## Weighted Averages I

### Transcript

Now we can talk about, weighted averages. Sometimes the test will ask us for the combined average of two or more different groups. Here's a sample problem, we're not gonna solve this yet but just to give you an idea of what this situation is like. We'll look at this problem.

On a ferry, there are 50 cars and ten trucks. The cars have an average mass of 1,200 kilograms, and the trucks have an average mass of 3,000 kilograms. What is the average mass for all the vehicles on the ferry? And so we have two different groups. We have an average for each group.

And we want to come up with the average for everything put together. There are a few different ways to approach a problem such as this. The first approach would be to think in terms of sums, as we discussed in the first lesson of this module, thinking in terms of sums can often simplify problems about averages. And it's very important to remember, we can't add averages, we usually can't average averages, but we can add sums.

And we know the sum of one group and the sum of the other group, all we have to do is add them, and we would get the sum of the whole group. And so, thinking about sums is often the key to working through average problems, and that includes weighted average problems. In that problem with vehicles, we have the actual number of each type of vehicles. Since all the masses are in thousand, we do the calculations with everything divided by 1,000 just for simpler numbers.

So here's the problem again. And so how would we find the sums? Well, sum is average times number. So for the cars, that's going to be 50. And instead of writing as 1,200, I'm gonna divide by 1,000, we just write it as 1.2 just so we don't have a bunch of extra zeros floating around.

50 times 1.2, that's the same as 5 times 12 which is 60. For the trucks, we have ten trucks and then the mass, we'll just call that three, 10 times 3 of course, is 30. Those are the two sums. We add up the sums the total sum is 90. And we have a sum of 90, and we want the average overall 60 vehicles.

Well, 90 divided by 60 is three-halves, or 1.5. And again, that is the mass divided by 1,000, so we just multiply by 1,000 again, and 1,500 is the answer. 1,500 is the average mass of all 60 cars on the ferry. And notice incidentally, first of all that, that average is between these two numbers, it would have to be between those two numbers.

And in fact, it is a little closer to cars because there are more cars. So that kind of makes sense. If you find the average of two groups and the number you get is not between them, then you probably have done something wrong. The average, the weighted average always has to be between the average of the two groups, so we're somewhere in the middle, if it's three or more groups.

So using sums is one way to approach weighted average problems. Another approach, a very different approach, involves thinking about proportions. In some problems, each group of individuals is some proportion or percent of the whole. And so they, in the problem about cars on a ferry, we actually gave you counts.

We said, this is the number of cars, this is the number of trucks. Instead, it might be that the groups are broken down by percentage. This is the percent of group one, this is percent of group two, percent or proportion. We can simply multiply the average of each group by this proportion. Find the sum of these, and that sum is the average of the entire set.

So just, algebraic, let's talk about this first. Suppose there are three groups, and they each have averages, A1, A1, A3. Those are the averages of our three groups, or our three categories. The first group is P1 of the whole, that's it's proportion. The second is P2, the third is P3. In other words, if they're given as percents, we would just change the percents to decimals.

Then we'd have the proportions. And of course, all the proportions should add up to one. All we have to do is take the product of each average times its own proportion. So average of the first group times the proportion of the first group. Plus average of the second group times the proportion of the second group, plus, and do that for each group and that sum will be the average of the whole.

Here's a practice problem, pause the video, try to do that on your own and then we'll talk about this. Okay, so notice here, we don't know the actual number of people of the company. We don't care about the number of people of the company. All we're given are percents, which are very easy to change the proportions, 70%, 20%, 10%, 0.7, 0.2, 0.1.

Those are the proportions at this company. And we have the average for each group. So again, all these numbers are in a thousand. We don't wanna deal with the extra zeros, so we're just gonna call these averages 40, 80, and 120, and leave off the extra zeros to save ourselves writing all those extra zeros.

So all we're gonna do for the average. We're gonna multiply the proportion. So for the marketers, they're 0.7, multiply that by 40. The programmers, 0.2, multiply that by 80, and then the managers, 0.1, multiply that by 120. Well, 0.7 times 40 is the same as 7 times 4, and 0.2 times 80 is the same as 2 times 8.

So that's 28 plus 16, plus 12. Of course, 12 plus 28 is 40, plus 16 is 56. And again, this is divided by 1,000, we have to tack on the extra zeros to get the actual number. That actual number is 56,000, and that is the average salary of all employees at the company.

So one way to find the weighted averages is to find the sums, find the sum of the first group, add the sum of the second group. Add over all the groups, if we have more than two groups, then you'll get the total sum. You can just divide the total sum by the number of individuals. And if the problem gives it to you in proportions, all we have to do is multiply each average times the proportion.

