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A
RATIO is a relationship between two quantities expressed as a
fraction.
Ratio Example:
or we can write 10 pencils : 1 package
Spoken, this is “10 pencils per 1
package” or “10 pencils in 1 package.”
Ratio Example: Given the
previous ratio, if I have 5 packages of pencils, how many pencils
do I have?
Well, start with the ratio and multiply by given
information to arrive at the answer like this:
* 5 packages = 10 * 5 = 50 pencils
Ratio Example:
Using the same ratio, if I have 40 pencils, how many packages did
I start with?
First of all,
is equivalent to 
Each package has ten pencils, or ten pencils are in each package.
* 40 pencils =
 = 4 packages
Notice in these last
two examples, to answer the question, we write the known ratio so that the quantity
in question appears in the numerator of the fraction, then we
multiply by the other known quantities, 5 packages and 40 pencils respectively.
This method is the basis of what later will be called unit analysis
in Chemistry and in Physics (discussed below).
Ratio Example:
12 inches are in a foot, and in 1 mile we have 5280 feet, how many inches are
in 3 miles?
The ratios we have are:
 and
The question asks
how many inches are in 3 miles; inches should be in the numerator and miles
should be in the denominator. Well, the first ratio has inches in the
numerator; the second ratio has miles in the numerator as well, so we must
invert the second ratio to place miles in the denominator to arrive at the
answer like so:
*  =
But the question is
asking about 3 miles. Multiplying by 1 in terms of 3/3 we write the previous
result into the form that answers the question, like so:
*
 = 
however, the
question is not asking for the ratio, it is asking for the number of inches in
3 miles, that number is 190080, which is the answer.
Ratio Example:
2.54 centimeters are in 1 inch. How many yards are in a meter?
The ratio given us
is  , we know that 12 inches
are in a foot and 3 feet are in a yard, which gives us yards in the numerator
of the requested ratio. This means we need to invert the given ratio before we
begin to get inches in the numerator, which will eventually lead to feet then
to yards in the numerator.
Also, we know that
100 centimeters are in a meter, and this ratio gives us the denominator in the
answer. So let's begin:
 *
 *
 *
 =
 =
The question asks
how many yards are in ONE meter, so we need to multiply this result by 1
written as (1/91.44) / (1/91.44) like so:
*
 =
 =
1.09 is the
answer to the question.
Note that whenever
the question desires a quantity in terms of a unit of another quantity, simply
divide the numerator in the answer by the denominator in the answer. In this
case we divide 100 by 91.44 to arrive at 1.09.
We start with the
known ratio, possibly inverted, so the units are in the proper position for the
answer, to do this we had to invert the given ratio. Then we multiple by one
ratio after the other to get the answer, canceling units along the way as is
shown below:
inches cancels
with inches, feet cancel with feet, and finally centimeters cancels with
centimeters, leaving yards in the numerator and meters in the denominator.
This method of
multiplying one ratio after the other to convert units is called
“Unit Analysis.”
A
PROPORTION is the statement of equality between two or more ratios.
Since ratios are
fractions, and a proportion is the statement “one fraction equals another
fraction,” then we're simply working with equivalent fractions. ...equivalent fractions review.
2 : 3 = 16 : 24 “2 is to 3 as 16 is to 24”
1 / 2 = 3.5 / 7 “1 is to 2 as 3.5 is to 7”
1 : 3 : 4 = 5 : 15 : 20 “1 is to 5 as 3 is to 15
as 4 is to 20”
In general we have a
: b as c : d means 
=
At
this point it may help to review
Rational Numbers.
We can cross multiply and get a * d = b * c.
As long
as a and c are not zero, we can invert both ratios and get
 =
These
two operations allow us to solve conditional proportions (also called
conditional equations.)
Ratio Example:
 =
 ==> x * 45 = 12 * 15
= 180
x = 180 / 45 = 4
or we
could simply multiply both sides of the equation by 15 and immediately
get: x
=  =  = 4
Ratio Example:
 =
7 * y = 8 * 3 (cross multiply)
y = 24 / 7
or
invert:  =
24/7 = y (multiply both sides by 3)
Ratio Example: John
bought 30 music cds at a flea market for $12. How much would he have paid for 6
cds?
 =
invert:  =
$12/5 = $2.40 (multiply both sides by 6 cds)
Ratio Example:
The visiting team attracted 1/3 of the fans that the home team
attracted. If the visiting team attracted 300 fans, how many fans
did the home team attract?
1 is to 3 as 300 is to X
1 : 3 = 300: X
900 = x (invert then multiply by 300)
answer:
900 (As a side note here, the total number of fans that came to the game
would be 300 + 900 = 1200.)
Ratio Example:
Sarah can run 100 yards in 2 minutes. How far can she run in 3 minutes?
This is a rate problem, Sarah's rate is 100/2 =
50 yards per minute.
= 150 yards (note how minutes cancel)
this problem can also be done using a proportion:
we have
3 * 50 = 150 yards = x (multiplying both sides by 3)
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