Find 𝑥 divided by 𝑦, given that 𝑥 is equal to three and six-sevenths and 𝑦 is equal to six and five-sevenths.
So what we need to do here, is evaluate this expression here. Now we know that 𝑥 is equal to three and six-sevenths. So we know that 𝑦 is equal to six and five-sevenths. So the first thing we need to do is to write out that expression but using numbers instead of letters. So we can say 𝑥 divided by 𝑦 is equal to three and six-sevenths divided by six and five-sevenths.
Now, if this was a calculator question, you’d just pop those numbers into the calculator and get your answer out. But, sadly it’s not. So we’re gonna have to do this longhand. Now the first thing that we need to do is to convert those mixed numbers into improper fractions or top heavy fractions.
Well remember, three and six-sevenths means three plus six-sevenths, and six and five-sevenths means six plus five-sevenths. So there’s two little calculations we’ve got to do first inside those parentheses. Now we can rewrite three as three over one; that’s the same thing. Three divided by one is the same as three, and six divided by one is the same as six. So we’ve got some fraction addition to do before we can do our fraction division.
Now in order to do fraction addition, you do need to have common denominators. And three over one plus six over seven, we don’t have a common denominator. But if I multiply three over one by seven over seven, well seven over seven is just one. So I’m-I’m doing one times three, so I’ve still got three there. It’s not changing the size or the magnitude of that number. But I’m gonna get it in a slightly different format. So I’m gonna have seven times three, which is 21, over seven. And 21 over seven is the same as three. But more importantly, it has the same denominator as six over seven. So I can now do 21 over seven plus six over seven, is 27 over seven.
But wait, I need to go back and do the other calculation, six over one plus five over seven. Again, I need to have a common denominator. So to get six over one having a common denominator of seven with five over seven, I’m gonna multiply six over one by seven over seven. Again, seven over seven is just one. So I’m not changing the magnitude or the size of six. I’m keeping it the same number, but in a slightly different format. Seven times six is 42 and seven times one is seven. So this becomes 42 over seven. And that means, the second parentheses there, I’ve got 42 over seven plus five over seven. So I’ve now got my common denominator. And 42 sevenths plus five-sevenths is 47 sevenths.
So this calculation has come down to 27 over seven divided by 47 over seven. Now whenever I have to do fraction division, I just remember my old rhyme: Dividing fractions is easy as pie; flip the second and multiply. So the equivalent calculation to 27 over seven divided by 47 over seven, is to turn that divide into a multiply and to flip the second fraction so it becomes seven over 47. And now I could just do your 27 times seven over seven times 47, but I can do a bit of cancelling first. So seven is going to seven once, seven is going to seven once, so I can cancel the seven from the top and the bottom. And 27 and 47 don’t have any common factors. Well, not other than one.
So 27 times one is 27 and 47 times one is 47. So our answer is 27 over 47. So let’s just draw a nice big box around it and make it stand out nice and clear on the page.