The weights of three boxes are one and one-ninth kilograms, two and one-third kilograms, and two-ninths of a kilogram. What is the total weight of all the boxes?
Well here are three boxes. Now what we’re gonna do is add up all of those three weights and see what the total weight of all three boxes is. So the calculation we’re doing is one and one-ninth plus two and one-third plus two-ninths. Now let’s look at two different ways of doing that calculation.
Now if you think about it, one and one-ninth means one plus one-ninth. And two and a third means two plus one-third. So when I add one and a ninth to two and a third is the same as doing one plus a ninth plus two plus a third. And then lastly, we’ve got to add two-ninths.
Now the first thing to notice is we’ve got a whole number here and a whole number here. And because addition is commutative, it doesn’t matter what order we add all these numbers together in. So first of all, I’m just going to add the one and the two to make three. So that’s equal to three plus a ninth plus a third plus two-ninths. And next, it’s very easy to add fractions together if they’ve got the same denominator like a ninth and two-ninths. So one-ninth plus two-ninth is three-ninths. So that becomes three plus three-ninths plus a third.
Now you should notice that three-ninths can be simplified. Three divided by three is one and nine divided by three is three. So three-ninths simplifies to a third, and we have three plus a third plus a third. And now I’ve got two fractions with common denominators again. So one-third plus one third is two-thirds. So that becomes three plus two-thirds. And as we’ve said before, three plus two-thirds is the same as three and two-thirds. And since our units are kilograms, the answer is that the total weight of the boxes is three and two-thirds kilograms.
Now another approach would be to convert all of our numbers to top heavy fractions or improper fractions then find common denominators and then add them together. Let’s try that. Well one and one-ninth, the one is one whole one or nine-ninths. So one plus a ninth is nine-ninths plus one-ninth which is ten-ninths. And now two and a third, that’s two whole ones plus a third. Now each whole one is three-thirds. So I’ve got two of those: two times three is six-thirds. So that’s gonna be six-thirds plus one-third is seven-thirds. And the fraction two-ninths can’t be simplified, so we just copy that down.
And now if I look at our calculation, we’ve got ten-ninths. And we’re adding two-ninths, so that’s gonna make twelve-ninths. And hopefully, we can spot the twelve and nine are both divisible by three, so I can simplify twelve-ninths. And twelve divided by three is four and nine divided by three is three, so twelve-ninths is the same as four-thirds. And we’ve got to add seven-thirds. So again, we’ve got a common denominator of three, so four-thirds plus seven-thirds is eleven-thirds.
Now technically, eleven-thirds of a kilogram is a perfectly acceptable answer, but in the question they gave us mixed numbers for our weights. So I want to calculate the mixed number version of eleven-thirds. And to do that, I need to know how many times does three go into eleven. Well three times three would be nine, four times three would be twelve. So that would be too many. So it’s gonna be three whole ones. And three whole ones counts for nine-thirds. So I’ve got another ten- eleven. I’ve got another two-thirds to go. So that’s three plus two-thirds or three and two-thirds. And again, our units are kilograms. So whichever way we do the calculation, we get the same answer.