# 0960 chapter 3

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```					3.1 Changing App Probs into Equations
How do different words and phrases relate to
mathematical operations?

What words do you know that indicate a
certain operation?

Refer to the chart on p 180

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3.1 Changing App Probs into Equations
One of the things you will be asked to do in
this section is translate from words to
symbols and from symbols to words:
Given a mathematical expression, provide
an expression in words that would be
equivalent:

2x + 3

2
Given a mathematical expression, provide an
expression in words that would be equivalent:

2x + 3                 • Three more than
twice a number
• The sum of twice a
number and three
• Twice a number,
increased by three
a number

3
Given an expression in words, provide a
mathematical expression that would be equivalent:

• Four less than three   • 3x – 4
times a number
• Five more than twice   • 5 + 2x
a number
• Three times the sum    • 3(x + 8)
of a number and 8
• Twice the difference
of a number and 4      • 2(x – 4)

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Consider this:           • 2(x – 4)
Two times a number
minus 4
Which expression does
that represent?        • 2x – 4
What if I read it with
inflection and
pauses?
Be careful!!!

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Given a sentence in words, provide a
mathematical equation that would be equivalent:

Three times the difference between a
number and two is four less than six times
the number
3(x – 2) = 6x – 4
When four is added to a number, the sum
will be twenty
x +4 = 20

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Steps for solving a word problem
understand it (not until you know how to solve
it, but until you understand what the story is.)
2. Take notes. (they only have to make sense to
you, but they should include all the important
information in the problem)
3. Define the variable (and any other unknown
quanities.
4. Write an equation.
5. Solve the Equation.
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3.1 Changing App Probs into Equations

In this section, you are learning how to pick
out the single variable and how to define

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Steps for solving a word problem
understand it (not until you know how to solve
it, but until you understand what the story is.)
2. Take notes. (they only have to make sense to
you, but they should include all the important
information in the problem)
3. Define the variable (and any other unknown
quanities.
4. Write an equation.
5. Solve the Equation.
3.2 Solving Application Problems
Two subtracted from 4 times a number is 10.
Find the number.
n = number
4n – 2 = 10
+2 +2
4n = 12
4    4
n=3

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3.2 Solving Application Problems
The sum of two numbers is 26. Find the two
numbers if the larger number is 2 less than
three times the smaller number.

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NOTES: sum is 26
Larger is 3(sm) -2
?find both numbers?

3n-2 =Larger number
n     =Smaller number
smaller + larger = 26
n + (3n – 2)= 26
4n – 2 = 26
4n = 28
n=7
smaller is 7; larger is 19
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3.2 Solving Application Problems
A company currently produces 1200 widgets
per year. They plan to increase production
by 550 widgets each year until they reach
their goal of 4500 widgets produced in a
year. How many years will it take them to
reach their goal?

13
1200/yr starting point
550/yr increase
4500/year goal
?how long will it take?
y = number of years
starting + inc = goal
1200 + 550y = 4500
550y = 3300
y=6
it will take 6 years.

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3.2 Solving Application Problems
Joe is moving and needs to rent a truck. It
will cost him a flat fee of \$60/day plus
\$0.40/mile. He has \$92 budgeted for the
truck rental. How many miles can he drive
without going over the amount he has
budgeted?

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\$60/day
\$0.40/mile
\$92 total
?how many miles?
m = miles
60 + 0.40 m = 92
0.40 m = 32
m = 80
he can drive 80 miles.

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3.2 Solving Application Problems
If I went out to eat and spent exactly \$20
including tax (7.5%), how much was the
pretax price of my meal?

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spent 20
including 0.075 tax
?pretax price?

p = pretax price
price + tax = total spent
p + 0.075p = 20
1.075p = 20
p = 18.60
my meal was \$18.60 before tax.
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3.2 Solving Application Problems
I received a raise in 2007 of 18% over what I
what did I make in 2006?

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18% raise (2007)
43,000 in 2007
?make in 2006?

m = 2006 salary
06salary + raise = 07salary
m + 0.18m = 43,000
1.18m = 43,000
m = 36,440.68
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3.2 Solving Application Problems
I am considering two sales jobs: one where I
would earn \$450 base and 3%
commission on my sales. The second I
would earn 0 base but 10% commission
on sales. What would my weekly sales
need to be in order for my earnings to be
equal?

21
450 base + 3% com
0 base + 10% com
?sales to be equal?
w=weekly sales
450 + 0.03w = 0+0.10w
450=0.07w
6428=w
if I had weekly sales of \$6428, my earnings
would be equal.

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3.3 Geometric Problems
I plan to build a rectangular patio where the
length is 8 feet more than the width. The
perimeter of the patio will be 56 feet. Find
the length and width. P=2L + 2W

w

w+8
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Length is 8 more than width
Perimeter is 56 feet
?find L and W?
W + 8 = length
W = width
2L + 2W = P
2(w + 8) + 2w = 56
2w + 16 + 2w = 56
4w + 16 = 56
4w = 40
w = 10
Width =10 ft; length = 18 ft   24
3.3 Geometric Problems
A triangle has two congruent angles. The
third angle is 30°greater than the two
equal angles. Find all three angles.
(angles of a triangle add up to 180°)

2

1        3

25
There are two = angles
3rd angle is 30 more than other two
?find all three angles?
Angle1=x
Angle2=x
Angle3=30 + x
x + x + (30 + x) = 180
3x + 30 = 180
3x = 150
x = 50°
The angles measure
50°, 50° and 80°            26
3.3 Geometric Problems
In a trapezoid, the bottom angles are 15
degrees less than twice the top angles.
Find all the angles.

