Transcript 1.1 Problem Solving with Fractions • Addition words  Plus, more, more than, added to, increased by, sum, total, sum of, increase of, gain.

1.1 Problem Solving with Fractions
•
Addition words
 Plus, more, more than, added to, increased by,
sum, total, sum of, increase of, gain of
•
Subtraction words
 Less, subtract, subtracted from, difference,
less than, fewer, decreased by, loss of, minus,
take away
1.1 Problem Solving with Fractions
•
Multiplication words
 Product, double, triple, times, of, twice, twice
as much
•
Division words
 Divided by, divided into, quotient, goes into,
divide, divided equally per
•
Equals
 Is, the same as, equals, equal to, yields, results
in, are
1.1 Problem Solving
•
Changing word phrases to expressions:
The sum of a number and 9
7 minus a number
Subtract 7 from a number
The product of 11 and a number
5 divided by a number
x+9
7-x
x–7
11x
5
x
The product of 2 and the sum of a 2(x + 8)
number and 8
1.1 Problem Solving
•
Equation: statement that two algebraic
expressions are equal.
Expression
x–7
No equal sign
Can be evaluated or
simplified
Equation
x–7=3
Contains equal sign
Can be solved
1.1 Problem Solving with Fractions
• Solving Application Problems
1. Read and understand the problem
2. Know what is given and work out a plan
to answer what is to be found.
3. Estimate a reasonable answer
4. Solve the problem by using the facts
given and your plan
1.1 Problem Solving with Fractions
• Estimating a reasonable answer:
which of the following would be a
reasonable cost for a man’s shirt?
1.
2.
3.
4.
$.65
$1
$20
$1000
1.2 Adding and Subtracting
Fractions
• Adding fractions with the same denominator:
a c
ac
 
b b
b
• Subtracting fractions with the same denominator:
a c
ac
 
b b
b
1.2 Adding and Subtracting
Fractions – Factor Trees
18
6
3
2
3
1.2 Adding and Subtracting
Fractions
•
To add or subtract fractions with different
denominators - get a common denominator.
• Using the least common denominator:
1. Factor both denominators completely
2. Multiply the largest number of repeats of each
prime factor together to get the LCD
3. Multiply the top and bottom of each fraction
by the number that produces the LCD in the
denominator
1.2 Adding and Subtracting Fractions –
no common factors in denominator
• Adding fractions with different denominators:
a c
ad  bc
 
b d
bd
• Subtracting fractions with different denominators:
a c
ad bc
 
b d
bd
1.2 Adding and Subtracting
Fractions
• Try these:
1 5
 ?
9 9
5
2

?
7 21
5 1
 ?
9 4
1.2 Adding and Subtracting
Fractions
• Proper fraction – numerator is less than the
denominator
• Improper fraction - numerator is greater
than the denominator
• Mixed fraction – sum of a fraction and a
whole number
1.2 Adding and Subtracting
Fractions
• Converting a mixed fraction to an improper
fraction: 3 3  3  8  3  27
8
8
8
• Converting an improper fraction to a mixed
fraction: 35
3
9
Divide 9 into 35:
35
8
3
9
9
9
35
27
8
1.3 Multiplying and Dividing
Fractions
•
•
Multiplying or dividing the numerator
(top) and the denominator (bottom) of a
fraction by the same number does not
change the value of a fraction.
Writing a fraction in lowest terms:
1. Factor the top and bottom completely
2. Divide the top and bottom by the greatest
common factor
1.3 Multiplying and Dividing
Fractions
• Multiplying fractions:
a c
a c
 
b d
bd
• Dividing fractions (multiply by the reciprocal):
a c
a d
ad
   
b d
b c
bc
1.3 Multiplying and Dividing
Fractions
• Try these:
12
(simplify)
16
7 3

?
9 14
9
3
 ?
10 5
1.3 Multiplying and Dividing Fractions
• Converting decimals
125
25
5
1
fractions:
0.125 



