**Ratios and Rates**

- Rates are ratios with different units in the numerator and denominator, essential for solving various ACT math problems.
- To solve rate problems, set up an equation where ratio equals ratio, ensuring the units match on both sides.
- Operations with proportions, including simplification and cross-multiplication, are fundamental techniques in solving these problems.
- Practical examples include converting units, calculating time for tasks, and determining quantities in scientific contexts.
- Understanding unit conversion and the ability to manipulate proportions are key skills for success on the ACT.

**Q:** In the last question about gold, the solution did not follow the pattern of setting up the numerators and denominators to match (mentioned in the summary). Actually you crossed the symbols out diagonally. Any advice on how to distinguish when to use this method versus the other of matching the numerators and denominators? The formula seemed quite different than the others and I am just wondering is there any method of knowing which approach to take.

**A: **I can see how this would be a bit confusing. The most important point here is that the advice to "make sure the units of the numerators and denominators match" only applies to *proportions*—ratios set equal to each other. Take a look back at the question at about 2:00 into the video to see an example of this:

*8 grams / 1 hour = 30 grams / H hours.*

When they are equal to each other like that, you need to match grams with grams and hours with hours (or whatever other units).

When you *multiply* ratios, on the other hand, as shown in the question you asked about, you don't need to have the same units in each ratio. Consider this simple case:

*John drives 40 mph for 2 hours. How far does he drive?*

Well, that's 40m / 1 hour * 2 hours, which gives us 80 miles. They are different types of measurements, but we can multiply them together when one measurement is "unit A per unit B" and the second measurement is "unit B." For example, 20 dollars per book and 5 books? That's 100 dollars. :-)

20 dollars/book * 5 books = 100 dollars <-- the "books" cancel because we have (book/book) = 1

For the last problem, we don't have a proportion. We are multiplying the ratios together to get the ratio we want:

1) The density of gold — or, in other words, the weight per unit of volume, in this case cm³

2) The price per weight 3) the measurements of a cube

It asks us to find the price of gold in the cube.

First, we find the volume of the cube — 8 cm³

Second, since we know the weight per volume — 20g/cm³, we can find the weight of the cube = 8cm³ * 20g/cm³ = 160g (the cm³'s cancel)

Third, since we know the price per weight, we can find the price of 160g = 160g * $50/g = $8000

So the question you should be asking yourself:

- Am I setting up a proportion to solve such as (A/B = C/x)? In this case the units of the numerator and denominator need to line up. Otherwise the statement doesn't make sense.

More examples:

- Proportions on the GRE
- Or, am I multiplying ratios together to get a target ratio? In this case, we want the units to line up diagonally such that all but the units of the final answer we are looking for cancel out. That's what we did in the gold problem.

Or, take this example:

10 bags per 4 bugs 5 bears per bug 5 donuts per 2 bears.

How many donuts per bag?

(5 donuts / 2 bears) * (5 bears/ 1 bug) * (4 bugs / 10 bags) =

100 donuts / 20 bags = 5 donuts / bag

I hope that makes sense! :)

**Q:** In the block of ice question, I don't understand why the grams canceled out. I understand why the numbers were simplified, but why do the grams cancel?

**A: **Great question! That's definitely not something that Mike explains clearly in the video. I'll try to be more explicit here.

The video sets up the equation:

8 grams / 1 hour = 30 grams / H hours. (I should note here that the "1" hour is implied -- the number 1 here is not really necessary, but it will make this more obvious later).

We simplify that to 4 grams / 1 hour = 15 grams / H hours.

Now we cross multiply, keeping the units along with the numbers. We get:

4 grams * H hours = 15 grams * 1 hour

At this point, we can actually think of the units (the grams and hours) as variables that we can divide out. If we "divide" both sides by grams, we'll get:

4 * H hours = 15 * 1 hour

The grams have canceled out here. Now, if we we divide both sides by 4, we get:

H hours = 15/4 * 1 hours

Evaluating 15/4 * 1 we'll get H = 3.75 hours, or 3 hours and 45 minutes.

Now, I'm oversimplifying a little bit here, because "grams" and "hours" aren't variables, they are units. However, the underlying thinking is the same -- if we divide two numbers with the same units, the units cancel out. This is actually a really important concept in chemistry called dimensional analysis. It's not something we cover in our course, but here's a good video that explains dimensional analysis.