1.5 A Problem Solving Plan Using Models

GOAL 1Translate verbal phrases into algebraic expressions.

GOAL 2Use a verbal model to write an algebraic equation or inequality to solve a real-life problem, such as making a decision about an airplane’s speed.

What you should learn

To solve real-life problems such as finding out how many plates of dim sum were ordered for lunch.

Why you should learn it

GOAL 1 TRANSLATING VERBAL PHRASES

1.5 A Problem Solving Plan Using Models

Learn the vocabulary in the table on page 32!!!

NOTE: Order will be important, even in addition and multiplication. The question to ask is, “What number do I start with?”

Example: “A number increased by 3” is written as n + 3.

“3 increased by a number” is written as 3 + n.

EXAMPLE 1

Extra Example 1

Translate the phrase into an algebraic expression.

a. Six less than 4 times a number y.

b. Three more than the difference of five and a number n.

c. A number y decreased by the sum of 8 and the square of another number x.

4y – 6 4y

5 – n + 3

y – (8 + x2)

CheckpointTranslate the phrase into an algebraic expression.

a. Eight more than the sum of two times a number y and three.

b. Four less than the sum of five and twice a number n.

(2y + 3) + 8

(5 + 2n) - 4

GOAL 2 USING A VERBAL MODEL

When translating a verbal phrase or sentence look for the word “is.” If it is there, you are nearly always looking at an equation or inequality.

1.5 A Problem Solving Plan Using Models

“is”: equation

“is less than” or “is greater than”: inequality

Remember to look for this 2-letter word!

We will write mathematical models to represent real-life situations. These models may be expressions, equations, or inequalities. We will use the steps shown in the text to solve these situations.

EXAMPLE 2

Writing an Algebraic Model

VERBAL MODEL

LABELSALGEBRAIC

MODELSOLVE CHECK

Extra Example 2

You and your friends go to a music store to buy CD’s on sale for $6 each. Together you spent $77.76, which included a tax of $5.76. How many CD’s were bought? Use the problem solving plan shown in the text.

VERBAL MODEL

Cost per CD •

Number of CDs = Bill – Tax

LABELS $6 n $77.76 $5.76

ALGEBRAIC MODEL

6n = 77.76 – 5.76

SOLVE6n = 72 n = 12

CHECK Is 12 a reasonable answer? Yes.

ANSWER: You bought 12 CDs.

CheckpointEight friends went to a restaurant for dinner. The waiter gave them a bill for $130. At the register a tax and tip of $30 was added to the bill. Use a verbal model and mental math to find about how much each person should contribute to pay an equal share of the bill.

$20

# of friends •

Amount each pays

= Bill +Tax and tip

8 a $130 $30

8a = $130 + $30

Extra Example 3

An investor finds a mutual fund that has an annual return of 10%. The investor wants to earn $250 in simple interest by the end of one year. Use the formula I = Prt.

a. What should be the minimum amount invested?

b. If the investor wants to earn twice the amount of simple interest, should the investment or the amount of time be doubled? Explain.

EXAMPLE 3

VERBAL MODEL

LABELS

ALGEBRAIC MODEL

SOLVE

CHECK

Interest = Principal • • Rate Time

$250 p 1 yr0.10

yr0.10

$250 1 yryr

p

$250 = p • 0.10 Use mental math: $250 is one-tenth of what number?$2500 = p

Is $2500 reasonable? YES.

ANSWER: The amount to invest is $2500.

Extra Example 3 (cont.)

Extra Example 3 (cont.)To earn twice as much, the investor has two options. Either twice as much money can be invested ($5000), or the $2500 can be invested for twice as long.

Checkpoint1. A salesperson drives at a speed of 50 miles per hour.

When he is 175 miles from his destination, he remembers he has a meeting in 3 hours.

a. At his current speed, will he be on time for his meeting?

b. At what minimum speed should he travel from this point on if he wants to be on time for the meeting?

No

about 60 mi/h

QUESTIONS?