Adding and subtracting fractions How to add and subtract fractions.
Adding and subtracting fractions
- Welcome to the presentation on adding and subtracting fractions.
- Let's get started.
- Let's start with what I hope shouldn't confuse you too much.
- This should hopefully be a relatively easy question.
- If I were to ask you what one fourth plus one fourth is.
- Let's think about what that means.
- Let's say we had a pie and it was divided into four pieces.
- So this is like saying this first one fourth right here,
- let me do it in a different color.
- This one fourth right here,
- let's say it's this one fourth of the pie, right?
- And we're going to add it to another one fourth of the pie.
- Let's make it this one-- let me change the color-- pink.
- This one fourth, this pink one fourth is this one fourth of the pie.
- So if I were to eat both one fourths,
- or one fourth and then I eat another one fourth,
- how much have I eaten?
- Well, you could look from just the picture,
- I have now eaten two out of the four pieces of the pie.
- So if I eat one fourth of a piece of pie or one fourth of a pie,
- and then I eat another one fourth of a pie,
- I will have eaten two fourths of the pie.
- And we know from the equivalent fractions module
- that this is the same thing as that I've eaten one half of the pie,
- which makes sense.
- If I eat two out of four pieces of a pie, then I've eaten one half of it.
- And if we look at it mathematically, what happened here?
- Well the denominators or the bottom numbers,
- the bottom numbers in the fraction stayed the same.
- Because that's just the total number of pieces that I have in this example.
- Well, I added the numerators, which makes sense.
- I had one out of the four pieces of pie, then I ate another one out of the four pieces of pie,
- so I ate two out of the four pieces of pie, which is one half.
- Let me do a couple more examples.
- What is two fifths plus one fifth?
- Well we do the same thing here.
- We first check to make sure the denominators are the same.
- We'll learn in a second what we do when the denominators are different.
- If the denominators are the same, the denominator of the answer will be the same.
- And we just add the numerators.
- two fifths plus one fifth is just two plus one over five, which is equal to three over five.
- And it works the same way with subtraction.
- If I had three over seven minus two over seven, that just equals one over seven.
- I just subtracted the three, I subtracted the two from the three to get one
- and I kept the denominator the same.
- Which makes sense.
- If I have three out of the seven pieces of a pie,
- and I were to give away two out of the seven pieces of a pie,
- I'd be left with one of the seven pieces of a pie.
- So now let's tackle-- I think it should be pretty straightforward
- when we have the same denominator.
- Remember, the denominator is just the bottom number in a fraction.
- Numerator is the top number.
- What happens when we have different denominators?
- Well, hopefully it won't be too difficult.
- Let's say I have one fourth plus one half.
- Let's go back to that original pie example.
- Let me draw that pie.
- So this first one fourth right here, let's just color it in,
- that's this one fourth of the pie.
- And now I'm going to eat another one half of the pie.
- So I'm going to eat one half of the pie.
- So this one half.
- I'll eat this whole one half of the pie.
- So what does that equal?
- Well, there's a couple of ways we could think about it.
- First we could just re-write one half.
- one half of the pie, that's actually the same thing as two fourths, right?
- There's one fourth here and then another one fourth here.
- So one half is the same thing as two over four,
- and we know that from the equivalent fractions module.
- So we know that one fourth plus one half,
- this is the same thing as saying one fourth plus two fourths, right?
- And all I did here is I changed the one half to a two fourths,
- by essentially multiplying the numerator and the denominator of this fraction by two.
- And you can do that to any fraction.
- As long as you multiply the numerator and the denominator by the same number,
- you can multiply by anything.
- That makes sense because one half times one is equal to one half.
- You know that.
- Well another way of writing one is one half times two over two.
- two over two is the same thing as one, and that equals two over four.
- The reason why I picked two is because I wanted to get the same denominator here.
- I hope I'm not completely confusing you.
- Well, let's just finish up this problem.
- So we have one fourth plus two fourths,
- so we know that we just add the numerators, three,
- and the denominators are the same, three fourths.
- And if we look at the picture, true enough,
- we have eaten three fourths of this pie.
- Let's do another one.
- Let's do one half plus one third.
- Well once again, we want to get both denominators to be the same,
- but you can't just multiply one of them to get --
- there's nothing I can multiply three by to get two,
- or there's no, at least, integer I can multiply three by to get two.
- And there's nothing I can multiply two by to get three.
- So I have to multiply both of them so they equal each other.
- It turns out that what we want for,
- what we'll call the common denominator,
- it turns out to be the least common multiple of two and three.
- Well what's the least common multiple of two and three?
- Well that's the smallest number that's a multiple of both two and three.
- Well the smallest number that's a multiple of both two and three is six.
- So let's convert both of these fractions to something over six.
- So one half is equal to what over six.
- You should know this from the equivalent fractions module.
- Well if I eat one half of a pizza with six pieces, I would have eaten three pieces, right?
- That make sense.
- One is one half of two, three is one half of six.
- Similarly, if I eat one third of a pizza with six pieces,
- it's the same thing as two over six.
- So one half plus one third is the same thing as three over six plus two over six.
- Notice I didn't do anything crazy.
- All I did is I re-write both of these fractions with different denominators.
- I essentially changed the number of pieces in the pie,
- if that helps at all.
- Now that we're at this point then the problem becomes very easy.
- We just add the numerators, three plus two is five,
- and we keep the denominators the same.
- Three over six plus two over six equals five over six.
- And subtraction is the same thing.
- One half minus one third, well that's the same thing as three over six minus two over six.
- Well that equals one over six.
- Let's do a bunch more problems and hopefully you'll start to get it.
- And always remember you can re-watch the presentation,
- or you can pause it and try to do the problems yourself,
- because I think sometimes I talk fast.
- Let me throw you a curve ball.
- What's one tenth minus one?
- Well, one doesn't even look like a fraction.
- But you can write it as a fraction.
- Well that's the same thing as one tenth minus--
- how could we write one so it has the denominator of ten?
- It's the same thing as ten over ten, right?
- ten over ten is one.
- So one tenth minus ten over ten is the same thing as one minus ten--
- remember, we only subtract the numerators,
- and we keep the denominator ten, and that equals negative nine over ten.
- one tenth minus one is equal to negative nine over ten.
- Let's do another one. Let's do one more.
- I think that's all I have time for.
- Let's do minus one ninth minus one over four.
- Well the least common multiple of nine and four is thirty-six.
- So that's equal to thirty-six.
- So what's negative one ninth where we change the denominator from nine to thirty-six?
- Well, we multiply nine times four to get thirty-six.
- We have to multiply the numerator times four as well.
- So we have negative one, so it becomes negative four.
- Then minus one over thirty-six.
- Well to go from four to thirty-six, we have to multiply this fraction by nine,
- or we have to multiply the denominator by nine,
- so you also have to multiply the numerator by nine.
- One times nine is nine.
- So this equals minus four minus nine over thirty-six,
- which equals minus thirteen over thirty-six.
- I think that's all I have time for right now.
- And I'll probably add a couple more modules.
- But I think that right now that you might be ready to do the adding and subtracting module.
- Have fun.
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