Sophie and Simon are peeling a pile of potatoes for lunch in the cafeteria. Sophie can peel all the potatoes by herself in 45 minutes, while it would take Simon 30 minutes to do the job working alone.

Sophie and Simon are peeling a pile of potatoes for lunch in the cafeteria. Sophie can peel all the potatoes by herself in 45 minutes, while it would take Simon 30 minutes to do the job working alone. If Sophie and Simon work together to peel the potatoes, How long will it take them?

It will take them 18 minutes, if Sophie and Simon work together to peel the potatoes.

To solve this problem, we can use the concept of rates. Sophie's rate of peeling potatoes is 1/45 of the pile per minute, and Simon's rate is 1/30 of the pile per minute.

When they work together, their rates add up:

Sophie's rate + Simon's rate = Their combined rate.

So, it would be:

(1/45) + (1/30) = 1/x

Where x is the time it takes them to peel the potatoes together.

To find x, we need to solve for x in the equation:

(1/45) + (1/30) = 1/x

First, let's find a common denominator for 45 and 30, which is 90.

(2/90) + (3/90) = 1/x

(5/90) = 1/x

Now, cross multiply:

5x = 90

Divide both sides by 5:

x = 90/5

x = 18

So, it will take them 18 minutes to peel the potatoes together.

Time and Work in Mathematics

Time and work problems in mathematics deal with finding the amount of work done by individuals or groups working together over a period of time. These problems typically involve finding rates of work, determining how long it takes to complete a task, or calculating how many people are needed to complete a job within a given time frame.

Here are some key concepts and methods used in solving time and work problems:

1. Rate of Work: This refers to the amount of work done by an individual or a group in a unit of time. It is usually expressed as work per unit time (e.g., tasks per hour, widgets per day).

2. Reciprocal of Rate: In many problems, it's helpful to work with the reciprocal of the rate of work. For example, if someone can complete a task in 5 hours, their rate of work is 1/5 of the task per hour.

3. Basic Formula: The basic formula for time and work problems is:

   Work = Rate × Time

   This formula states that the total work done is equal to the rate at which work is done multiplied by the time taken to do it.

4. Inverse Variation: In some scenarios, the time taken to complete a task is inversely proportional to the number of workers. This means that if the number of workers increases, the time taken to complete the task decreases, and vice versa.

5. Simultaneous Work: When multiple people are working together on a task, their individual rates of work can be added to find the combined rate of work.

6. Fractional Work: In cases where only a part of the work is completed, fractional work can be used to represent the portion of the task completed in a certain amount of time.

7. Word Problems: Time and work problems are often presented as word problems, where the problem must be translated into mathematical equations and solved using the concepts mentioned above.

Here's a simple example:

Example: If it takes 6 hours for a group of 4 workers to complete a task, how many hours would it take for 3 workers to complete the same task?

Solution: Let the rate of work for each worker be 1/x of the task per hour.

For 4 workers, the total rate of work is 4 * 1/6 = 2/3 of the task per hour.

Now, for 3 workers, the time taken can be calculated using the formula:

Time = Work / Rate = 1 / Rate

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

So, it would take 3 workers 2/3 hours to complete the same task.