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Video: Finding Unknowns in Proportions

Tim Burnham

Through a series of examples, we use the technique of finding equivalent fractions in order to identify the value of the unknown in the equations (e.g., 39/𝑥 = 13/7).

13:15

Video Transcript

In this video we’re gonna be looking at proportions and finding the value of unknowns. Another way of saying this is that we’ll be finding equivalent fractions.

Given that thirty-nine over 𝑥 equals thirteen over seven. Find the value of 𝑥. So we’ve got two fractions: thirty-nine over something is equal to thirteen over seven. Now if we could find an equivalent fraction for thirteen over seven which had thirty-nine on the numerator here, then we’d instantly know that the denominator was the same as 𝑥. And if we take a quick look at those fractions. If I do thirteen times three, I get thirty-nine. So if I multiply the top and the bottom of my fraction by three, I will get an equivalent fraction.

Remember three over three is one. So I am multiplying thirteen over seven by one, which is just gonna leave me with a number with the same magnitude, thirteen over seven. But it’d be in a slightly different version of that number, so thirteen times three is thirty-nine and seven times three is twenty-one. This fraction is equivalent to this fraction. So those two things are still equal. But now look we’ve got thirty-nine over 𝑥 is equal to thirty-nine over twenty-one. So 𝑥 must be equal to twenty-one. And that’s our answer.

Given that 𝑥 over twenty-five is equal to eighteen over one hundred and fifty. Find the value of 𝑥. So in this question the unknown is on the numerator. So if I can find an equivalent fraction to eighteen over one hundred and fifty that has twenty-five on the denominator, then I’ll instantly be able to read off what the value of 𝑥 is.

And if I look carefully, this six times twenty-five is a hundred and fifty. So if I find an equivalent fraction of eighteen over one hundred and fifty by dividing the top by six and dividing the bottom by six — well a hundred and fifty divided by six is twenty-five because that was the whole point of doing this and eighteen divided by six is three — so I’ve got two fractions that are equal. They both got the same denominator. So 𝑥 must be equal to three.

Now an alternative way of doing this question would have been to try to get 𝑥 on its own and 𝑥 is already on the numerator of a fraction. So if I just multiply that side by twenty-five over one, I’ll be able to divide — so cross-cancel here — so divide twenty-five by five to get one and divide twenty-five by five to get one. So on the left-hand side, I’ll have one times 𝑥 over one times one; that would just be 𝑥.

But the problem is it won’t be equal to eighteen over a hundred and fifty anymore because we’ve got on the left-hand side is now twenty-five times bigger than it was. So what I need to do is the same to the right-hand side: twenty-five over one times that as well. And now I can divide twenty-five by twenty-five to get one and a hundred and fifty by twenty-five to get six. So 𝑥 is equal to eighteen over six. Well six can be divided by six to make one and eighteen can be divided by six to make three. So 𝑥 is equal to three over one. 𝑥 is just equal to three.

So two different methods you choose which one you prefer: we could either find the equivalent fractions and then just read off the answer or we could do a bit of algebra and make 𝑥 the subject to also get the same answer.

Now here is a bit of a cheeky question. Given that fifteen over 𝑥 is equal to two point five over seven. Find the value of 𝑥. We don’t really like having decimals in fractions, so that’s why I say this is a bit of a cheeky question. But nonetheless we can apply the same methods in order to solve it.

Now looking at this question, two point five, if I double two point five, I get five. And then if I triple five, I get fifteen. So two point five times six is fifteen. So by multiplying the top and the bottom of this fraction here by six, six over six is one remember, so we’re gonna find an equivalent fraction to two point five over seven. But the top — the numerator — will be fifteen instead of two point five. So as we said two point five times six is fifteen and seven times six is forty-two. So this fraction and this fraction are equivalent. And now we can say we’ve got the same numerators in each fraction. So 𝑥 must be equal to forty-two.

Now an alternative approach if you didn’t really like working with decimals might have been to start off by multiplying the top and the bottom of the right-hand side by two in order to find an equivalent fraction to two point five over seven, which didn’t involve decimals. And now I can see look five all I need to do is times that by three to get fifteen. So if I multiply fourteen by three, I’ll get 𝑥. So 𝑥 equals forty-two.

And let’s look at yet another way of doing it using cross-multiplication. So if I multiply both sides by 𝑥 over one, then the two sides will still be equal. We’ve done the same thing to both sides but on the left-hand side look I can divide the top by 𝑥 and I can divide the bottom by 𝑥. So we’ve got fifteen times one over one times one; that’s just fifteen. So fifteen is equal to two point five 𝑥 over seven. Now if I multiply both sides by seven. And because the right-hand side is a fraction, I’m gonna multiply it by seven over one, the fraction equivalent of seven. And on the left-hand side, it’s not a fraction; so I’m just gonna literally multiply that by seven.

Then seven times fifteen is a hundred and five. So on the right-hand side I can divide the top by seven and the bottom by seven to get rid of that fraction. So this’s just two point five times 𝑥. And now I’m gonna divide both sides by two point five so that I can cancel the two point fives on the right-hand side, just leaving me with the 𝑥. So I’ve got 𝑥 is equal to a hundred and five over two point five. Well if you don’t feel comfortable about doing that division, what you could do is multiply a hundred and five over two point five by one. And of course multiplying by one doesn’t change it, but the version of one that we’re gonna multiply by is two over two. And the advantage of doing that is that we end up with the denominator no longer being a decimal.

