pre-activity solving percent problems · pdf file491 pre-activity preparation while shopping...

491

PRE-ACTIVITY

PREPARATION

While shopping at the end-of-winter 60%-off clearance sale, you fi nd a great sweater to purchase, originally priced at $70. If you also use a coupon for 20% off the discounted price, how much will you pay for the sweater? Many similar practical applications may be familiar to you as a consumer. Discounts, tips, interest on loans, earnings on investments, and taxes all require percent calculations.

Percents are also used for presenting growth and loss fi gures in business and industry, for determining markups on merchandise in retail, for describing the components of solutions in science and medicine, for sports statistics, and for test scores and grades, to name just a few more of their many applications.

In this section, you will combine your knowledge and understanding of proportions with that of percents, and learn two methodologies you can use to solve a basic percent problem.

Solving Percent Problems

LLEARNINGEARNING OOBJECTIVESBJECTIVES

• Determine the unknown and known quantities in a percent problem.• Use the percent proportion or the percent equation to solve a percent problem.• Test the validity of the solution to a percent problem.

TTERMINOLOGYERMINOLOGY

NEW TERMS TO LEARN

amount

base

percent equation

percent proportion

relative size

PREVIOUSLY USED

equation

of

percent number

proportion

ratio

unknown

variable

Section 5.2

492 Chapter 5 — Percents

BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE

Recall that the term of, as in “calculate a percent of another number,” or “calculate a fraction of another number” means to multiply the percent or fraction by the other number.

Think about a familiar application of percents—calculating 6% sales tax on a $200 purchase:

$12 = 6% of $200

The parts of this statement can be matched to the basic percent equation for percent problems:

It is also true that the amount divided by the base results in the percent:

Using the fact that a percent might also be written as the ratio of the percent number to 100,

the equation can also be written as the percent proportion:

For example, 25% of 40 means = of 825 40 103

4% ( ).× 00 means =

3

480 60× ( ).

sales tax sales tax rate

cost of purchase

Amount = Percent × Base

$12 6% $200

$

$%

12

2006= or Amount

BasePercent=

In the sales tax example, $

$

12

200

6

100=

AmountBase

=percent number

100

12 ÷ 200 = 0.06 = 6%THINK

Note: The denominator for the percent number will always be 100.

493Section 5.2 — Solving Percent Problems

In a percent problem,

• The percent is easily recognizable:

Example 1: 50% of 124 is 62 Example 2: 110% of 80 is 88

• The base follows the phrase “…percent of” (or “…% of” ):


• The amount is the result of multiplying the percent times the base. In a percent problem, the amount is usually associated with the word “is.”


The three quantities in the percent equation or the percent proportion—the amount, the base, and the percent—will take on different values for different situations. In a percent problem, any one of the three might be the unknown quantity to solve for.

The keys to solving a percent problem are to identify the unknown quantity and to determine the two known quantities.

Consider Example 1 again: 50% of 124 is 62

If the amount is unknown, the following are the possible ways the percent problem might be stated:

“What (number) is 50% of 124?” 50% of 124 is how much?

“50% of 124 is what number?” “Find 50% of 124.”

“How much is 50% of 124?”

If the base is unknown, the problem might be stated:

“62 is 50% of what number?” or “50% of what number is 62?”

If the percent is unknown, the problem might be stated:

“62 is what percent of 124?” or “What percent of 124 is 62?”


Steps in the Methodology Example 1 Example 2

Step 1

Identify the unknown.

Identify the unknown quantity. Percent is unknown.

Step 2

Write the percent proportion.

Write the percent proportion.

Step 3

Set up the proportion.

Identify and substitute the two known quantities into the percent proportion.

Use a variable for the unknown quantity.

Amount = 11.52

Base = 256 (follows “of”)

Percent number = n

Solve for the percent number.

Step 4

Solve the proportion.

Solve the proportion.

Solving a Percent Problem with the Percent Proportion

Example 1: 11.52 is what percent of 256?

Example 2: What percent of 136 is 47.6?

►►

►► Try It!

MMETHODOLOGYETHODOLOGY

The fi rst option for solving for the unknown quantity in a percent problem is to use the percent proportion. You will be familiar with this from your work with solving proportions.

AmountBase

=percent number

100

THINK

11.52256

= n100

11 52 100 256

256

25611522564 5

1

1

.

.

× = ×

× = ×

=

=

n

n

n

n

11.52 100256

the percent number

)256 1152 0

1024

1280

1280

0

4 5..

−

−



Step 5

Present the answer.

Present your answer.

If the unknown quantity in Step 1 is a percent, present your answer as a percent.

4.5%

Step 6

Validate your answer.


In your original proportion, replace the variable with your answer and cross-multiply.

If you were solving for a percent, substitute the percent number into the percent proportion.

Note: An alternate way to validate is to substitute your answer into the percent equation. (See Models 1, 2, and 3.)

11.52256

=

× = ×=

4.5100

11 52 100 4 5 2561152 1152

. .

?

?

MMODELSODELS

Model 1

What is 2.5% of 280?

Step 1 Amount is unknown.

Step 2

Step 3 Amount = a Base = 280 percent number = 2.5

Step 4

Step 5 Answer: 7

Step 6 Validate:

AmountBase

percent number100

=

a280

2 5100

= .

a

a

a

× = ×

= × =

=

1

1100

100

2 5 280100

2 5 280100

700100

7

.

