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