5. You are scheduling your classes for next year. There are 4 required classes: a physical education class, a history class, an english class, and a math class. You can choose from 5 physical education
classes, 6 history classes, 5 english classes, and 3 math classes. In how many different ways could
you select the 4 required classes?

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Explanation:

For now, let's consider we can only select 2 classes instead of 4. Let's say we only had to worry about PE and history.

Let A,B,C,D,E represent the five PE classes.

Let F,G,H,I,J,K represent the six history classes.

If you arrange the letters in a table, you should get 5*6 = 30 different inner boxes of the table. Refer to the diagram below.

Each two letter code in the table represents the 2 class combo. For example, BG means you picked PE class B and history class G.

This idea can be extend to selecting 3 classes. Imagine that each of the codes in the table are laid along the side of a new table. So we'd have a table of 30 rows. Then lay out 5 columns (say with letters L,M,N,O,P) to represent the 5 English classes. You would get a table of 30*5 = 150 different combos here. Of course you shouldn't actually make such a table since it's so large, but I'm simply just saying for you to imagine one. All of the tables mentioned are completely optional. A tree diagram is another way to organize things, though tree diagrams tend to get cumbersome.

Anyway, we would have 5*6*5 = 30*5 = 150 different ways to pick the PE, history and English classes. We can extend this out to include the math classes as well.

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In short, all we do is multiply the given values together:

5*6*5*3 = 30*15 = 450

This idea is known as the counting principle.

or

step-by-step explanation:

we need to solve for 'x' the following equation:  

we have that:

⇒

⇒ ⇒

factorizing, we get that:

  ⇒

then, the solutions are: or

nothing extraneous has been found!

step-by-step explanation:

the ratio of red paint to blue paint is always 9/7. that means that, whatever x's value, (4x+12)/(4x-4) has that same ratio. so, set up an equation and cross-multiply to solve for x:

9/7 = (4x+12)/(4x-4)

9(4x-4) = 7(4x+12)

36x - 36 = 28x + 84

8x - 36 = 84

8x = 120

x = 15