## Intro to Ratios

- Ratios are fractions that compare part to part or part to whole, presented in simplest form without indicating the absolute size of the groups.
- Ratios can be represented in four forms: p to q, fraction, colon, and idiom, with fraction form being most useful for mathematical operations.
- The concept of scale factor (n) is crucial for understanding and working with ratios, allowing for the simplification of problems involving absolute quantities.
- Proportions, equations of the form fraction equals fraction, are essential for solving ratio problems, especially when dealing with part to whole relationships.
- Ratios involving three or more terms are handled similarly, with the test typically presenting these in colon form for simplicity.

**Q: ****Can you explain what a scale factor is? In the problem with boys and girls, why do we take 7n - 3n?**

A ratio problem may give us ratios and one or more absolute amounts. A ratio tells us about the ratio, but we don't know the absolute amounts.

By using a **scale factor**, we can "convert" a ratio in absolute (I mean the actual) amounts.

Let's take this problem.

The ratio of boys to girls is 3/7. Great, so I know that *for every 3 boys, there are seven girls.* But that doesn't tell me the actual amounts I need.

But I make a scale factor "n" and say the number of boys is 3n and the number of girls is 7n. Notice I have preserved the ratio. I just tagged an "n" onto the end of the numbers in the ratio. But now, these are actual amounts. While 3 and 7 were just part of the ratio of boys to girls, **3n is the actual number of boys and 7n is the actual number of girls.**

You may say: So what? We don't know what n is!

Right! But these actual amounts 3n and 7n may be *useful in setting up an equation.*

And if we can set up an equation and solve for n, then we know the exact number of boys and girls!

So let's set up an equation:

There are 32 more girls than boys.

Girls = 7n

Boys = 3n

"Boys + 32 = Girls"

or

32 = Girls - Boys

**32 = 7n - 3n**

32 = 4n

n = 8

So n = 8 is the scale factor. And know we know the girls is 7n = 7(8) = 56 and the boys is 3n = 3(8) = 24.

We call it a "scale factor" because it relates the ratio to the actual numbers.

A scale factor isn't the only approach or always easiest approach, but it's often a good way to work with these problems.

**Q: Can you explain the last problem? Why do we add the parts?**

We are told each part of concrete is made up of cement, sand, and gravel in the ratio of 1:2:3

So we have:

cement : sand : gravel

1 : 2 : 3

The point of a ratio is to compare things using **equal parts.**

This ratio of 1:2:3 tells me that for every 1 part of cement, I have 2 parts of sand and 3 parts of gravel. All of these parts are **equal,** and make up the concrete.

1 + 2 + 3 = 6

So for every 1 part of cement, I have 5 other parts of sand and gravel, and a total of **6 parts of concrete.**

Why do we have 6 parts and not 1 part concrete? Well, say each part is 1 kg and we have 1 kg cement, 2 kg sand, and 3kg gravel. If we mixed these together, would we get 1 kg of concrete? No -- *we have mixed together 6kg total of materials.* It would not make sense to mix 6 kg of materials and end up with 1 kg of concrete. We know have a *6kg mixture of concrete.*

Going back to "parts," we can see that sand makes up 2 of the 6 total parts in each part concrete. We mixed 2 parts sand with other ingredients to get 6 parts concrete.

That's why we can say:

sand : concrete =

2 : 6