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Addition and subtraction of fractions

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Shelagh and Luana realize that they could express their ages more specifically if they used fractions. Doing this will allow them to calculate the difference in their ages more precisely. If Shelagh describes her age as 15 and 1/2 years old and Luana is 7 and 7/12-- I'm assuming years old-- what is the difference in their ages? Write the answer as a simplified mixed number. So Shelagh is older, so she's 15 and 1/2 years old, so we literally want to find the difference between 15 and 1/2 and 7 and 7/12. Now we can see immediately-- or maybe it's not obvious to you, but 1/2 is the same thing is 6/12, which is a less than 7/12. So if we separated out the whole number parts and the fraction parts, this fraction is actually smaller. The 1/2 is smaller than the 7/12, so it might be easier to just do this as mixed numbers so we don't have to do all of that borrowing and regrouping. So let's write both of them as mixed numbers. So we have 1 and 1/2. We're still going to-- sorry, let's write both of them as improper fractions, so we're going to keep 2 in the denominator. And as an improper fraction, 2 times 15 is 30 plus 1 is 31. So 31/2 the same thing as 15 and 1/2, so these are equivalent. And then 7 and 7/12, if we write it as an improper fraction, so we're still going to have the minus sign out there, the denominator is 12. 12 times 7 is 84. 84 plus 7 is 91. It is 91. Did I do that right? 91. Yep, that looks right. 91/12. Now, we've written both of them as improper fractions, but we still have different denominators, so we have to have a common denominator. So what is the least common multiple of 2 and 12, or what's the smallest number that's divisible by both of them? Well, 12 is divisible by 12, and 12 is also divisible by 2, so 12 is the least common multiple. So we want both of them to be over 12. And this fraction already is over 12 so we don't have to change it: 91/12. Now, on the blue one, to go from 2 to 12 in the denominator, we have to multiply by 6, so we also have to multiply by 6 in the numerator. So if we multiply 31 by 6, what do we get? You can even do this in your head. 6 times 30 is 180, 6 times 1 is 6, so it's going to be 186. And now we subtract. So the difference in their ages is going to be-- the denominator is going to be 12. It's going to be 186 minus 91, which is equal to-- over 12. Let's see, 186 minus just 90 would be 96, but we're subtracting 91, so it's going to be 95. 95, right? 6 minus 1 is 5, and then 180 minus 90 is 90, so that works out. So the way we've expressed it right now, it isn't is an improper fraction, but they want us to write it as a simplified mixed number, so let's do that. So how many times does 12 go into 95? Let's see, 8 times 12 is 96, so that's too big, so 12 goes seven times. So this is equal to 7. And if we take-- well, let me write it out. We don't want to do too much in our head. If we go 12 into 95, I'm saying it goes seven times. So if that's true, if that's the largest number, then the remainder should be less than 12. 7 times 12 we know is 84 from our times tables, or you could just think it's going to be 14 plus 10 times 7, which is 70. Either way. And let's see, 95 minus 84: 5 minus 4 is 1, 9 minus 8 is 1, so we have a remainder of 11. So 12 goes into 95 seven times. You get 7 wholes and you have 11 left over. So 95/12 is 7 and 11/12. And we're done! The difference between Shelagh and Luana's age is 7 and 11/12, or Shelagh is 7 and 11/12 years older than Luana.