Video: GCSE Mathematics Foundation Tier Pack 4 • Paper 1 • Question 9

GCSE Mathematics Foundation Tier Pack 4 • Paper 1 • Question 9

06:13

Video Transcript

Part a), calculate three-fifths divided by seven-fifths. Part b), calculate three- sevenths plus two-thirds.

So in order to actually calculate three-fifths divided by seven- fifths, we have to use our rule for dividing fractions. And that rule tells us that if we’re going to divide fractions, so in this case we’ve got 𝑎 over 𝑏 divided by 𝑐 over 𝑑, then this is equal to 𝑎 over 𝑏 and then multiplied by 𝑑 over 𝑐. So it’s multiplied by the fraction, but with the numerator and denominator swapped. And this is actually known as a reciprocal. So we’ve actually swapped the numerator and denominator. Okay, so let’s apply this rule to our calculation. So therefore, if we do that with ours, we’ve now got three over five or three-fifths multiplied by five-sevenths or five over seven. And that’s because if we swap the numerator and denominator around, we had seven over five, so we’ve now got five over seven. And like I said, this is the reciprocal.

Now as we’re gonna be multiplying fractions, we’ve got a rule for multiplying fractions. And that is if we have fractions, so have 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑, then this is going to be equal to 𝑎𝑐 over 𝑏𝑑. I’ve actually written it here with the multiplication sign in just to show you what happens. Well you actually multiply each of the numerators by each other, so 𝑎 multiplied by 𝑐, and each of the denominator together, so 𝑏 multiplied by 𝑑. So in that case, we go back to ours. What we’re going to do is multiply three by five for our numerator, which gives us 15. And then for the denominator, we’ve got five multiplied by seven, which gives us 35. So therefore we’ve got 15 over 35. So we can say that the answer to the calculation three-fifths divided by seven-fifths is 15 over 35.

Now in the question, it hasn’t actually asked us to simplify our answer. So you actually get the mark for this. However, I will show you how you’d actually simplify 15 over 35 as this can often be asked to do. And it’s also good practice for you. Well we can see that actually five is going to be the factor of 15 and 35, and it’s the highest factor that’s shared between the two numbers. So therefore we’re gonna divide the numerator and denominator by five. And when we do this, we get three over seven or three- sevenths. And that’s because 15 divided by five is three, and 35 divided by five is seven. So therefore, this will be the fully simplified answer to our calculation.

Okay great, let’s move on to part b. So in part b, we’re asked to calculate three-sevenths plus two-thirds. So in order to actually calculate this, what we want to do is actually find a common denominator. And to do that, we need to find the lowest common multiple of seven and three. So in order to do this, one way you can do it is actually by writing out the first few multiples of seven and then seeing when you write out the multiples of three if any of them is shared. So we do that, we’ve actually got the first few. We’ve got seven, 14, 21, and 28. And then write out the multiples of three. We can see that yes there is actually one that shared, and that’s 21. So therefore, what we want to do now is actually make this the denominator for each of our fractions. So we’re gonna actually convert them into fractions with the denominator of 21.

So therefore, our first fraction is going to become nine over 21. And that’s because to get from seven to 21, we need to multiply the denominator by three. So we get 21 on the denominator. And then whatever you do to the bottom, so whatever you do to the denominator, we’ve gotta do to the top or the numerator. So therefore, what we do is we multiply the three by three, which gives us nine. Great, so we’ve got nine over 21. And then our second fraction is going to be 14 over 21. And this is because again we look what we needed to do to the denominator to get from three to 21, and what we do is we multiply it by seven, so then we do the same to the numerator, and two multiplied by seven gives us 14. Okay, so we’ve got nine over 21 plus 14 over 21.

Now we’re gonna quickly remind ourselves of the rules when we’re actually adding fractions with the same denominator. Well all we do when we’re adding fractions with the same denominator is we add together the numerators and then put that over the denominator. So for instance if we have 𝑎 over 𝑏 plus 𝑐 over 𝑏, this is equal to 𝑎 plus 𝑐 over 𝑏. So therefore, we’re gonna get nine plus 14 over 21, which is gonna give us an answer of 23 over 21, which can also be converted into a mixed number, which would be one and two twenty-oneths or one and two over 21. And the way we did this was we look at how many times the denominator goes into the numerator, so how many times 21 goes into 23, which is once. And we got a remainder of two. And then we put this remainder over the original denominator. Okay, great! So this is the answer to the calculation three-sevenths plus two-thirds.

So now we’ve got the answer, but what I wanna do is actually check it use an alternate method. The reason I’m showing you this is because if you have a bit of trouble trying to find the lowest common multiple of the denominators or you forget how to do that, this is a method that will always work. It might involve some cancelling later on, but it’s something that some students do find easier. This method has been called many different things. Some people call it the fishing chair. I like to call it cross cross smile, and I’ll show you how it works. So first of all, what you do is you actually do multiplication across. So this is our first cross. So you do the top left numerator by the bottom right denominator, so three multiplied by three gives us nine. Then you include the sign in the calculation you’re doing, so it’s plus. Then we do the second cross, which is seven multiplied by two, which is 14, written like this. And then for our denominator, what we do is we multiply seven by three. So we multiply the two denominators we had in the calculation, so this is our smile, which gives us 21. And we can see that we’ve actually arrived at the same stage we had in the original calculation. And therefore, we’ll get the same result because we’ve got nine plus 14 over 21, which give us 23 over 21 or one and two over 21.

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