A proportion is an equation
in the form of a over b equals c over d,
where b and d are not equal to zero.
There are two common ways to solve a proportion.
We can solve a proportion just like we solve
a rational equation by clearing the fractions
from the equation by multiplying both sides by the LCD.
And we can also solve a proportion by cross multiplying.
Let's take a look at both methods.
Let's first solve this proportion
like we would solve a rational equation
by multiplying both sides of the equation
by the least common denominator.
Because the least common multiple of six and 24 is 24,
we can clear the fractions from the equation
by multiplying both sides by 24.
This would gives us 24 times two over six
equals 24 times x over 24.
If it's helpful we can write 24 as a fraction
with a denominator of one.
And now let's simplify out all the common factors
between the numerators and denominators before multiplying.
Notice 24 and six share a common factor of six.
There is one six in six and four sixes in 24.
We now just have four times two,
which is equal to eight.
On the right, we have a common factor of 24.
There is one 24 in 24 here and here.
We now just have one times x, which is equal to x.
So we have eight equals x,
which is equivalent to x equals eight,
which is the solution to the proportion.
Now let's solve the same proportion
using cross multiplication.
One property of proportions is
that the cross products are always equal.
For the proportion a over b equals c over d,
the cross products are a times d and b times c.
And for proportions, the cross products are always equal,
and, therefore, a times d will always equal b times c.
For example, if we have one over two equals three over six,
or 1/2 equals 3/6,
notice that one times six is equal to two times three.
Let's use this property and solve the proportion again.
Again, the cross products are two times 24
and six times x,
which means two times 24 or 48
must equal six times x which is 6x.
And now, to solve this equation we divide both sides by six.
48 divided by six is 8.
On the right, six divided by six simplifies to one.
One times x is x, and we get eight equals x,
which is equivalent to x equals eight.
And of course we get the same solution.
Before we look at a second example, though,
there are a couple other ways
that we could have solved this proportion.
Using what we know about equivalent fractions,
notice how looking at the denominators,
because six times four is equal to 24,
we could've multiplied two by four
and if two times four is eight,
which would also give us x equals eight.
We could've also noticed that
since two times three is equal to six,
we could've asked, what number times three is equal to 24,
which would've also given us eight.
But these two methods are only convenient
when we have nice numbers, which is not always the case.
Let's look at a second example.
Let's solve this proportion.
Again, let's solve this proportion,
first, as we solve rational equations,
which means we multiply both sides of the equation
by the LCD to clear the fractions from the equation.
Well, here we have a denominator of three
and a denominator of x,
and, therefore, the LCD is equal to 3x.
Let's multiply both sides of the equation by 3x.
And, again, if it's helpful,
we can write 3x with a denominator of one,
and now we simplify before multiplying.
Here we have a common factor of three.
Three divided by three simplifies to one.
Where there's one, three, and three here and here,
we now just have x times eight, which is 8x.
Equals, on the right,
we have common factor of x.
x divided by x simplifies to one.
And we have three times 12, which is equal to 36.
Now to solve for x, we divide both sides by eight.
Eight divided by eight simplifies to one.
We have x equals 36/8ths, but this does simplify.
36 and eight share a common factor of four.
To simplify,
we divide the numerator and the denominator by four.
36 divided by four is equal to nine,
eight divided by four is equal to two,
which gives us x equals 9/2.
And now let's solve is again using cross multiplication.
The cross products are eighth times x
and three times 12,
which means eight times x or 8x
must equal three times 12, which is equal to 36.
And just like over here, we now divide both sides by eight.
Simplifying, we get x equals 36/8ths,
which we know is equal to 9/2,
giving us the same solution of x equals 9/2.
Notice how for this proportion,
it would not have been as easy to determine
what times eight is equal to 12
to determine what to multiply three by to determine x
and it also would not have been as easy to determine
what times eight is equal to three
to determine what times 12 is equal to x.
I hope you found this helpful.