6. Alice and Alma have to fill a workbook. Alice has already completed 3 pages and can do 7 pages per hour. Alma has completed 11 pages, but can only work at a rate of 3 pages per hour. Eventually Alice will catch up and the two will be working on the same page. How long will that take? How many pages will each of them have finished?

This problem is an example of solving equations with variables on both sides. To solve, we must first set up an equation for both Alice and Alma.

Since Alice can do 7 pages per hour, we can represent this part of the equation as 7p. She has already completed 3 pages, so we just add the 3 to the 7p:

7p + 3

Since Alma can do 3 pages per hour, we can represent this part of the equation as 3p. She has already completed 11 pages, so we just add 11 to the 3p:

3p + 11

To determine when both girls will have written the same amount of pages, we set the two equations equal to each other:

7p + 3 = 3p + 11

Then, we solve for p. First, the variables must be on the same side of the equation. We can do this by subtracting 3p from both sides of the equation:

4p + 3 = 11

Next, we must get p by itself. We work towards this by subtracting 3 from both sides of the equation:

4p = 8

Last, we divide both sides by 4. So p = 2.

This means that it will take 2 hours for Alice and Alma to have read the same amount of pages. If we want to know how many pages they will have read, we simply plug the 2 back into each equation:

7p + 3

= 7 (2) + 3

= 14 + 3

= 17

3p + 11

= 3 (2) + 11

= 6 + 11

= 17

After 2 hours, Alice and Alma will have read the same amount of pages: 17 pages.