27
Bottom angles are 15 less than twice the top
Bottom angle = 2x - 15
Bottom angle = 2x - 15
Top angle = x
Top angle = x
Sum of all angles = 360
x+x+(2x-15)+(2x-15)=360
6x – 30 =360
6x = 390
X=65°
The angles are 65°, 65°, 115° and 115°       28
3.4 Motion, Money, Mixture Problems
A family went canoeing. The parents are in
one canoe and the children in another.
Both canoes start at the same time from
the same point and travel in the same
direction. The parents paddle at 2 miles per
hour and the children at 4 miles per hour.
How long until the canoes will be 5 miles
apart?

parents 2 mph   5 miles

kids   4 mph                                  29
3.4 Motion, Money, Mixture Problems
?how long until they are 5 miles apart
R         x      T = D
(P) 2 mph X          T = 2T
(K) 4 mph X          T = 4T
5 miles

4T – 2T = 5
2T = 5
T = 2.5 hours

It will take 2 ½ hours for them to be 5 miles apart.
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3.4 Motion, Money, Mixture Problems
My husband and I go running at the same
time but starting from different locations
and running toward each other. We are 6
miles apart when we start . My husband
runs 0.5 mph faster than I run. After twenty
minutes, we meet. How fast were we each
running?
Me                Hubby
20 min
R                 R + 0.5

6 miles                         31
?how fast is each running?
R         x       T = D
R mph        X     1/3 = (1/3)R
R+0.5mph X         1/3 = (1/3)(R+0.5)
6 miles
My distance + his distance = 6 miles
(1/3) R + (1/3) (R + 0.5) = 6
R + R + 0.5 = 18 (after clearing fractions)
2R + 0.5 = 18
2R = 17.5
R = 8.75 mph (my rate)
r+ 0.5 = 9.25 (hubby’s rate)
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3.4 Motion, Money, Mixture Problems

I have \$15,000 to invest. The two
investments I am considering are:
-a mortgage investment that has an 11%
return
-a cd investment that has a 5% return
How much should I put into each investment
if I want to earn \$1500 interest in one
year?

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?how much should I put into each investment?

P X          R x          T =       I
(M) X              0.11         1         0.11x
(CD) 15000-x       0.05         1   0.05(15000-x)
\$1500
Int(m) + int(cd) = 1500
0.11x + 0.05(15000-x) = 1500
11x + 5(15000-x) = 150000        (after clearing dec)
11x + 75000 – 5x = 150000
6x + 75000 = 150000
6x = 75000
x = 12,500
\$12,500 goes into the mortgage and \$2,500 into the CD
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3.4 Motion, Money, Mixture Problems
The community playhouse is putting on a
play. Tickets are \$15 for adults and \$8 for
students. They have sold 361 tickets for a
total of \$4841. How many adult tickets and
how many student tickets have they sold?

35
?how many adult tickets and student tickets were
sold?
price x number sold = income from tickets
(A) 15     361-x        15(361-x)
(S) 8      x                  8x
\$4841
15(361-x) + 8x = 4841
5415 – 15x + 8x = 4841
5415 – 7x = 4841
-7x = -574
X= 82 student tickets sold
361-x = 279 adult tickets sold
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3.4 Motion, Money, Mixture Problems
A candy store sells licorice sticks for
\$4.50/lb and licorice balls for \$2.75/lb. If
they want to sell a new mix of the two for
\$3.75/lb, how many pounds of licorice
balls will they need to put with 5 pounds of
licorice sticks?

37
?how many pounds of licorice balls will they put
with 5 pounds of licorice sticks?
price     x     quantity     =    value
(st) 4.50               5           4.5(5)
(bl) 2.75               x           2.75x
(mx)3.75                5+x         3.75(5+x)
4.5(5) + 2.75x = 3.75(5+x)
450(5) + 275x = 375(5+x)
2250+275x = 1875 + 375x
2250 = 1875 + 100x
375 = 100x
3.75 = x
3.75 pounds of licorice balls would be needed.
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3.4 Motion, Money, Mixture Problems

A Chemistry teacher is preparing for
tomorrow’s lab when he realizes that he
needs a 15% base solution but only has
on hand the same type base solution in
10% and 25%. How much of the 10%
solution will he need to add to 6 liters of
the 25% solution in order to make a 15%
base solution?

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?how much 10% solution shall be added to the 6
liters of 25% solution to create 15%?
strength x quantity (L) =         total
0.10                    x                 0.10x
0.25                    6                 0.25(6)
0.15                    x+6            0.15(x+6)
0.10x + 0.25(6) = 0.15(x+6)
10x + 25(6) = 15(x+6)
10x + 150 = 15x + 90
150 = 5x + 90
60 = 5x
12=x so 12 liters of 10% solution
40

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