MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
Course Web Site
http://www.papademas.net/occ/math114
Textbook Companion Web Site
http://wps.prenhall.com/chet_cleaves_cmupdate_7
Welcome to Week 8
This Week’s Agenda
•
•
•
Review Chapter 8 in the Course Textbook ( Ratio and Proportion )
Commence completion of Lab Assignments 15 and 16
( download from the course Web site )
Commence completion of Homework Assignment(s) 8
( download from the course Web site )
Chapter Sections
http://www.interactmath.com
© Copyright 2012 by P.E.P.
1
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
Ch 8: Ratio and Proportion
Section 1: Ratio and Proportion
Section 2: Direct and Joint Variation
Section 3: Inverse and Combined Variation
Ch 8: Ratio and Proportion
Section 1: Ratio and Proportion
Section 2: Direct and Joint Variation
Section 3: Inverse and Combined Variation
Chapter 8 Ratio and Proportion ( Glossary Terms )
combined variation
Variables that are related through both direct and inverse variation are known as a
combined variation. Here two variables are necessary for a third to be determined.
An example equation of combined variation is: z = k x 2 / y
Here z varies directly as the square of x and inversely as y .
direct variation
A direct variation equates two variables through some constant of proportionality.
y = k x is an example of a direct variation where variable y is directly proportional
to variable x via the constant of proportionality k .
direct variation
x 
y 
both variables increase with each other
x 
y 
both variables decrease with each other
If your years of service with the company increase, then your salary tends to increase.
If temperature decreases then pressure also decreases.
inverse variation ( indirect variation )
An indirect variation equates two variables through some constant of proportionality.
y = k / x is an example of an indirect variation where variable y is indirectly
proportional to variable x via the constant of proportionality k .
© Copyright 2012 by P.E.P.
2
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
indirect variation
x 
y 
one variable increases while the other variable decreases
x 
y 
one variable increases while the other variable decreases
inverse square variation
This is a variation of the form y = k / x 2
Examine Newton’s Inverse Square Law which is such that when the radius r between
two bodies tends to increase, then the force of attraction between them tends to
decrease.
Newton’s Inverse Square Law
F =(Gm1m2)/r2
Here
F is the force of attraction
G is Universal Gravitation constant 6.674 × 10 − 11 Newton m 2 kg − 11
m 1 is the mass of the first object
m 2 is the mass of the second object
r is the distance between the two masses
joint variation
A joint variation is an equation of the form z = k x y where z is the dependent
variable, x and y are the independent variables and k is known as the constant of
proportionality. For an equation of this type, z is said to vary jointly as x and y .
if x and y are > 1
x 
y 
z 
proportion
An equating of two ratios. It is a common type of equation that contains fractions as a
proportion. ( example: 25 / 100 = x / 150 )
ratio
a fraction comparing a quantity in the numerator or the denominator ( example mpg or
miles per gallon ( actual and not stated ) = typical is 18 mpg )
unit fraction
Equates to 1 but perhaps not units – wise. 2.2 pounds / 1 kg is a unit fraction.
4 quarts / 1 gallon is a unit fraction.
© Copyright 2012 by P.E.P.
3
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
Chapter 8 Ratio and Proportion
Section 8.1 Section 1: Ratio and Proportion
Property of Proportions
Given the proportion
a
b
=
c
d
then a d = b c , provided that b ≠ 0 and d ≠ 0 .
An example proportion is: 25 / 100 = x / 150
Here if we process 25 customer invoices out of 100 or we work at a rate of
25 % , what is your particular number of invoices processed at that same rate
if they give you 150 invoices?
Answer 25 % × 150 or 37.5 invoices.
Solve Equations that are Proportions
[ Example ]
Verify whether or not the following proportion is true.
3
8
=
11
32
[ Solution ]
Perform cross multiplication.
( 3 ) ( 32 ) = ( 11 ) ( 8 )
Verify if the products equate to each other.
96 ≠ 88
Hence an invalid proportion.
© Copyright 2012 by P.E.P.
4
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
[ Example ]
Solve the following proportion.
x
16
=
9
4
[ Solution ]
Perform cross multiplication.
( x ) ( 4 ) = ( 9 ) ( 16 )
or
4 x = 144
or
x = 36
[ Example ]
Solve the following proportion.