1000
• Converting fractions
to decimals:
200
40
8
.3 7 5
3
 8 3.0 0 0
8
2.4
.6 0
.5 6
.0 4 0
.0 4 0
2.1 Solving Equations
• A linear equation in one variable can be written in the
form: Ax + B = 0
• Linear equations are solved by getting “x” by itself on
one side of the equation
• Addition Property of Equality: if A=B then
A+C=B+C
• Multiplication Property of Equality: if A=B and C is
non-zero, then AC=BC
• General rule: Whatever you do to one side of the
equation, you must also do it to the other side.
2.1 Solving Equations
• Some equations have more than one term
with the same variable. These are called
“like terms”
8x  2x  5x
• Like terms can be combined by adding the
coefficients: 8 x  2 x  5 x  5 x
9 y  6 y  3y
3 z  12z  15z
2.1 Solving Equations
• Example of solving an equation:
7 k  12  2k  8
5k  12  8
5k  20
k 4
2.2 Applications of Equations
•
1.
2.
3.
4.
5.
Translate the following:
The sum of a number and 16
Subtract a number from 5.4
The product of a number and 9
The quotient of a number and 11
Four-thirds of a number
2.2 Applications of Equations
•
When 5 times a number is added to twice the
number, the result is 10. Find the number.
1. x is the variable representing the number.
2. Equation: 5 x  2 x  10
3. Solve:
7 x  10
x
4. Check:
10
7
 1 73
10
5( 10
)

2
(
)
7
7
50
7

20
7

70
7
 10
2.3 Formulas
• I = PRT
• Interest = principal x rate x time
• M = P(1 + RT) • Maturity value
• G = NP
• Gross sales = number of items
sold x price per item
• S=C+M
• Selling price = cost of the item +
markup
2.3 Formulas
• Example: Solve for T in the formula:
M  P(1  RT )
1. Distribute:
M  P  PRT
2. Subtract P from M  P  PRT
both sides
3. Divide by PR
M P
T 
PR
2.4 Ratios and Proportions
• Ratio – quotient of two quantities with the
same units
Examples: a to b, a:b, or
a
b
Note: percents are ratios where the second
number is always 100:
35
35%  100
2.4 Ratios and Proportions
• Proportion – statement that two ratios are
equal
a
c
Examples: b
d

Cross multiplication:
if
a
b

c
d
then
ad  bc
2.4 Ratios and Proportions
• Solve for x:
81
x

Cross multiplication:
81  7  9  x
567  9  x
so x = 63
9
7
3.1 Writing Fractions and
Decimals as Percents
• Write a decimal as a percent by moving the
decimal point 2 places to the right and
attaching a percent sign:
• Example:
0.382  38.2%
3.1 Writing Fractions as Percents
• Write a fraction as a percent by converting
the fraction to a decimal and then
converting the decimal to a percent:
.3 7 5
• Example:
3
8
0.375  37.5%
 8
3.0 0 0
2.4
.6 0
.5 6
.0 4 0
.0 4 0
3.1 Writing Fractions and
Decimals as Percents
• Write a percent as a decimal by moving the
decimal point 2 places to the left and
removing the percent sign:
• Example:
3.41  341 %
3.1 Writing Fractions and
Decimals as Percents
• Write a percent as a fraction by first
changing the percent to a decimal then
changing the decimal to the fraction and
reduce:
• Example:
45
95
9
45%  0.45 


