Use the digits one, one, two, three, four, and six to create two mixed numbers with a sum of five and five-sixths. There are five possible answers. One and one-sixth and two and three-quarters. Or one and one-sixth and four and two-quarters. Or one and two-thirds and four and one-sixth. Or one and a quarter and two and three-sixths. Or finally, one and four-sixths and four and two-thirds.
The sum of two numbers is what we get when we add them together. It’s their total. And in this problem, we’re being asked to create two numbers that have a sum, or a total, of five and five-sixths. Five and five-sixths is a mixed number. Mixed numbers are special types of numbers that have a whole number part — in this case, it’s the whole number five — and also a fraction part. That’s the five-sixths on the end.
So, we need to find two numbers that add together with a sum of five and five-sixths. But if we read the question carefully, we can see that these are not just any numbers. We need to create two mixed numbers. That’s two numbers with a whole number part and a fraction part. And we can’t just find any two mixed numbers. We’re given six digits to use: one, one, two, three, four, and six. So, we have six digits. And if we think about two mixed numbers, there are six places where those digits could go.
Now, we’re given five possible pairs of mixed numbers. Without given some answers to choose from, we’d be having to try lots and lots of different pairs of mixed numbers to see which ones added together to make five and five-sixths. But as it stands, we know that our answer is going to be one of these five pairs. The first thing that we can do to find the answer quickly is to go through each pair of mixed numbers and see whether they do use the digits one, one, two, three, four, and six.
Let’s begin by looking at the first one: one, one, two, three, four, six. And the second: one, one, two. There’s no three digit in these mixed numbers. In fact, the four digit has been used twice. We do have a six. So, we can’t say that this pair of mixed numbers uses the digits one, one, two, three, four, and six. And we have to cross it through.
Looking at our third pair, we have one, one, two, three, four, six. Our fourth pair is made up of one, one, two, three, four, six. But if we look at our final pair of mixed numbers, we can see that there’s only one one. We have a two, a three, and a six, but we actually have two fours. So, these mixed numbers have not been made from the digits one, one, two, three, four, and six. We’re gonna have to cross these through. So, just by looking at the digits that each pair of mixed numbers is made from, we’ve managed to narrow our choices down. There are only three possible answers.
What else do we know about a pair of fractions that might add together to make five and five-sixths? Well, we know that when we add our two fraction parts together, the chances are it’s going to make another fraction less than one. It could possibly make a mixed number value worth one and a bit. But it’s not going to make a value of two or more. And we can use this fact to help us because we know the whole number part of our mixed number needs to equal five.
So, we know if we add the two whole numbers in our two mixed numbers together, they’re going to have to make a total of five. Or possibly four if the two fractions added together to make a total more than one. But they can’t make anything other than five or four. We just won’t be able to make five and five-sixths that way. So, the next thing we can do is to look at the whole-number parts of each pair of fractions.
In the first pair of fractions, we have one and two. These are going to make a total of three. So, we can see that even if we did add our fraction parts together and they made a mixed number that had a value of one and a bit, it would still not be enough to make a total of five and five-sixths. The total is just not going to be big enough.
The next possible pair shows whole numbers of one and four. And these add together to make five. So, as long as our two fractions add together to make five-sixths, then we may have found the correct answer. Let’s just check our final option. Here, we can see that the whole-number parts are worth one and two again. We’re not going to be able to make a total that’s worth five and five-sixths. It’s going to be too small. And so, we’re only left with one possible pair to check.
What do we get if we add one and two-thirds and four and one-sixth? Let’s start by adding our two whole-number parts. One plus four equals five. Now, we need to find the total of the two fractions. But if we look carefully at these fractions, we can see that they both have different denominators, two-thirds and one-sixth.
Now, in order to add these two fractions together, we need to convert one or both of them so that they both have the same denominator. Let’s convert our two-thirds into a number of sixths. To change from thirds into sixths, we multiply the denominator by two; we double it. And so that our fraction keeps the same value, we need to do the same to the numerator. Two doubled is four. And so, we can say two-thirds is the same as four-sixths.
Let’s change the fraction part of our first mixed number because one and two-thirds is exactly the same as one and four-sixths as we’ve just found out. Now, we can add our two fraction parts much more easily. Four-sixths plus one more sixth equals five-sixths. We can see that the sum of the two mixed numbers that we were left with does equal five and five-sixths. We were asked to use the digits one, one, two, three, four, and six to create two mixed numbers with a sum of five and five-sixths. And the correct two mixed numbers are one and two-thirds and four and one-sixth.