Introduction to Problem Solving

```How much is a Right Answer worth?
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This is a course in Algebraic Skills
It is not a course in Right Answers
It is not a course in Good Guessing
Skills are proven by showing Work
Word Problems exercise both your language
Work hard in this section – it’s important
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Sect. 1.4
Introduction to Problem Solving
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The 5-Step Strategy
with letters for unknowns
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Familiarizing means
Understanding a Word Problem
BOOK
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Skim over the whole problem statement.
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Reread it: Verbalize it, perhaps aloud.
Find the Question asked and think how to
reword it. Pick meaningful variable letters.
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HW: write problem number and brief
summary of the problem
23. Day Care. A family living in Boston,
Massachusetts pays \$1089 per month for
full-time day care for their toddler. This is
11/4 of what they paid for comparable day
care when they lived in Billing, Montana.
How much did child care cost in Billings?
HW: Write it as a partial Answer
[gap(s) where information will go]
HW: Put the variables(s) next to the gap(s)
HW
23. Day Care - \$1089 Boston, find cost
(b) \$
for Billing care
Billings probably cheaper
As needed, look up formulas & definitions
HW: Sketch diagrams if needed, make notes
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HW: Set up tables that correspond to
formulas, label each column & row
Understanding is a critical step …
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You still need to Translate, Solve and Check
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25. Photography -
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It will take you (t)
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hrs to go 25 miles upstream.
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It will take you (t) 2 79 hrs to go 25 miles upstream.
2. Translate to an Algebraic Formula
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What kind of formula should be used?
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Choose meaningful letters for variables,
and refine the table according to the formula
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Write the formula: 25 = 9t and Solve
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Number-Value Problems
Some problems deal with quantities that have a value. In these
problems, we must distinguish between the number of and the
value of the unknown quantity. For problems such as these, we
will use the relationship
Number  value = total value
Examples:
1. How much did 4 used Xbox games cost if they are \$22.50 each?
2. IBM stock rose \$2.40/share. What is my profit for 50 shares?
3. It cost me \$120 for DVDs priced at \$6. How many did I buy?
4. Quon bought 83 old records for \$180. What’s each one worth?
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3. Price before tax.
The price of the home theater system is (p) \$ 549
Price + Tax = Total Price
p + 0.06p = 581.94
1.06p = 581.94
p = 581.94/1.06
p = 549
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Geometric Problems
These types of problems rely on formulas from geometry.
Examples:
1. Perimeter: Of a rectangle 2W + 2L = P
Of a triangle P = S1 + S2 + S3 Of a circle P = πd
2. Area: Of a rectangle W(L) = A
Of a triangle A = B(H)/2 Of a circle A = πr2
3. Triangles: Sum of 3 angles = 180 degrees
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4. Cutting the skylight glazing.
The glazing should be cut into two pieces,
one (a) 80 in. long, the other (b) 120 in. long
Perimeter of a square= 4 times length of one side
If s = length of a side of the little skylight,
a = 4s = cut length of the shorter piece of glazing and
b = 4(1.5s) = 6s = length of the longer glazing
Total glazing length= sum of the two perimeters
200 = 4s + 6s
200 = 10s
20 = s
a = 4s = 4(20) = 80
b = 6s = 6(20) = 120
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Number Problems
Some problems deal with several numbers or quantities, but
only give you the total number. They also give you some
relationship between the numbers. For problems such as
these, we will need to express both numbers using different
expressions using a single variable name.
Examples:
1. Two sequential mile-markers add up to 249. What are they?
2. One number is three times another. Their sum is 76. They are?
3. Stocks are \$3 more than bonds. Ten each cost \$230. Each cost?
4. Ten coins are worth \$1.70. How many quarters and nickles?
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5. Three different numbers.
The largest of the three numbers is 93 .
first number = x
second number = y = 3x – 6
third number = z = (2/3)x + 2
x + 3x – 6 + (2/3)x + 2 = 150
(14/3)x – 4 = 150
(14/3)x = 154
(3/14)(14/3)x = (154) (3/14)
x = 33
y = 3(33) – 6 = 99 – 6 = 93
z = (2/3)(33) + 2 = 22 + 2 = 24
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Rate/Time/Distance Problems
These problems deal with the formula (rate)(time)=distance.
Sometimes, wind and river current speed are present. For
problems like these, it is often helpful to build an RT=D table
to organize the known numbers and unknown quantities.
Examples:
1. Bill walks north at 3mph, Sue walks south at 4mph. After 4 hours,
how far apart are they?
2. It takes Pat 6 hours to row upstream and 4 hours to row back
downstream. The current is 2mph. What is Pat’s rowing speed?
3. A UFO goes west coast-to-coast in 22 minutes, against the 100mph
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jet stream. Going east only takes 18 minutes. UFO speed?
Rate · Time = Distance
Example – SEARCH AND RESCUE
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Two search-and-rescue teams leave base camp at the same time,
looking for a lost child. The first team, on horseback, heads north
at 3 mph, and the other team, on foot, heads south at 1.5 mph.
How long will it take them to search a distance of 18 miles
between them?
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Meaningful variable names:
t – time
State the form of the answer:
It will take the search teams ______
4 hours to search all 18 miles.
Formula?
distance = (combined rate) times (time)
18 = (3 + 1.5)t
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t = 18 / 4.5 = 4
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Next time
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1.5 Formulas, Models & Geometry
1.6 Properties of Exponents
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