Seven children measured their heights. And then we’re shown a table with a list of seven children on the left-hand side and their heights on the right-hand side. What is the mean height of the children in centimetres?
So in this problem, we’re provided with a set of information: seven children and their heights in centimetres. And we’re asked to calculate the mean height of the children. But what does the word “mean” mean?
The mean is a special way to calculate the average of a set of information. And the way to find the mean is to first of all find the total of a set of numbers and then divide it by the number of those numbers. So to calculate the mean height from our table, first of all, we need to find the total of all the heights together and then divide it by the number of heights that there are, which of course is seven because there are seven children.
So let’s go through these two steps carefully to find the mean height, the average height, of the children. As we’ve said, our first step is to find the total of all of the heights. How can we add these seven numbers? Well, we could list them as one large vertical addition. Or we could look at the numbers more carefully to try to find pairs that make nicer numbers to add together.
Let’s look at the first two numbers: 144 and 136. The two digits in the ones column combine to make 10, a round number. And so we can say that 144 plus 136 equals 280. With the next two numbers, the two digits in the ones place combine together to make five. And so if we add them together, we can make a multiple of five. 142 plus 143 equals 285.
Looking at the next two numbers, we can see that they’re going to make a multiple of 10. Two and eight ones equals 10. 152 plus 148 equals 300. We can think of it as taking the two from the first number and adding it to the second before adding them. That would be the same as adding 150 to 150, which is 300.
So by combining the numbers mentally, we’ve now got four numbers that we can add together. Let’s add the first two numbers together. We know that 280 doubled or 280 plus 280 equals 560. So 280 plus 285 equals five more than this, 565.
Now we can add our next number. We can see the value of making these into nice numbers. All we have to do is add 300. 565 plus 300 equals 865. And finally, we can add the 150. 865 plus 150 equals 1015. So the total height of all of the children is 1015 centimetres.
But of course, the question doesn’t ask us to find the total height. It asked us to find the mean height. We need to divide 1015 by the number of heights that there are or the number of children that there are, which of course is seven. Let’s use short division to find the answer.
There are no sevens in one. So we can move the one into the next column. How many sevens are in 10? Well, there’s only one seven in 10. And there’s a remainder of three. How many sevens are in 31? Seven, 14, 21, 28, 35. 35 is too large. So there must be four sevens in 31. It takes us to 28. Four sevens in 31, and there’s a remainder of three. And finally, how many sevens are in 35? Well, we know five times seven is 35. And so the mean height of all of the children is 145 centimetres.
To start with, we thought carefully about what the word “mean” means. It was a type of average. And we remembered that we found the mean by taking the total of all the heights and dividing it by the number of heights.
Next, we found the total of all of the heights. And instead of adding all seven heights at once, we found ways to pair them up, to turn them into easier numbers to add together. Then we divided that total by the number of heights or the number of children, which was seven. And this gave us the mean height.
We know this is a type of average. Some of the children are shorter than 145 centimetres and some of the children are taller than 145 centimetres. But the average height, the mean height, of these seven children is 145 centimetres.