If it's a percent, we just have to remember change that percent to a decimal. A1 times p1, plus A2, times p2, plus etc. Add all those up and that will be the average of the whole.

Read full transcriptOn a ferry, there are 50 cars and ten trucks. The cars have an average mass of 1,200 kilograms, and the trucks have an average mass of 3,000 kilograms. What is the average mass for all the vehicles on the ferry? And so we have two different groups. We have an average for each group.

And we want to come up with the average for everything put together. There are a few different ways to approach a problem such as this. The first approach would be to think in terms of sums, as we discussed in the first lesson of this module, thinking in terms of sums can often simplify problems about averages. And it's very important to remember, we can't add averages, we usually can't average averages, but we can add sums.

And we know the sum of one group and the sum of the other group, all we have to do is add them, and we would get the sum of the whole group. And so, thinking about sums is often the key to working through average problems, and that includes weighted average problems. In that problem with vehicles, we have the actual number of each type of vehicles. Since all the masses are in thousand, we do the calculations with everything divided by 1,000 just for simpler numbers.

So here's the problem again. And so how would we find the sums? Well, sum is average times number. So for the cars, that's going to be 50. And instead of writing as 1,200, I'm gonna divide by 1,000, we just write it as 1.2 just so we don't have a bunch of extra zeros floating around.

50 times 1.2, that's the same as 5 times 12 which is 60. For the trucks, we have ten trucks and then the mass, we'll just call that three, 10 times 3 of course, is 30. Those are the two sums. We add up the sums the total sum is 90. And we have a sum of 90, and we want the average overall 60 vehicles.

Well, 90 divided by 60 is three-halves, or 1.5. And again, that is the mass divided by 1,000, so we just multiply by 1,000 again, and 1,500 is the answer. 1,500 is the average mass of all 60 cars on the ferry. And notice incidentally, first of all that, that average is between these two numbers, it would have to be between those two numbers.

And in fact, it is a little closer to cars because there are more cars. So that kind of makes sense. If you find the average of two groups and the number you get is not between them, then you probably have done something wrong. The average, the weighted average always has to be between the average of the two groups, so we're somewhere in the middle, if it's three or more groups.

So using sums is one way to approach weighted average problems. Another approach, a very different approach, involves thinking about proportions. In some problems, each group of individuals is some proportion or percent of the whole. And so they, in the problem about cars on a ferry, we actually gave you counts.

We said, this is the number of cars, this is the number of trucks. Instead, it might be that the groups are broken down by percentage. This is the percent of group one, this is percent of group two, percent or proportion. We can simply multiply the average of each group by this proportion. Find the sum of these, and that sum is the average of the entire set.

So just, algebraic, let's talk about this first. Suppose there are three groups, and they each have averages, A1, A1, A3. Those are the averages of our three groups, or our three categories. The first group is P1 of the whole, that's it's proportion. The second is P2, the third is P3. In other words, if they're given as percents, we would just change the percents to decimals.

Then we'd have the proportions. And of course, all the proportions should add up to one. All we have to do is take the product of each average times its own proportion. So average of the first group times the proportion of the first group. Plus average of the second group times the proportion of the second group, plus, and do that for each group and that sum will be the average of the whole.

Here's a practice problem, pause the video, try to do that on your own and then we'll talk about this. Okay, so notice here, we don't know the actual number of people of the company. We don't care about the number of people of the company. All we're given are percents, which are very easy to change the proportions, 70%, 20%, 10%, 0.7, 0.2, 0.1.

Those are the proportions at this company. And we have the average for each group. So again, all these numbers are in a thousand. We don't wanna deal with the extra zeros, so we're just gonna call these averages 40, 80, and 120, and leave off the extra zeros to save ourselves writing all those extra zeros.

So all we're gonna do for the average. We're gonna multiply the proportion. So for the marketers, they're 0.7, multiply that by 40. The programmers, 0.2, multiply that by 80, and then the managers, 0.1, multiply that by 120. Well, 0.7 times 40 is the same as 7 times 4, and 0.2 times 80 is the same as 2 times 8.

So that's 28 plus 16, plus 12. Of course, 12 plus 28 is 40, plus 16 is 56. And again, this is divided by 1,000, we have to tack on the extra zeros to get the actual number. That actual number is 56,000, and that is the average salary of all employees at the company.

So one way to find the weighted averages is to find the sums, find the sum of the first group, add the sum of the second group. Add over all the groups, if we have more than two groups, then you'll get the total sum. You can just divide the total sum by the number of individuals. And if the problem gives it to you in proportions, all we have to do is multiply each average times the proportion.

If it's a percent, we just have to remember change that percent to a decimal. A1 times p1, plus A2, times p2, plus etc. Add all those up and that will be the average of the whole.