Working together, Alex and Millie can file all the papers in the file cabinets in 4 hours. It would have taken alex 6 hours to do the job alone. What is the missing value in the table ? 

Working together, Alex and Millie can file all the papers in the file cabinets in 4 hours. It would have taken alex 6 hours to do the job alone. What is the missing value in the table that represents the part of the papers that Alex would file if Alex and Millie worked together?

A.4r

B.4/6

C.6r

D.6/4

Let's denote Alex's rate as "a" (papers filed per hour) and Millie's rate as "m" (papers filed per hour).

From the given information, we know that:

1. When they work together, their combined rate is enough to complete the job in 4 hours.
2. Alex alone takes 6 hours to complete the job.

Using the formula "Rate = Work / Time", we can express the rate of work for both Alex and Millie:

For Alex: a = 1/6 For Millie: m = 1/4

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

a + m = 1/6 + 1/4

To find the common denominator, which is 12, we rewrite the fractions:

a + m = 2/12 + 3/12 a + m = 5/12

So, together, they complete 5/12 of the job in one hour.

Now, let's look at the options:

A. 4r B. 4/6 C. 6r D. 6/4

We need to find the part of the job Alex would complete in one hour when working together with Millie. Since Alex's rate is a = 1/6, we can substitute "a" with "1/6" in option A:

4r = 4 * (1/6) = 4/6

Therefore, the correct answer is option B. 4/6.

Work and Time in Mathematics

In mathematics, work and time problems typically involve calculating the amount of work done by one or more individuals working together or separately over a certain period of time. These types of problems often revolve around scenarios such as painting a house, filling a tank, or constructing a building. The key concepts involved in solving work and time problems are:

1. Rate of Work: This refers to the amount of work done per unit time by an individual or a group. It is often expressed as work per hour, work per day, etc.

2. Work Formula: The basic formula used to solve work and time problems is: Work = Rate × Time This formula states that the total amount of work done is equal to the rate at which the work is done multiplied by the time spent working.

3. Inverse Variation: In some cases, as more workers are added, the amount of time needed to complete the work decreases inversely.

4. Parallel Work: When multiple people are working on the same task simultaneously, their individual rates of work can be added together to find the total rate of work.

5. Sequential Work: When tasks are performed one after another, the total time needed is the sum of the times needed for each task.

6. Fractional Work: Sometimes, part of a job is completed, and then the remainder is completed later. In such cases, fractional parts of the work are calculated.

To solve work and time problems, you need to carefully analyze the given information, identify the rate of work for each person or group involved, and then use the appropriate formula or method to calculate the total work done or the time taken to complete the task.

Here's a simple example:

Example: If it takes 6 hours for a painter to paint a room, and another painter joins to help after 2 hours, how long will it take for them to finish painting the room together?

Solution: Let's denote the rate of the first painter as R1 (work done per hour) and the rate of the second painter as R2. We know that R1 = 1/6 (since it takes 6 hours for the first painter to complete the job).

After 2 hours, the first painter completes 2 × R1 of the work, leaving 1 - 2 × R1 of the work remaining.

Now, both painters work together. The combined rate of work is Rcombined = R1 + R2.

We have R1 = 1/6 and Rcombined = 1/6 + R2.

Since both painters finish the remaining work together, we can set up the equation:

Time taken to finish remaining work = Remaining work / Rcombined

= (1 - 2 × R1) / (1/6 + R2)

Substitute R1 = 1/6 and solve for R2.

Once you find R2, you can calculate the time it takes to finish the remaining work.