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Today we begin our study of mathematical models by looking at how we can build mathematical expressions from English descriptions of situations. That is, the dreaded word problems.

A model is a representation of something, like a model airplane. A mathematical model is a representation of something in the form of mathematical expressions or equations or systems of equations.

Here are the top three uses of models:

1. Predict the future (where will a hurricane be an hour or a day from now?)

2. Save money on experiments (use a model of an airplane rather than a real one).

3. Perform unethical or impossible experiments (what will be the side effects of a new drug).

To describe real-world situations using algebra we must build algebraic expressions that represent English phrases.

10 more suggests we add 10 but we must be careful to translate correctly.

Is it c + 10 or c - 10? Are you sure?

ANS: c - 10


To make sure the mathematical model matches the verbal model, it's a good idea to check the translations. Below are two methods that can be used to check the correctness of translations:

• Relative sizes check In this method, we use intuition to determine which item should have more and then see if the algebra is consistent with this. In the example where we have ten more chairs than students, intuition tells us there should be more chairs. Comparing the algebraic expressions, we see that c – 10 (the expression for the number of students) is always smaller than c (the variable for the number of chairs). Since intuition and the algebra are in agreement, we are more confident that the translation is correct.

• Concrete numbers check In this method, we make up a number for the given variable and then use intuition and the algebraic expression to calculate the other number. If the numbers agree, our confidence in the translation will increase. In the students and chairs example, let's say we have 30 students. Intuition tells us that there should be 40 chairs (10 more than the number of students). The translation, c – 10, also gives 30 when we replace c with 40. Since intuition and the algebra match, we think the translation is correct.

Number of women Number of men 88m


Number of studentsNumber of professors12

12s

Here we can write the relation as follows:

Total costLet represent

Then represents

cost of juice

40 cost of juiSo 5

cost of cancost

ce400

50

of c

2

an

4010 2

5

xxx

x

x

xx

The can costs 5 cents. The juice costs 5 + 40 = 45 cents. Together they cost 50 cents.


We will look at several different "types" of traditional word problems:

Simple Interest: When you borrow money for school or a car you have to pay the person you borrowed the money from for the use of their money. What you pay them is called interest.

Remember to convert the 1.5% per month into percent per year by multiplying by 12.

0.015$50001

i Prt

Imonth

12 months

1 year

0.75 year

$675


Total interest interest from 5% interest from 9%

First, we must figure out how much interest we want (6% of $10,000 for 9 months)

1year9monthsTotal In 0.0$10, 6t 00012mont

erest $hs

450r

tP

P trI

So, we want to earn $450 in interest after 9 months.

Now, we figure out how much interest we will get from the CDs at 5%:

Let x = amount invested at 5%

0.0Interest at 5% 0.03752

5 91

PrI

x

t

x

Now, figure out how much interest we will get from the bonds at 9%

Since we have a total of $10,000 and x of it is invested in CDs the remainder must be invested in bonds. So, 10,000 - x = amount invested at 9%

Interest at 9% 675 0.00.10 09,000 67912

5

P

x

r

x

tI

Putting these together we have:

Total interest interest from 5% interest from 9%450 0.0375 675 0.0675450 675 0.03

7500

x xx

x

So, we should invest $7,500 at 5% and 10,000 - 7,500 = 2,500 at 9%

CHECK:

Total amount invested = 7,500 + 2,500 = 10,000

Total interest at 5% = (0.05)•(7,500)•(0.75) = 281.25

Total interest at 9% = (0.09)•(2,500)•(0.75) = 168.75

Total interest = 168.75 + 281.25 = $450


We will do more difficult problems where we are not simply evaluating a formula. Let's use this procedure:


1. Read problem carefully.

2. Draw diagram.

3. Identify formulas needed. None.

4. Assign a variable to represent what you are looking for.

x = # gal cream

5. Express other unknown quantities in terms of your variable.

50-x = # gal milk

6. Write an equation that models a relationship between your expressions.

Butterfat in Cream + Butterfat in Milk = Butterfat in Mix

0.25x + 0.035(50 - x) = 0.125(50)

7. Solve equation.

x = 20.93 (rounded)

8. Write final answer, including units.

20.93 gal cream

50 – 20.93 = 29.07 gal milk

9. Check answer with the facts in the problem.

Gal cream + Gal milk = 50 gal

20.93 + 29.07 = 50 CHECKS

Butterfat in Cream + Butterfat in Milk = Butterfat in Mix

0.25(20.93) + 0.035(29.07) = 0.125(50) ?

5.23 gal + 1.02 gal = 6.25


Here is one more problem if we have time:

1. Read problem carefully.

2. Draw diagram.

3. Identify formulas needed.

4. Assign a variable to represent what you are looking for.

x = # tens I chose # tens for x because the # fives is given in terms of the # tens.

5. Express other unknown quantities in terms of your variable.

2x = # fives

31 - x - 2x = number of ones

6. Write an equation that models a relationship between your expressions.

Value of all bills = value of ones + value of fives + value of tens

133 = (31 - x - 2x)•1 + 2x•5 + 10x

7. Solve equation.

x = 6

8. Write final answer, including units.

6 tens, 2(6) = 12 fives, 31 – 6 – 12 = 13 ones

9. Check answer with the facts in the problem.

6 + 12 + 13 = 31 bills

6 * 10 + 12 * 5 + 13 * 1 = 133

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