Succeed with maths – Part 1

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# 1.1 The process of adding and subtracting

You might like to watch this video before attempting Activity 1, which will guide you through the process of adding and subtracting fractions. Note, that in the video where there is reference to ‘fourths’, this is more usually known in the UK as quarters. There is also a section towards the end of the video where negative numbers are introduced. You will cover this in Weeks 7 and 8 of the course, so there is no need to watch this at the moment, unless you are interested.

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#### Transcript

Narrator
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 1/4 plus 1/4 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 1/4 right here - let me do it in a different colour. This 1/4 right here, let's say it's this 1/4 of the pie, right? And we're going to add it to another 1/4 of the pie. Let's make it this one - let me change the colour - pink. This 1/4, this pink 1/4 is this 1/4 of the pie. So if I were to eat both 1/4s or 1/4 and then I eat another
1/4, how much have I eaten? Well, you could look from just the picture, I have now eaten 2 out of the 4 pieces of the pie. So if I eat 1/4 of a piece of pie or 1/4 of a pie, and then I eat another 1/4 of a pie, I will have eaten 2/4 of the pie. And we know from the equivalent fractions module that this is the same thing as I've eaten 1/2 of the pie,
which makes sense. If I eat 2 out of 4 pieces of a pie, then I've eaten 1/2 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 I have in this example. Well, I added the numerators, which makes sense. I had 1 out of the 4 pieces of pie, then I ate another 1 out of the 4 pieces of pie, so I ate 2 out of the 4 pieces of pie, which is 1/2. Let me do a couple more examples.
What is 2/5 plus 1/5? 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. 2/5 plus 1/5 is just 2 plus 1 over 5, which is equal to 3 over 5. And it works the same way with subtraction. If I had 3 over 7 minus 2 over 7, that just equals 1 over 7. I just subtracted the 3, I subtracted the 2 from the 3 to get 1 and I kept the denominator the same. Which makes sense. If I have 3 out of the 7 pieces of a pie and I were to give away 2 out of the 7 pieces of a pie, I'd be left with 1 of the 7 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 1/4 plus 1/2. Let's go back to that original pie example. Let me draw that pie.
So this first 1/4 right here, let's just colour it in, that's this 1/4 of the pie. And now I'm going to eat another 1/2 of the pie. So I'm going to eat 1/2 of the pie. So this 1/2. I'll eat this whole 1/2 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 1/2. 1/2 of the pie, that's actually the same thing as 2/4, right?
There's 1/4 here and then another 1/4 here. So 1/2 is the same thing as 2/4, and we know that from the equivalent fractions module. So we know that 1/4 plus 1/2, this is the same thing as saying 1/4 plus 2/4, right?
And all I did here is I changed the 1/2 to a 2/4 by essentially multiplying the numerator and the denominator of
this fraction by 2. 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 1/2 times 1 is equal to 1/2, you know that. Well another way of writing 1 is 1/2 times 2/2. 2 over 2 is the same thing as 1, and that equals 2 over 4. The reason why I picked 2 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 1/4 plus 2/4, so we know that we just add the numerators, 3, and the denominators are the same, 3/4. And if we look at the picture, true enough, we have eaten 3/4 of this pie. Let's do another one.
Let's do 1/2 plus 1/3. 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 3 by to get 2, or there's no, at least, integer I can multiply 3 by to get 2. And there's nothing I can multiply 2 by to get 3. 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 2 and 3. Well what's the least common multiple of 2 and 3? Well that's the smallest number that's a multiple of both 2 and 3. Well the smallest number that's a multiple of both 2 and 3 is 6. So let's convert both of these fractions to something over 6. So 1/2 is equal to what over 6. You should know this from the equivalent fractions module. Well if I eat 1/2 of a pizza with 6 pieces, I would have eaten 3 pieces, right? That make sense. 1 is 1/2 of 2, 3 is 1/2 of 6. Similarly, if I eat 1/3 of a pizza with 6 pieces, it's the same thing as 2 over 6. So 1/2 plus 1/3 is the same thing as 3/6 plus 2/6. Notice I didn't do anything crazy. All I did is I re-wrote 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, 3 plus 2 is 5, and we keep the denominators the same. 3 over 6 plus 2 over 6 equals 5/6.
And subtraction is the same thing.1/2 minus 1/3, well that's the same thing as 3 over 6 minus 2 over 6. Well that equals 1 over 6. 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 1/10 minus 1? Well, one doesn't even look like a fraction. But you can write it as a fraction. Well that's the same thing as 1/10 minus - how could we write 1 so it has the denominator of 10? Right. It's the same thing as 10 over 10, right? 10 over 10 is 1. So 1/10 minus 10 over 10 is the same thing as 1 minus 10 - remember, we only subtract the numerators and we keep the denominator 10, and that equals negative 9 over 10. 1/10 minus 1 is equal to negative 9 over 10. Let's do another one. Let's do one more. I think that's all I have time for. Let's do minus 1/9 minus 1 over 4. Well the least common multiple of 0 and 4 is 36. So that's equal to 36. So what's negative 1/9 where we change the denominator from 9 to 36? Well, we multiply 9 times 4 to get 36. We have to multiply the numerator times 4 as well. So we have negative 1, so it becomes negative 4. Then minus over 36. Well to go from 4 to 36, we have to multiply this fraction by 9, or we have to multiply the denominator by 9, so you also have to multiply the numerator by 9. 1 times 9 is 9. So this equals minus 4 minus 9 over 36, which equals minus 13 over 36.
I think that's all I have time for right now, and I'll probably add a couple more modules, but I think you might
be ready now to do the adding and subtracting module. Have fun.
End transcript

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## Activity 1 Adding and subtracting fractions

Timing: Allow approximately 10 minutes

Now try these examples, showing your answers in the simplest form or mixed number where relevant. Remember to make sure before you add or subtract to make the denominators the same. If you need a hint to help, click on ‘Reveal comment’.

• a.

### Comment

Are both fractions out of the same number of parts? Remember as always to show your answer in the simplest form.

• a.Both the given fractions are eighteenths, so they can be added together directly:

To simplify to , divide the numerator (top) and the denominator (bottom) by 6.

• b.

### Comment

Both fractions are eighths, so again you can add them directly.

If this is converted to a mixed number, the answer is:

• c.

### Comment

Can you find equivalent fractions for each given fraction that all share the same denominator? What number can be divided by both 6 and 7?

• c.This sum involves sixths and sevenths, which are different types of fraction. However, you can change both into forty-seconds, since both 6 and 7 evenly divide into 42. So, by multiplying by 7 and by multiplying by 6.

Thus, the sum is .

• d.

### Comment

Try adding the whole numbers first, and then add the fractional parts together.

• d.In this calculation you can add the whole numbers first (2 + 3 = 5) and then add the fractions. First, you must convert each fraction into twenty-fourths, as both 3 and 8 divide exactly into 24. So, the sum is:
• e.

• e.Both the fractions are sixteenths so you subtract straightaway:
• f.

### Comment

First ensure both fractions have the same denominator.

• f.You need both fractions to be out of the same number of parts (the denominators).

Since , you can multiply the top and bottom of by 3 to make the equivalent fraction of and then carry out the subtraction.

Therefore:

Well done – you’ve completed your first activity involving carrying out calculations with fractions! In the next activity you will look at a more practical application of fractions.