3x
14
=
11
36
[ Solution ]
Perform cross multiplication.
( 3 x ) ( 36 ) = ( 11 ) ( 14 )
or
108 x = 154
or
x = 154 / 108
or
x = 77 / 54
[ Example ]
Solve the following proportion.
x + 5
x − 5
=
5
2
[ Solution ]
© Copyright 2012 by P.E.P.
5
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
Perform cross multiplication.
(x + 5 )(2) = (5)(x − 5 )
or
2 x + 10 = 5 x − 25
or
2 x − 5 x = − 10 − 25
or
− 3 x = − 35
or
x = − 35 / − 3
or
x = 35 / 3
Section 8.2 Section 2: Direct and Joint Variation
Solve Problems of Direct Variation Using Proportions
For problems and exercises involving direct proportion, as one quantity
increases ( decreases ) , the other quantity increases ( decreases ) .
Steps to Follow to Construct a Direct Proportion:
(1)
Establish two pairs of related data.
(2)
Write one pair of data in the numerators of the two ratios.
(3)
Write the other pair of data in the denominators of the two ratios.
(4)
Form a proportion using the two ratios.
[ Example ]
Marilyn earns $ 95 for 7 hours of labor in a retail business office. How many
hours must she work to earn $ 285 ?
[ Solution ]
These pairs are directly related since as the hours increase, the pay also
increases.
Construct a direct proportion.
© Copyright 2012 by P.E.P.
6
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
$ 95
$ 285
=
7
x
Perform cross multiplication.
( $ 95 ) ( x ) = ( 7 ) ( $ 285 )
or
95 x = 1,995
or
x = 1,995 / 95
or
x = 21 hours
Solve Problems of Direct Variation Using a Constant of Variation
An example of a direct variation is:
y = $ 1.85 x
Here total the price y is directly related to the items purchased, namely x .
If 10 items are purchased the total price is $ 18.50 . Here the factor $ 1.85 is
considered to be a constant of proportionality.
[ Example ]
An architect's drawing is scaled at 0.25 inches = 4 feet. Find the actual
measurement for a wall that is 2.75 in. on the drawing.
[ Solution ]
0.25 in
4 ft
=
2.75 in
x
Perform cross multiplication.
( 0.25 ) ( x ) = ( 2.75 ) ( 4 )
or
0.25 x = 11
or
x = 11 / 0.25
or
x = 44 feet
© Copyright 2012 by P.E.P.
7
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
Solve Problems of Joint Variation Using a Constant Of Variation
[ Example ]
If w varies jointly as x and y , find w if the constant of variation is 3 ,
x = 56.8 and y = 12.5 .
[ Solution ]
Since w varies jointly as x and y we write:
w = kxy
Substitution yields.
w = ( 3 ) ( 56.8 ) ( 12.5 )
w = ( 3 ) ( 56.8 ) ( 12.5 )
w = 2,130
Solve Problems of Inverse Variation Using Proportions
[ Example ]
If y varies inversely as x , find the constant of variation when x = 42 and
y = 0.05 .
[ Solution ]
Since y varies inversely as x we write:
y = k/x
0.05 = k / 42
or
k = 42 × 0.05
or
k = 2.10
© Copyright 2012 by P.E.P.
8
MATH 114
Applied Mathematics I
Week 8
Ratio and Proportion
Lecture Notes
Student Name
Solve Problems of Inverse Variation Using a Constant of Variation
[ Example ]
In a relationship, y varies inversely as x and the constant of variation is 3.5 .
Find y when x = 0.08 .
[ Solution ]
Since y varies inversely as x we write:
y = k/x
Substitution yields:
y = 3.5 / 0.08
y = 43.75
Solve Problems of Combined Variation Using a Constant of Variation
[ Example ]
z varies directly as the square of x and inversely as y . Find z when x = 3 ,
y = 24 , and the constant of combined variation is 48 .
[ Solution ]
Write an equation of combined variation
z = kx2/y
Here k is used in conjunction with x and y to determine z .
z = ( 48 ) ( 3 ) 2 / ( 24 )
z = ( 48 ) ( 9 ) / ( 24 )
z = 18
© Copyright 2012 by P.E.P.
9