F. Ratio
and Proportion
A
ratio is a fraction that compares two numbers. The ratio of x
to y is written as x : y or x/y. Ratios are usually used to compare
quantities of the same type, for example, the ratio of the length
of a Toyota to the length of a Cadillac. We would not form the
ratio of the length of a Toyota to the cost of a Cadillac.
A proportion
states that two ratios are equal. Two ratios involve four numbers:
two numerators and two denominators. Most often, one of these
four numbers is not known; it is found by equaling the two ratios,
such as
2 = 6
15 x
The unknown x is then found by
cross multiplying:
2x = 15(6)
therefore, x = 45
Two
quantities are directly proportional if one is a constant
times the other: x = cy (where c is a constant). They are inversely
proportional (or indirectly proportional) if one is
a constant divided by the other: x = c/y, or equivalently, xy
= c. If a quantity is stated to be proportional to another, the
word directly is implied, so if x is stated to be proportional
to y, it means x = cy.
To
decide if two quantities are directly or inversely proportional,
we ask the question, "Do the quantities both increase or
decrease or does one increase while the other decreases?"
If they both increase or decrease, they are directly proportional;
if one increases while the other decreases, they are inversely
proportional.
To
solve an equation that represents a direct proportion, such as
x = cy, we set up the equation as
x1 = y1
x2 y2
where the subscript 1 refers
to the first situation and the subscript 2 to the second situation.
If the equation results from an inverse proportion, such as xy
= c, we have
x1 = y2
x2 y1
To
solve problems involving proportions, 3 of the 4 numbers will
usually be known and the problem will be to calculate the fourth.
Example 21
Calculate x if 4:15 = 16:x
Solution
The equation is written in a more obvious form as
Example 23
If an airplane travels 1,200 miles in 2.5 hours, how far will
it travel in 10 hours?
Solution
This represents a direct proportion: both the distance traveled
and time increase. Consequently, if we let x = distance the airplane
will travel, we have
1200 = 2.5
x 10
12,000 = 2.5x
x = 4,800 miles
Example 24
What is the ratio of
2/3 to 5/4?
Solution
The ratio is (2/3)/(5/4) which is equal to 2/3 × 4/5 =8/15.
