### Video Transcript

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

Given that thirty-nine over đť‘Ą
equals thirteen over seven. Find the value of đť‘Ą. So weâ€™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 weâ€™d instantly know that the denominator was the same as
đť‘Ą. 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 itâ€™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 weâ€™ve got
thirty-nine over đť‘Ą is equal to thirty-nine over twenty-one. So đť‘Ą must be equal to
twenty-one. And thatâ€™s our answer.

Given that đť‘Ą over twenty-five
is equal to eighteen over one hundred and fifty. Find the value of đť‘Ą. 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 Iâ€™ll instantly be able to read off what the value of đť‘Ą
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 â€” 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 â€” so Iâ€™ve got two fractions that are equal. They both got the same
denominator. So đť‘Ą must be equal to
three.

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

But the problem is it wonâ€™t be
equal to eighteen over a hundred and fifty anymore because weâ€™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 đť‘Ą 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 đť‘Ą is equal to three over
one. đť‘Ą 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 đť‘Ą the
subject to also get the same answer.

Now here is a bit of a cheeky
question.

Given that fifteen over đť‘Ą is
equal to two point five over seven. Find the value of đť‘Ą. We donâ€™t really like having
decimals in fractions, so thatâ€™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 weâ€™re
gonna find an equivalent fraction to two point five over seven. But the top â€” the numerator â€”
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 weâ€™ve got
the same numerators in each fraction. So đť‘Ą must be equal to
forty-two.

Now an alternative approach if
you didnâ€™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 didnâ€™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, Iâ€™ll get đť‘Ą. So đť‘Ą equals forty-two.

And letâ€™s look at yet another
way of doing it using cross-multiplication. So if I multiply both sides by
đť‘Ą over one, then the two sides will still be equal. Weâ€™ve done the same thing to
both sides but on the left-hand side look I can divide the top by đť‘Ą and I can
divide the bottom by đť‘Ą. So weâ€™ve got fifteen times one
over one times one; thatâ€™s just fifteen. So fifteen is equal to two
point five đť‘Ą over seven. Now if I multiply both sides by
seven. And because the right-hand side
is a fraction, Iâ€™m gonna multiply it by seven over one, the fraction equivalent
of seven. And on the left-hand side, itâ€™s
not a fraction; so Iâ€™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 thisâ€™s just two point five
times đť‘Ą. And now Iâ€™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 đť‘Ą. So Iâ€™ve got đť‘Ą is equal to a
hundred and five over two point five. Well if you donâ€™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 doesnâ€™t change it, but the version of one that weâ€™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 Iâ€™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 thatâ€™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 arenâ€™t quite as nice as youâ€™d like them to be. So this method doesnâ€™t quite
work so well. So sometimes you have to resort
to using full algebra and rearranging that equation.

Given that đť‘Ą over three equals
five over four. Find the value of đť‘Ą. Well in this question, thereâ€™s
no obvious simple multiple for each side. So weâ€™ll use the
cross-multiplication technique. So I wanna get đť‘Ą on its
own. So Iâ€™m gonna multiply that by
three over one. So if I do that to one side,
Iâ€™ve got to do the same to the other side. So now theyâ€™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 Iâ€™ve got đť‘Ą times one
divided by one times one; thatâ€™s just đť‘Ą. And on the right-hand side,
five times three is fifteen over four. So đť‘Ą 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 hereâ€™s another cheeking up. Weâ€™ve got a fraction within a
fraction. So we wouldnâ€™t normally let you do
that, but itâ€™s in the question.

Fill in the missing number:
seven over five is equal to something over nine and four-fifths. Again thereâ€™s no obvious
multiple to create an equivalent fraction and never got a mixed number on the
denominator. So first of all Iâ€™m gonna
simplify that; Iâ€™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 Iâ€™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 thatâ€™s the number Iâ€™m gonna
use.

So Iâ€™ve got seven over five is
equal to something divided by forty-nine over five. Well we made dividing
fractions. So I donâ€™t know if youâ€™ve heard
the phrase â€śdividing fractions is as easy as pie, flip the second and
multiply.â€ť 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 Iâ€™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 donâ€™t
work out very nicely, but you just have to plough through them.

And hereâ€™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 Iâ€™m just gonna use
cross-multiplication to try and sort this one out. So Iâ€™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, Iâ€™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 Iâ€™m gonna do is find an
equivalent fraction to one point one over nought point six, which doesnâ€™t have
decimals in it. So Iâ€™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 Iâ€™m not changing the
magnitude of this-of this number here. Iâ€™m not changing the
calculation, but what I do is get an equivalent fraction which doesnâ€™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 Iâ€™ve got five times eleven
over two. So our blank is equal to
fifty-five over two. Or depending on the question,
we couldâ€™ve said thatâ€™s a twenty-seven and a half or twenty-seven point
five.

So hereâ€™s a question where weâ€™ve
got three fractions that are equal to each other. Weâ€™ve got two unknowns that weâ€™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 weâ€™re just gonna take them
in pairs. So weâ€™re just gonna look at the
first two fractions together and then weâ€™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 couldâ€™ve said sixty over
thirty-five is equal to something over thirty and a third. But if Iâ€™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 thatâ€™s probably not a good
idea. So the first thing Iâ€™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 Iâ€™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 weâ€™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 weâ€™ve done and apply those to your finding unknowns
in proportions questions.