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Succeed with maths: part 1
Succeed with maths: part 1

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1.1 The process of adding and subtracting

You might like to watch this next video before attempting Activity 1, which will guide you through the process of adding and subtracting fractions.

Download this video clip.Video player: swm_1_s4_adding_subtracting_fractions.mp4
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NARRATOR:
Let's start with a 1/4 plus a 1/4. To work this out, let's say we have a pie which we divide into four pieces. If our first 1/4 in our calculation was this quarter of the pie, and we were going to add it to another quarter of the pie, if I was to eat both of those quarters, so a quarter and then another quarter, how much would I have eaten? You could work this out by looking at the pie, and seeing that I've eaten two of the four pieces.
So 2/4 of the pie, which is the same as half of the pie. If we look at it mathematically, you can see that the denominators or the bottom numbers in the fractions stayed the same. This is because they represent the total number of pieces we had in this example. For the numerators, the top numbers in the fraction, we added them together. I had one of the four pieces of pie, and then I had another one of the four pieces of pie. So I ate two of the four pieces of pie, which is one half of the whole pie. It works the same way for subtraction.
If we had 3/7 take away 2/7, that would equal 1/7. As we just subtract the two from the 3 to get one, and we keep the denominator the same. Thinking again about this as pieces of pie, if you had three of the seven pieces of pie and you were to give away two of those seven pieces, you would be left with just one of the seven pieces of pie.
You can see that it is fairly straightforward to add and subtract fractions, when the denominator is the same. But what happens when we have different denominators? Let's say we have 1/4 plus 1/2. If you take the example of the pie again, for the first quarter of the calculation, let's colour this quarter of the pie in. And then if I was to eat another half of the pie, I would eat this much. So what does that equal in total?
Well, there's a couple of ways we can think about this. First of all, we could rewrite one half. One half of the pie is the same as 2 quarters. There's one quarter here, and another quarter here. So one half is the same as 2 quarters, 2 over 4. That means 1/4 plus 1/2 is the same as 1/4 plus 2 quarters. All we did here was change a half to two quarters, by multiplying the numerator and the denominator, the top and the bottom of the 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. The reason I picked two here, is because I wanted to get the same denominator as a quarter. To finish off this calculation, we just add the numerators to give 3 and keep the denominator the same of course. So the answer is 1/4 plus 2/4 equals 3/4. If you look at the image of the pie, you can see 3/4 of it has been shaded in.
Now let's try 1/2 plus a 1/3. With this, it isn't possible to multiply the denominator 2 by anything to get 3, or to multiply the denominator 3 by anything to get 2. So instead, in order to find the common denominator, we need to find the least common multiple of 2 and 3. So what's the least common multiple of 2 and 3?
Well, that's the smallest number that's a multiple of 2 and a multiple of 3, or the smallest number that both 2 and 3 divide into exactly. The smallest number that's a multiple of both 2 and 3 is 6, so let's convert both these factions to something over 6. So 1/2 is equal to what over 6? Well, if I ate a half of a pizza which had been divided into six slices, I would have eaten three slices. And that makes sense, because one is a half of 2 and 3 is a half of 6.
Similarly, if I ate a third of that pizza with six slices, it would be the same as 2/6. So 1/2 plus a 1/3 is the same thing as 3/6 plus 2/6. All we've done is write the original calculation with different denominators. We've essentially just changed the number of pieces in the pie. And now at this point, the problem becomes easy. We add the numerators to give 5, 3/6 plus 2/6 gives us 5/6. And we've kept the denominator the same as always.
And it works the same for subtraction. 1/2 take away 1/3, is the same as 3/6 take away 2/6. And that is equal to 1/6.
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Activity 1 Adding and subtracting fractions

Timing: Allow approximately 10 minutes

Have a go at these questions, 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.

  • a.five divided by 18 plus seven divided by 18

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

Answer

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

equation sequence part 1 five divided by 18 plus seven divided by 18 equals part 2 five plus seven divided by 18 equals part 3 12 divided by 18 equals part 4 two divided by three

To simplify 12 divided by 18 to two divided by three, divide the numerator (top) and the denominator (bottom) by 6.

  • b.three divided by eight plus six divided by eight

Hint: both fractions are eighths, so again you can add them directly.

Answer

The answer is:

equation sequence part 1 three divided by eight plus six divided by eight equals part 2 three plus six divided by eight equals part 3 nine divided by eight

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

nine divided by eight equals one and one divided by eight
  • c.five divided by six plus three divided by seven

Hint: 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?

Answer

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, five divided by six equals 35 divided by 42 by multiplying by 7 and three divided by seven equals 18 divided by 42 by multiplying by 6.

Thus, the sum is equation sequence part 1 five divided by six plus three divided by seven equals part 2 35 divided by 42 plus 18 divided by 42 equals part 3 53 divided by 42 equals part 4 one and 11 divided by 42.

  • d.

Hint: try adding the whole numbers first, and then add the fractional parts together.

Answer

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: equation sequence part 1 two and two divided by three plus three and one divided by eight equals part 2 sum with 3 summands five plus two divided by three plus one divided by eight equals part 3 sum with 3 summands five plus 16 divided by 24 plus three divided by 24 equals part 4 five and 19 divided by 24

  • e.15 divided by 16 minus five divided by 16

Answer

Both the fractions are sixteenths so you subtract straightaway: equation sequence part 1 15 divided by 16 minus five divided by 16 equals part 2 10 divided by 16 equals part 3 five divided by eight

  • f.19 divided by 24 minus three divided by eight

Hint: first ensure both fractions have the same denominator.

Answer

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

Since eight multiplication three equals 24, you can multiply the top and bottom of three divided by eight by 3 to make the equivalent fraction of nine divided by 24 and then carry out the subtraction.

Therefore: equation sequence part 1 19 divided by 24 minus three divided by eight equals part 2 19 divided by 24 minus nine divided by 24 equals part 3 10 divided by 24 equals part 4 five divided by 12

  • g.Two children were squabbling about chocolate. Josie had been given five divided by 12 of a bar by her aunt and one divided by four of a bar by her dad. Tim had been given one divided by two a bar by his mum and one divided by six by his friend. All the original bars were the same size. Was Josie or Tim given more chocolate, or were they given equal amounts?

Hint: add up what fraction of a bar each child had. To compare your two fractions, cancel each to its lowest terms.

Answer

Josie had five divided by 12 plus one divided by four of a bar. To add these, we need to put them over a common denominator. The common denominator is 12. So five divided by 12 plus one divided by four becomes five divided by 12 plus three divided by 12 equals eight divided by 12. eight divided by 12 cancels down to two divided by three of a bar.

Tim has one divided by two plus one divided by six of a bar. The common denominator is 6. So one divided by two plus one divided by six becomes three divided by six plus one divided by six equals four divided by six. four divided by six cancels down to two divided by three of a bar.

Therefore, Josie and Tim had the same amount of chocolate.

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.