So I’ve got two hundred and ten over five. Well fives going to one hundred and twenty times, so two hundred will be forty times. And then five is going to ten twice, so that’s forty-two. Now that was a bit more long and drawn out than the other method, but sometimes on this method over here the numbers aren’t quite as nice as you’d like them to be. So this method doesn’t quite work so well. So sometimes you have to resort to using full algebra and rearranging that equation.

Given that 𝑥 over three equals five over four. Find the value of 𝑥. Well in this question, there’s no obvious simple multiple for each side. So we’ll use the cross-multiplication technique. So I wanna get 𝑥 on its own. So I’m gonna multiply that by three over one. So if I do that to one side, I’ve got to do the same to the other side. So now they’re still balanced. Now on the left-hand side if I divide the bottom by three, I get one. If I divide the top by three, I get one. Now I’ve got 𝑥 times one divided by one times one; that’s just 𝑥. And on the right-hand side, five times three is fifteen over four. So 𝑥 equals fifteen over four. And obviously depending on the question, you might need to change that to a mixed number or you might need to change it to a decimal.

And here’s another cheeking up. We’ve got a fraction within a fraction. So we wouldn’t normally let you do that, but it’s in the question. Fill in the missing number: seven over five is equal to something over nine and four-fifths. Again there’s no obvious multiple to create an equivalent fraction and never got a mixed number on the denominator. So first of all I’m gonna simplify that; I’m gonna create a top heavy fraction.

Well nine and four-fifths is nine plus four-fifths. And creating a common denominator that nine is the same as forty-five divided by five. So I’ve got forty-five fifths plus four-fifths, which is forty-nine fifths. So nine and four-fifths is the same as forty-nine fifths. So that’s the number I’m gonna use.

So I’ve got seven over five is equal to something divided by forty-nine over five. Well we made dividing fractions. So I don’t know if you’ve heard the phrase “dividing fractions is as easy as pie, flip the second and multiply.” This means that this right-hand side is equivalent to something times five over forty-nine. So we just return the divide into a multiply and then we flip the numerator and the denominator. So now on the right-hand side I can cancel off, so forty-nine divided by forty-nine is one and five divided by five is one. So I’ve just got one times that something. And on the denominator five times five is twenty-five and forty-nine times seven is three hundred and forty-three. So sometimes the numbers don’t work out very nicely, but you just have to plough through them.

And here’s a question that puts decimals in fractions which is again a little bit cheeky. Fill in the blank: one point one over nought point six is equal to something over fifteen. Now I’m just gonna use cross-multiplication to try and sort this one out. So I’m gonna multiply both sides by fifteen over one so that I can isolate that little blank on the right-hand side. Now on the right-hand side dividing the top and bottom by fifteen, I’ve got one over one. So I just got the blank. And on the left-hand side, I got fifteen times one point one over nought point six. So what I’m gonna do is find an equivalent fraction to one point one over nought point six, which doesn’t have decimals in it. So I’m just gonna multiply the top and the bottom by ten.

Obviously multiplying individual ten by ten over ten is just multiplying it by one. So I’m not changing the magnitude of this-of this number here. I’m not changing the calculation, but what I do is get an equivalent fraction which doesn’t have decimals in it. So this gives me that my blank is equal to fifteen times eleven over six. Now fifteen is divisible by three and six is divisible by three. And fifteen divided by three is five and six divided by three is two. So I’ve got five times eleven over two. So our blank is equal to fifty-five over two. Or depending on the question, we could’ve said that’s a twenty-seven and a half or twenty-seven point five.

So here’s a question where we’ve got three fractions that are equal to each other. We’ve got two unknowns that we’ve got to find. Fill in the blanks twelve over seven is equal to sixty over something is equal to something else over thirty and a third. So we’re just gonna take them in pairs. So we’re just gonna look at the first two fractions together and then we’re gonna look at the first and the third fractions together.

And looking at the first pair, twelve over seven is equal to sixty over something, I can straight away say that look if I do twelve times five, I get sixty. So I could come up with an equivalent fraction here by multiplying the top and the bottom by five and find out what that missing denominator is.

So twelve over seven by multiplying top and bottom by five gives me sixty over thirty-five. So twelve over seven is the same as sixty over thirty-five. So that missing number there must be thirty-five. Now looking at the first and the third fractions, twelve over seven is equal to something over thirty and a third.

Well I could’ve said sixty over thirty-five is equal to something over thirty and a third. But if I’ve made any mistakes in that first part of the question, I will be carrying forward that mistake into the second part of the question. So that’s probably not a good idea. So the first thing I’m gonna do is to convert thirty and a third into a top heavy fraction. And thirty and a third is the same as thirty plus a third. And thirty is the same as ninety over three. So that is equivalent to ninety over three plus one over three, which is ninety-one over three.

So twelve over seven is equal to something over ninety-one over three. So if I multiply both sides by ninety-one over three, then the ninety-one over three is cancelled on the right-hand side, just leaving me with my blank. Now I’ve got ninety-one over three times twelve over seven on the left-hand side. Well I can do a bit of cancelling there. So twelve and three both divisible by three and seven and ninety-one are both divisible by seven because seven times thirteen is ninety-one. So my blank is equal to thirteen times four over one times one. And that is fifty-two. So I can fill in that answer as well.

So we’ve seen a range of different quite tricky questions in some cases and a few different techniques for tackling them. In some cases we can just find equivalent fractions and read off the relevant answer. And in other cases we had to maybe do some cross-multiplication and some algebra and some rearranging and some quite complicated calculations in order to work out our answers. Hopefully you can pick your favourite techniques from what we’ve done and apply those to your finding unknowns in proportions questions.