A proportion is when you have one ratio set equal to another ratio so it�s an equality
of two ratios so it looks something like this we will have one ratio maybe like a over b
and we have that set equal to another ratio c over d and this entire equation is considered
a proportion the left hand side is a ratio the right hand side is called a ratio but
the entire equation is called a proportion.
Now what will typically happen in a proportion is you will have some unknown somewhere on
the left to right side of the equation and in often times these will be attached to word
problems or application problems.
Let me give you a simple example: let�s say there is a young boy in a neighborhood
and he cut grass and he could cut three yards in 2 hours so he could cut 3 neighbors yards
in 2 hours but then we wanted to know how many yards could he cut in 10 hours and so
what we do we set up a proportion that asks that question so the ratio of how many yards
he could cut relative to how many hours it takes to cut them would be 3 to 2, 3 yards
in 2 hours.
We will set that equal to an unknown number of yards in 10 hours and so as long as this
ratio stays 3 to 2 we can create a proportion and our goal will be to solve for x, now how
we do that we will talk about n just a minute.
This particular example is relative small we can probably guess the solution, in just
a minute I will give you a systematic way of finding this.
Obviously he would have to repeat 3 yards in 2 hours he would have to do this 5 times
for 10 hours because he is taking up 2 hours at a time for each of the 3 yards so we will
multiply 2 x 5 to get 10, will have to multiply 3 x 5 to get 15 so in other words he can do
15 yards in 10 hours and that�s the equivalent statement as saying 3 yards in 2 hours but
that�s not a great way to do it, that�s just using our good old fashion intuition,
the way we typically solve proportions in an algebraic manner is using something called
cross multiplication it�s an extremely helpful algebra procedure looks something like this.
If you have a over b set equal to c over d, if you have this proportion here then we could
actually rewrite this by taking a times d and setting it equal to b times c, so this
is equivalent to a times d equals b times c and this is obvious why this is preferable
this does not have any fractions and its usually easier to solve, please understand cross multiplication
is not magic or anything like that.
Its got solid fundamental reasons why you can do this if you just wanted to use straight
up algebra to rewrite this notice you could multiply both sides by b and notice what would
happen on the left hand side, the b�s would cancel and we could also multiply both sides
by d because as long as we do it to both sides of the equation it is legal algebra and notice
the d�s would cancel on the right hand side but not the left hand side so what� remaining
at the end of the day is a times d equals b times c so there�s good fundamental reasons
why cross multiplication works.
Let�s go back to this example here, if we wanted to solve this more systematically,
lets erase this here if we have this ratio 3 over 2 equals x over ten then what we can
do is cross multiply and get 3 times ten equals 2 times x, so that�s an equivalent equation
but the reason why I like this one is it doesn�t have any fractions, so if 30 equals 2x divide
both sides by 2 and x equals 15 and that�s the same answer we go before but we got this
one in a much more systematic way and this is how we will this when these proportion
problems get really really big with lots of terms and big fractions that equal to other
large fractions cross multiplication will be extremely helpful in solving these proportions.
Before I let you for I want to mention one caution this is a mistake that a lot of students
make that I want to guard you against, students get really good with cross multiplication
they have a over b equals c over d and no doubt this is absolutely equivalent to a times
d equals b times c that�s 100 % true but students wind up confusing that with multiplying
a over b times c over d and what students will want to do is somehow take a times d
and set it equal to b times c and that�s absolutely the wrong thing to do if there
is no equal sign between these 2 terms that does not a proportion you can�t do cross
multiplication, and so usually it is an oversight by students who is just not paying attention
but now that I have called your attention to it just make sure you don�t make that
mistake on homework or a test.