Matthew is thirty-three eighths feet tall. His brothers Noah and Daniel are twenty-seven fifths feet tall and seventeen quarters feet tall, respectively. Arrange the siblings in order from tallest to shortest.
In this problem, we’re given three brothers and three heights. Matthew is thirty-three eighths feet tall. His brothers Noah and Daniel are twenty-seven fifths feet tall and seventeen quarters feet tall, respectively. And the word “respectively” here means in the order that they were mentioned. In other words, the first height is Noah’s height, and then seventeen quarters is Daniel’s height. And we’re told to arrange the siblings in order. That’s in order of height from tallest to shortest. Now what do we notice about the heights that we’re given?
Well, firstly, they’re all in feet. So that makes them easy to compare. They’re all in the same unit of measurement. But they’re all a fraction of feet. In fact, each height is an improper fraction. An improper fraction, remember, is a fraction with a value greater than one. These are fractions where the numerator is larger than the denominator.
So if we just think of Matthew’s fraction as an example, 33 is larger than eight. And so thirty-three eighths is going to be larger than one, which of course we would expect. Matthew’s height is going to be more than one foot. We need to compare our three improper fractions together. And perhaps the best way to do this is to start by converting each improper fraction into a mixed number.
A mixed number is made up of a whole number and a fraction. Let’s see how it works with Matthew’s height. We know that eight-eighths are the same as one whole. So how many lots of eight-eighths are there in thirty-three eighths? We can count in eights to find out. Eight, 16, 24, 32. This is as close as we’re going to get to 33 without going above it.
Thirty-two eighths feet are the same as four whole feet. So the whole number part of our mixed number is four. Of course, we need to represent thirty-three eighths feet, not thirty-two eighths. And so we need to add our fraction part, which is one more eighth. Matthew’s height written as a mixed number is four and one-eighths feet.
Noah’s height is a number of fifths, twenty-seven fifths. We know that five-fifths are the same as one whole. So we can use the same method as we did to find Matthew’s height to turn Noah’s improper fraction into a mixed number. How many lots of five-fifths are there in twenty-seven fifths? We know five lots of five are 25. And so five whole feet are the same as twenty-five fifths. The whole number part of our mixed number is five. But we don’t want to find twenty-five fifths. We need to show twenty-seven fifths. So this is another two fifths. Noah’s height as a mixed number is five and two-fifths feet.
Finally, let’s consider Daniel’s height. Here we have seventeen quarters. We know that four-quarters make one whole. But how many lots of four-quarters are there in 17 quarters? We know four lots of four equals 16. And so sixteen quarters are the same as four whole feet. So the whole number part of Daniel’s height is four. But Daniel’s height isn’t sixteen quarters. It’s seventeen quarters. So the fraction part of Daniel’s height is one-quarter. Daniel’s height written as a mixed number is four and a quarter feet.
Now we need to look at the three heights and to put the three brothers in order from tallest to shortest. Now sometimes when comparing fractions, we need to convert them so that they have the same denominator. Sometimes they’re hard to compare. But in this case, we can compare them without doing this.
To start with, we can see that there’s one brother who’s definitely the tallest. If we look at the whole number of feet for each brother, we can see that Matthew and Daniel are both four foot something. But Noah is five foot something. It doesn’t matter what the fraction part of Noah’s height is. He’s definitely the tallest.
Now we just need to compare Matthew and Daniel’s heights. As we’ve just said, both of them are four foot something. So we need to compare the fraction part of each height. Matthew’s height is four and one-eighth. Daniel’s height is four and one-quarter. Which is larger, one-eighth or one-quarter?
We might think that one-eighth is greater than one-quarter because eight is a larger number than four. But remember that the denominator in a fraction tells us how many parts the whole has been split into. So one-eighth means that we split one whole foot into eight equal parts. And we choose one of them. One-quarter means we only split one foot into four equal parts. Each part is going to be larger. One-quarter is larger than one-eighth. And so four and a quarter, which is Daniel’s height, is larger than four and one-eighth, which is Matthew’s height. We now know the order of heights of the brothers.
To begin with, the three siblings’ heights were improper fractions. They all had a value greater than one. But because they all had different denominators, it was hard to compare them. So we converted each improper fraction into a mixed number. We were then able to use our knowledge of fractions to compare them and put them in order from tallest to shortest. And so the correct order of the three brothers is Noah then Daniel then Matthew.