Transcript Chapter 2 Section 7 - Lamar County School District

Chapter 2
Equations,
Inequalities and
Problem Solving
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Bellwork:
V
1. Solve N  R 
for V.
G
V = G(N – R)
F
2. Solve B 
for V.
P V
F
V P
B

2
S

2

r
2
3. Solve S  2rh  2r for h. h 
2

r


S  4lw
4. Solve S  4lw 2wh for h. h 
2w

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2.7
Percent, Ratio, and
Proportion
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Objectives:
Solve
percent equations
Solve problems involving
percents
Solve proportions
Solve problems modeled by
proportions
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Strategy for Problem Solving
General Strategy for Problem Solving
1. UNDERSTAND the problem.
• Read and reread the problem.
• Choose a variable to represent the unknown.
• Construct a drawing.
2. TRANSLATE the problem into an equation.
3. SOLVE the equation.
4. INTERPRET the result:
• Check proposed solution in problem.
• State your conclusion.
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Solving a Percent Problem
A percent problem has three different parts:
amount = percent · base
Any one of the three quantities may be unknown.
1. When we do not know the amount:
n = 10% · 500
2. When we do not know the base:
50 = 10% · n
3. When we do not know the percent:
50 = n · 500
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Solving a Percent Problem: Amount Unknown
Example: What is 9% of 65?
n  9%  65
n  (0.09)(65)
n  5.85
5.85 is 9% of 65
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Solving a Percent Problem: Base Unknown
Example: 36 is 6% of what?
36  6%  n
36  0.06n
36
0.06n

0.06 0.06
600  n
36 is 6% of 600
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Solving a Percent Problem: Percent Unknown
Example: 24 is what percent of 144?
24  n 144
24  144n
24 144n

144 144
0.16  n
2
16 %  n
3
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2
24 is 16 % of 144
3
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Solving Markup Problems
Example:
Mark is taking Peggy out to dinner. He has $66 to spend. If he wants
to tip the server 20%, how much can he afford to spend on the meal?
Let n = the cost of the meal.
Cost of meal n
100% of n
+
+
tip of 20% of the cost
20% of n
120% of n
1.2n  66
1.2n
66

1.2
1.2
n  55
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=
=
=
$66
$66
$66
Mark and Peggy can spend
up to $55 on the meal itself.
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Solving Discount Problems
Example:
Julie bought a leather sofa that was on sale for 35% off the original
price of $1200. What was the discount? How much did Julie pay
for the sofa?
Discount = discount rate  list price
= 35%  1200
The discount was $420.
= 420
Amount paid = list price – discount
= 1200 – 420
= 780
Julie paid $780 for the sofa.
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Solving Increase Problems
Percent of increase =
amount of increase
original amount
Example:
The cost of a certain car increased from $16,000 last year to
$17,280 this year. What was the percent of increase?
Amount of increase = original amount – new amount
= 17,280 – 16,000 = 1280
Percent of increase =
=
amount of increase
original amount
1280
16000 = 0.08 The car’s cost increased by 8%.
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12
Solving Decrease Problems
Percent of decrease =
Example:
amount of decrease
original amount
Patrick weighed 285 pounds two years ago. After dieting, he reduced
his weight to 171 pounds. What was the percent of decrease in his
weight?
Amount of decrease = original amount – new amount
= 285 – 171 = 114
amount of decrease
original amount
114
Patrick’s weight decreased
=
=
0.4
285
by 40%.
Percent of decrease =
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Proportions
The ratio of a number a to a number b is their
quotient.
A proportion is a mathematical statement that two
ratios are equal.
Examples
1 3

2 6
a c

b d
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x 7 x 9

5
3
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Cross Products
a c
 , then ad = bc.
If
b d
**This is cross-multiplication**
a
c


b d
means
ad = bc
**This is how we solve a proportion**
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Example
Solve for x:
a) 3
9

8 x
3 x  8 9
3x  72

3x 72

3
3
x  24
b)

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y
5

y 16 3
y 3  5y 16
3y  5y  80
3y  5y  5y  5y  80
2y  80
2y 80

2
2
y  40
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Example
To estimate the number of people in Jackson, population
50,000, who have no health insurance, 250 people were
polled. Of those polled, 39 had no insurance. How many
people in the city might we expect to be uninsured?
39
x

250 50000
39 out of 250 uninsured
39 50000  250x
How many out of 50,000?
1950000  250x
1950000 250x

250
250
7800  x
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Example
Which is a better buy for the same brand of toothpaste?
8 ounces for $2.59
or
10 ounces for $3.11
**Find the unit price, which is price divided by quantity**
$2.59
8oz
0.32375
or
or
$3.11
10oz
0.311
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32¢ per oz
or
31¢ per oz
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Closure:
1.
What is a proportion?
2.
How do we solve/simplify
proportional equations?
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