Math 101 Study Session Quiz 3 Chapter 6 Sections 1
through 4
July 28, 2016
Chapter 6 Section 1: Introduction to Equations
An equation expresses the equality of two mathematical expressions.
A solution of an equation is a number that when substituted for the variable, results in
a true equation. The equation 5 = 5 is a true equation, the equation 3 = 7 is not a true
equation.
To solve an equation means to find a solution of the equation.
Addition Property of Equations
The same number can be added to each side of an equation without changing its solution.
In symbols, the equation a = b has the same solution as the equation a + c = b + c.
Multiplication Property of Equations
Each side of an equation can be multiplied by the same nonzero number without changing
the solution of the equation. In symbols, if c 6= 0, then the equation a = b has the same
solutions as the equation ac = bc.
Uniform motion means that the speed and direction of an object do not change.
The uniform motion equation d = rt where d is the distance traveled, r is the rate of
travel, and t is the time spent traveling.
Note that if two objects are traveling in opposite directions, then the rate at which the
distance between them is increasing is the sum of their speeds.
Also note that if two objects are moving toward each other, the distance between them is
decreasing at a rate that is equal to the sum of the speeds.
Chapter 6 Section 2 Proportions
A proportion states the equality of two ratios or rates.
Definition of Proportion
c
a
c
a
If and are equal ratios or rates, then = is a proportion.
b
d
b
d
To solve a proportion, we will use the cross multiplication property.
1
Cross Multiplication Property
c
a
= if and only if ad = bc.
b
d
Chapter 6 Section 3 The Basic Percent Equation
of
translates to ·
is
translates to =
what translates to n
(multiplication)
(equals)
(the unknown number)
The Basic Percent Equation
Percent · base = amount
Note: Look for the number or phrase that follows the word of when determining the base
in the basic percent equation.
1
1
Also Note: 33 % = ,
3
3
2
2
66 % = ,
3
3
2
1
16 % = ,
3
6
1
5
83 % = .
3
6
Proportion Method
percent
amount
=
100
base
Simple Interest Formula
I = P rt
where I is the interest earned, P is the principal amount or the amount invested, r is the
simple interest rate, t is the time.
Percent Mixture Equation
Q = Ar
where Q is the quantity of a substance in a solution, A is the amount of the solution, and r
is the percent concentration of the substance.
Chapter 6 Section 4 Percent Increase and Percent Decrease
Percent increase is used to show how much a quantity has increased over its original value.
Percent decrease is used to show how much a quantity has decreased from its original value.
2
Below are some examples for us to try with solutions at the end of the study
guide:
Solve each of the following:
1. Is
3
a solution of
4
4x − 3 = 20x + 4?
2. The property tax on $180, 000 is $4320. At this rate, what is the property tax on a
home appraised at $290, 000?
3. Answer the following question using the basic percent equation: “0.05% of 130 is
what?”
4. Answer the following question using the basic percent equation: “36.4 is what percent
of 9.1?”
5. Answer the following question using the proportion method: “4.2 is what percent of
40?”
6. If Kachina Caron invested $1200 in a simple interest account and earned $63 in 9
months, what is the annual interest rate?
7. A carpet is made with a blend of wool and other fibers. If the concentration of wool
in the carpet is 75% and the carpet weighs 169 lb, how much wool is in the carpet?
8. The first occurrence of a sports competition included 12 competitive events. Plans for
the next competition will include 108 competitive events. Find the percent increase in
the number of competitive events in the sports competition.
Solutions to the Study Guide Questions
Solution to 1:
3
To answer the question, we replace each x with :
4
4x − 3 = 20x + 4
3
3
4
− 3 = 20
+4
4
4
12
60
−3=
+4
4
4
3 − 3 = 15 + 4
0 = 19
False
3
3
Hence, is not a solution of 4x − 3 = 20x + 4.
4
Solution to 2:
To find the property tax we write and solve a proportion using p to represent the property
tax.
p
$4320
=
$180000
$290000
4320 · 290000 = 180000 · p
1252800000 = 180000p
1252800000
180000
=
p
180000
180000
6960 = p
Hence, the property tax on a $290, 000 home is $6960.
Solution to 3:
.05
= .0005,
The basic percent equation is P ercent · base = amount, where P ercent =
100
base = 130 and amount = n.
P ercent · base = amount
.0005 · 130 = n
.065 = n
Hence, .065 is .05% of 130.
Solution to 4:
The basic percent equation is P ercent · base = amount, where P ercent = n, base = 9.1, and
amount = 36.4.
P ercent · base = amount
n · 9.1 = 36.4
9.1n = 36.4
36.4
9.1n
=
9.1
9.1
n=4
n = 4 · 100%
n = 400%
Hence 36.4 is 400% of 9.1.
Solution to 5:
The proportion method is
percent
amount
=
100
base
4
where percent = n, base = 40, and amount = 4.2.
percent
amount
=
100
base
4.2
n
=
100
40
40n = 4.2 · 100
40n = 420
40n
420
=
40
40
21
n=
2
n = 10.5%
Hence 4.2 is 10.5% of 40.
Solution to 6:
9
3
The simple interest formula is I = P rt where P = $1200, I = $63, t =
= and r is
12
4
unknown
I = P rt
63 = 1200 ·
3
·r
4
3600
r
4
63 = 900r
900r
63
=
900
900
7
=r
100
.07 = r
.07 · 100% = r
7% = r
63 =
Hence, the simple interest rate is 7%.
Solution to 7:
The percent mixture formula is Q = Ar where A = 169, r = .75, and Q is unknown.
Q = Ar
Q = 169 · .75
Q = 126.75
Hence, there is 126.75 lb of wool in the carpet.
Solution to 8:
5
Use the basic percent equation P ercent · base = amount where P ercent = n, base = 12, and
amount = 108 − 12 = 96.
P ercent · base = amount
n · 12 = 96
12n = 96
12n
96
=
12
12
n=8
n = 8 · 100%
n = 800%
Hence, the number of competitive events in the sports competition will increase by 800%.
Questions for You to Try
Below are some questions for you to try on your own.
1. Is 5 a solution of
z 2 + 2 = 12 + 3z
2. If a 24-gram serving of pasta contains 6 g of protein, how many grams of protein are
in a 454-gram box of the pasta?
3. Answer the question using the basic percent equation: “What percent of 16 is 6?”
4. Answer the question using the basic percent equation: “24 is 320% of what?”
5. Answer the question using the proportion method: “900 is what percent of 60?”
6. How much money must Andrea invest for 2 years in an account that earns an annual
simple interest rate of 8% if she wants to earn $900 from the investment?
7. The concentration of platinum in a necklace is 15%. The necklace weighs 12 g. Find
the amount of platinum in the necklace.
8. Data from the Federal Bureau of Investigation show that the number of hate crimes
in the United States decreased between two recent years, from 7792 incidents to 6647
incidents. What was the percent decrease in hate crimes between these two years?
6
Solutions
1. Yes
2. 113.5 g
3.
75
% or 37.5%
2
4. 7.5
5. 1, 500%
6. 5625
7. 1.8 g
8. 14.7%
7