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Ratios & Proportions. Lesson 8.1. Ratio: a ratio is a quotient of two numbers. a:ba to ba÷b. Always given in lowest terms. 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 is the first term - PowerPoint PPT Presentation
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Lesson 8.1
Ratio: a ratio is a quotient of two numbers.
a:b a to b a÷b
a
b
Always given in lowest terms.
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 = rsThen:
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
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 2bb
x 2yy
= ?
Cross multiply and simplify both sets.
ay = bx
b(x-2y) = y(a-2b)
bx-2by = ay-2by
bx = ay
Yes, they are equal.