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RATIOS AND PROPORTIONS
Ratio: a ratio is a quotient of two
numbers.
a:b a to b a÷b
a
b
Always write a ratio in lowest terms.
The slope of a line is a ratio between
two points. (rise over run)
Proportions: two or more ratios
set equal to each other.
a
b
c
d=
a:b = c:d
a is the first term
b is the second term
c is the third term
d is the fourth term
Product and Ratio Theorems
In a product containing four terms:
First and fourth terms are the extremes.
Second and third terms are the means.
Theorem 59: In a proportion, the product
of the means is equal to the product of the
extremes. (means-extremes product
theorem.)
a
b
c
d= ad = bc
If they aren’t equal, then the ratios
aren’t in proportion.
Theorem 60: If the product of a pair of
non-zero numbers is equal to the product
of another pair of non-zero numbers, then
either pair of numbers may be made the
extremes, and the other pair the means,
of a proportion. (means-extremes ratio
theorem.)
This theorem is harder to state than to use!
Given: pq = rs
Then:
p
r
s
q
p
s
r
q
r
p
q
s= = =
pq = rs pq = rs pq = rs
These proportions are all
equivalent since their cross
products are equivalent
equations.
In a mean proportion,
the means are the same.
4
16
1
4=
a
x
x
r=
4 is the
geometric
mean
x is the
geometric
mean of a
and r
Definition: If the means in a proportion are equal,
either mean is called a geometric mean or mean
proportional between the extremes.
Find the arithmetic & geometry means
between 3 and 27.
Arithmetic mean:
2
273
= 15
Geometric mean:
3
x
x
27=
x2 = 81
x = 9
Solve:
7
14
3
x=
You might want
to reduce the
fraction first.
7x = 42
x = 6
2
3
4
x=
2x = 12
x = 6
Find the fourth term
(sometimes called the
fourth proportional) of a
proportion if the first
three terms are 2, 3,
and 4.
Find the mean proportional(s)
between 4 and 16.
4
x
x
16=
x2 = 64
x = 8 If we are looking for the
length of a segment,
then only the positive
number works.
If 3x = 4y, find the ratio of x to y.
Make x and 3 the extremes and y
and 4 the means.
3x = 4y
x
y
4
3=
Is
a
b
x
y= equal to
a 2b
b
x 2y
y= ?
Cross multiply and simplify both sets.
ay = bxb(x-2y) = y(a-2b)
bx-2by = ay-2by
bx = ay
Yes, they are equal.