Put the following numbers shown in the grid in order of size from smallest to largest.
Now the numbers shown in the grid are currently in different formats. We have three fractions, one decimal, and one number which is in fact an integer or whole number. In order to compare these numbers, we first want them all to be in the same format. So as three of the numbers are fractions, it makes sense to convert the other two into the same form.
0.5 is probably a decimal that you’re quite familiar with as a fraction. It’s equal to one over two or a half. We’ll think about what to do with the number one a little bit later on. Now if we look at the fractions that we do have, so that’s three-sevenths, one-half, twenty-three twenty-eighths, and eight fourteenths, we can’t compare them at the moment as they all have different denominators.
In order to be able to compare these fractions, we need to convert them to equivalent fractions which all have the same denominator. To do this, we need to work out the lowest common multiple or LCM of seven, two, 28, and 14. This just means the smallest number that is in each of their times tables.
So to do this, we can start by listing out to their times tables. We won’t actually list out our two times table, as two is a small number relative to some of the others. So we’d have to write down quite a lot of our two times table. Instead, we know a number will be in our two times table if it’s an even number, so if it ends in zero, two, four, six, or eight. So we’ll just check that to see whether our number is in the two times table.
Our seven times table begins seven, 14, 21, 28. As 14 is two multiplied by seven, all of the numbers in the 14 times table are also in the seven times table. The 14 times table begins 14, 28. As 28 is equal to two multiplied by 14, all of the numbers in the 28 times table are also in the 14 times table and also in the seven times table. So we only need to write down the first number in the 28 times table, one times 28, which is 28. As 28 is the smallest number which appears in all three times tables, it’s the lowest common multiple of seven, 14, and 28. It’s also a multiple of two as it’s an even number, because its second digit is eight. This means we’ve found the lowest common multiple of seven, two, 28, and 14, and this is the denominator that we’re going to use for all of our factions.
Now one of our fractions, twenty-three twenty-eighths, already has the correct denominator, so we don’t need to worry about converting it. Let’s start with the decimal 0.5, which as a fraction was one-half. To get from two to 28 in the denominator, we need to multiply by 14. So to make sure that the fraction is equivalent, we need to do the same in the numerator. One multiplied by 14 is 14, so the fraction a half is equivalent to fourteen twenty-eighths.
Next, let’s consider the fraction three-sevenths. To get from seven to 28, we need to multiply by four, which we can see as 28 is the fourth number in our seven times table. So we need to do the same thing in the numerator. Three multiplied by four is 12, so the fraction three-sevenths is equivalent to twelve twenty-eighths.
Next, let’s consider eight fourteenths. To get from 14 to 28, we have to multiply by two, as 28 is the second number in the 14 times table. So again, we need to do the same thing in the numerator. Eight multiplied by two is 16, so the fraction eight fourteenths is equivalent to sixteen twenty-eighths.
Now for the number one, we could write this as the fraction twenty-eight twenty-eighths, as in order for a fraction to be equivalent to a whole one, it needs to have the same numerator and denominator. Or we could look at all of the fractions that we’ve already converted and see that their numerators are smaller than their denominators, which means all of these fractions are less than one. Either of these methods would tell us that the number one is in fact the largest of the five numbers.
Now let’s order the remaining numbers. And as they all have the same denominator, we can just compare the numerators. The smallest numerator is 12, so this means that the fraction twelve twenty-eighths or three-sevenths is the smallest. We must make sure that we write down the original values when we make our list. So the smallest value in the grid is three-sevenths. The next smallest numerator is 14, and the fraction fourteen twenty-eighths was equivalent to the decimal 0.5. So this is the next value in our list. Then we have the fraction sixteen twenty-eighths, which was equivalent to eight fourteenths, and then twenty-three twenty-eighths, which we didn’t change. Finally, we have the number one. The five numbers in order from smallest to largest are three-sevenths, 0.5, eight fourteenths, twenty-three twenty-eighths, one.