100 20  5 20
3.1 Writing Fractions and
Decimals as Percents
• Write a fractional percent as a decimal by
first changing the fractional part to a
decimal and leaving the percent sign. Then
move the decimal point 2 places to the left
and removing the percent sign:
• Example:
3 %  3.25%  .0325
1
4
3.2 Finding the Part
• B = Base – the whole or the total
• R = Rate – a number followed by “%” or
“percent”
• P = Part – the result of multiplying base
times rate
P( part)  B(base)  R(rate)
3.2 Finding the Part for a Business Problem
• B = Base – sales, R = Rate – sales tax rate,
P = Part – sales tax
• Example: If the sales tax rate is 5%, what is
the sales tax and total sale on $133 of
merchandise P  B  R
P  $133 5%  $133 .05
P  $6.65
3.2 Identifying the Base and the Part
Usually the Base
Sales
Investment
Savings
Retail Price
Value of Real Estate
Total Sales
Value of Stocks
Earnings
Original
Usually the Part
Sales Tax
Return
Interest
Amount of Discount
Rents
Commission
Dividends
Expenditures
Change
3.3 Finding the Base
• Using the Basic Percent Equation to solve
for Base:
22.5 is 30% of _____
22.5  30%  B
22.5  0.3B
22.5
B
0 .3
225
B
 75
3
3.3 Finding the Base
• Finding sales when sales tax rate is given:
The 5% sales tax collected by a store was
$380. What was the total amount of sales?
5% of B  $380
.05  B  $380
$380
B
 $7600
0.05
3.3 Finding the Base
• Finding the amount of an investment:
The yearly maintenance cost of an apartment is
2½% of its value. If maintenance is $37,000 per
year, what is the value of the apartment complex?
2 12 % of B  $37000
.025 B  $37000
$37000
B
 $1,480,000
0.025
3.3 Finding the Base
• Finding the base if rate and part are different
quantities:
United Hospital finds that 25% of its employees
are men and 720 are women are women. What is
the total number of employees?
First – if 25% are men, then the percent of women
= 100-25 = 75% 75% of B  720
.75  B  720
720
B
 960
0.75
3.4 Finding the Rate
• Using the percent equation to solve for rate:
45 is what percent of 180?
_____%of 180  45
180R  45
45
1
R
  0.25  25%
180 4
Note: Rate is always expressed as a percent
3.4 Finding the Rate
• Finding rate of return when the amount of
return and the investment are known:
$3400 is invested in a new computer
yielding additional income of $1700. What
is the rate of return?
_____%of $3400 $1700
3400R  1700
1700 1
R
  0.50  50%
3400 2
3.4 Finding the Rate
• Solving for the percent remaining:
A car is expected to last 10 years before it
needs replacement. If the car is 7 years old,
what percent of the car’s life remains?
To find the number of years remaining
subtract 7 from 10 to get 3 years left.
_____%of 10  3
10R  3
3
R
 0.30  30%
10
3.4 Finding the Rate
• Find the percent of increase/decrease:
Sales of digital cameras went from $40,000
to $100,000. Find the percent increase.
Increase = $100,000 - $40,000 = $60,000
_____%of $40,000 $60,000
40,000R  60,000
60,000
R
 1.5  150%
40,000
3.5 Increase and Decrease
Problems
• Increase Problem:
Original + Increase = New value
(base)
(part)
• Decrease Problem:
Original - Decrease = New value
(base)
(part)
3.5 Increase and Decrease Problems
• The value of a house is $143,000 this year. That is
10% more than last year’s value. What was the
value of the home last year?
Last year’s value + 10% of last year’s value = this
year’s value
100% B  10%  B  $143,000
110%  B  $143,000
1.1B  $143,000
143,000
B
 $130,000
1 .1
3.5 Increase and Decrease Problems
• Finding the base after 2 increases:
This year’s production of widgets was 144,000. It
is 20% more than last year’s production which
was also 20% more than the previous year’s
production. Find the number of widgets produced
2 years ago. To find last year’s # of widgets:
100% B  20%  B  144,000
120%  B  144,000
1.2 B  144,000
144,000
B
 120,000
1.2
3.5 Increase and Decrease Problems
• Widget problem (continued).
To get the # of widgets produced 2 years ago:
100% B  20%  B  120,000
120%  B  120,000
1.2 B  120,000
120,000
B
 100,000
1.2
3.5 Increase and Decrease Problems
• Decrease problem:
Craig paid $450 for an LCD TV set. The price he
paid was 10% less than the original price. What
was the original price?
100% B  10%  B  $450
90%  B  $450
.9 B  $450
$450
B
 $500
.9