a wall in 4 hours."

2. Identify the Variables: Typically, you'll need to identify the following variables:

   • The total amount of work to be done.
   • The rates at which individuals or machines can complete the work.
   • The time it takes for them to complete the work when working together or separately.

3. Use the Work Formula: The basic formula for work and time problems is: Work = Rate × Time This formula states that the amount of work done is equal to the rate at which work is done multiplied by the time it takes to complete the work.

4. Solve the Equations: Depending on the specific problem, you may need to set up and solve equations to find the unknown variables, such as the time it takes to complete the work.

5. Check Your Answer: Always check your solution to ensure it makes sense in the context of the problem. For example, make sure the time you've calculated is reasonable and that it aligns with the given rates and total work.

Let's consider an example:

Example: If it takes 6 hours for 4 workers to complete a job, how long would it take for 12 workers to complete the same job?

Solution: Let's denote:

• W as the total work to be done.
• R as the rate at which one worker completes the work.
• T as the time it takes for the workers to complete the work.

From the given information: W = 4R × 6 W = 24R

Now, when 12 workers are working together: W = 12R × T

Equating the work done: 24R = 12R × T

T = 24R / 12R T = 2 hours

So, it would take 12 workers 2 hours to complete the same job.

This is a basic example, but work and time problems can vary in complexity, requiring different approaches and techniques to solve them.