Calculate one and one-half minus one and one-third. Give your answer in its simplest form.
Here, we have two mixed numbers and we’re subtracting them. And we can subtract them in this form if we would like or we could change them to improper fractions and then subtract them from there. We can solve this both ways. So let’s go ahead and subtract them by keeping them in their mixed number form.
So when subtracting mixed numbers, we will subtract the whole numbers, which are the ones. So these places represent the whole part. And then the fraction portion, we need to subtract the one-half and the one-third. So this will be our fraction part. So the whole part of one minus one is simply zero.
Now, one-half minus one-third, in order to subtract fractions with different denominators, we have to find a common denominator between them. So what is the smallest number that two and three both go into? It would be six. So six will be our new common denominator. So the denominators will now be six. So how would we have gone from one-half to making it equal to something over six? Well, to get from two to six, we multiply by three. So we need to multiply the numerator by three and one times three is three.
Now, for one-third, to get from three to six, we multiply by two. So we also need to do the same to the numerator. And one times two is two. So when subtracting, we keep our denominator of six and then we subtract our numerators of three and two. So three minus two is one. So our whole part was zero and our fraction part is one-sixth. So there’s no need to write the zero part. So our final answer will be one-sixth.
Now, we could also solve this by turning our mixed numbers into improper fractions and then subtracting those. So let’s go ahead and do that. So we’re turning these mixed numbers into improper fractions. An improper fraction looks like a normal fraction, except the numerator is greater than the denominator. So we need to change one and one-half and one and one-third to improper fractions.
So we have the whole part, which is the one, and the fraction part, which is the one-half and also the one-third. So we need them to have the same denominator. And once they have the same denominator, we can add them together. So we need to change one to be something over two. One is the same thing as one over one. So to go from a denominator of one to two, we would multiply by two. So we multiply the numerator by two. So one times two for our numerator will be two. And two over two plus one over two, we add our numerators together and we keep our common denominator. So two plus one is three and our common denominator was two. So one and one-half is equal to three halves.
Now, let’s change the mixed number of one and one-third to an improper fraction. So we have the fraction portion of one- third, but we need to rewrite one as something over three. So one could be rewritten as one over one. And to go from the denominator of one to three, we’d have to multiply by three. And multiplying the numerator by three one times three is three. So three plus one is four. And we keep our common denominator of three. So we have four-thirds. So now, we have three-halves minus four-thirds. We’re almost there.
In order to subtract these fractions, we need to have a common denominator. So what is the smallest number that two and three can both go into? That would be six. So to go from two to six as a denominator, we would multiply by three. So we need to multiply the numerator by three and three times three is nine.
Now, for the second fraction, to go from three to six as a denominator, we multiply by two. So four times two is eight. And now that they have a common denominator — six will be our denominator. And we subtract our numerators. So nine minus eight is one. So just as we found before, our final answer in its simplest form would be one-sixth.