.

280

2 5

1400

5600

700 0

× .

.

THINK

7280

2 5100

7 100 2 5 280700 700

=

× = ×=

.

.

?

?7 = ×= ×=

2 5 2807 0 025 2807 7

. %

.

??

AlternateValidation:


Model 2

324 is 48% of what number?

Step 1 “of what number?” Base is unknown.

Steps 2 & 3

Step 4

Step 5 Answer: 675

Step 6 Validate:

AmountBase

percent number100

=

324 10048

48

4832 400

48675

1

1× = ×

=

=

b

b

b

,

324 48100b

=

)48 32400

288

360

336

240

240

0

675

−

−

−

324 48100

324 100 48 67532 400 32 400

675=

× = ×=, ,

?

?

324 48324 0 48 675324 324

= ×= ×=

%.

675??

Model 3

35 is what percent of 826? Round to the nearest hundredth percent.

Step 1 Percent is unknown. Solve for the percent number.

Steps 2 & 3

Step 4

Step 5 Answer: 4.24%

Step 6 Validate:

AmountBase

percent number100

= 35826 100

= n

35 100826

826

82635008264 24

1

1× = ×

=

≈

n

n

n.

)826 3500 000

3304

1960

1652

3080

2478

6020

5782

238

4 237 4 24.. .

−

−

−

−

≈

35826

35 100 4 24 8263500 3502 24

=

× = ×≈

4.24100.

.

?

?35 82635 0 0424 82635 35 0224

= ×= ×≈

4.24%..

??

the percent number

Close because of rounded percent number.





Step 1

Identify the unknown.

Identify the unknown quantity. Percent is unknown.

Step 2

Write the percent equation.

Write the percent equation. Amount = Percent × Base

Step 3

Set up the equation.

Identify and substitute the two known quantities into the percent equation.

Use a variable for the unknown quantity.

Amount = 11.52

Base = 256 (follows “of”)

Percent = n

11.52 = n × 256

Solve for the percent.

Step 4

Solve the equation.

Solve the equation.

When the percent is given, use the decimal form of the percent for computing the amount or base.

(See Models 1 and 2, on pages 498 and 499.)

Solving a Percent Problem with the Percent Equation

Example 1: 11.52 is what percent of 256?

Example 2: What percent of 136 is 47.6?

►►

►► Try It!

MMETHODOLOGYETHODOLOGY

Your second option for solving a percent problem is to use the percent equation. You will recognize its steps from solving proportions.

11 52256

256

2560 045

1

1.

.

= ×

=

n

n

the percent in its decimal form

)256 11 520

1024

1280

1280

0

0 045..

−

−



Step 5

Present the answer.

Present your answer.

If the unknown quantity in Step 1 is a percent, present your answer as a percent.

0.045 is the percent in its decimal form.

0.045% = 4.5%

Step 6



In your original equation, replace the variable with your answer and validate.

If you were solving for a percent, substitute the decimal form of your answer for the variable and validate.

Note: An alternate way to validate is to substitute your answer into the percent proportion. (See Models 1, 2, and 3.)

MMODELSODELS

Model 1

What is 2.5% of 280?

Step 1 Amount is unknown.

Step 2 Amount = Percent × Base

Step 3 Amount = a

Base = 280 a = 2.5% × 280

Percent = 2.5%

Step 4 Use the decimal form of the percent to compute the answer.

a = 0.025 × 280 a = 7

Step 5 Answer: 7

Step 6 Validate:

THINK

11 52 0 25611 52 11 52

.

. .= ×=

.045

256

0 045

1280

10240

11 520

× .

.

?

7 = ×=

0 025 2807 7

.? 7280

2 5100

7 100 2 5 280700 700

=

× = ×=

.

.

?

?

280

025

1400

5600

7 000

×.

.



Model 2

324 is 48% of what number?

Step 1 Base is unknown.

Steps 2 & 3 Amount = Percent × Base 324 = 48% × b

Step 4

Step 5 Answer: 675

Step 6 Validate:

Model 3

35 is what percent of 826? Round to the nearest hundredth percent.

Step 1 Percent is unknown.

Steps 2 & 3 Amount = Percent × Base 35 = n × 826

Step 4

Step 5 0.0424% = 4.24% Answer

Step 6 Validate:

3240 48

0 48

0 48675

1

1..

.= ×

=

b

b

)0 48 324 00

288

360

336

240

240

0

675. .−

−

−

675

0 48

5400

27000

324 00

× .

.

324 0 48324 324

= ×=

. 675? 324 48100

324 100 48 67532 400 32 400

675=

× = ×=, ,

?

?

)826 35 00000

3304

1960

1652

3080

2478

6020

5782

238

0 04237 0 04.. .

−

−

−

−

≈ 22435826

826

8260 0424

1

1= ×

=

n

n.

35826 100

35 100 4 24 8263500 3502 24

=

× = ×≈

4.24

..

?

?35 82635 0 0424 82635 35 0224

= ×= ×≈

4.24%..

??

0 0424

826

2544

8480

339200

35 0224

.

.

×

Alternate Validation:

Close because of rounded percent number.

324 = 0.48 × b

the decimal form of the percent



Model 4 A Comparison of the Two Methodologies When Solving for the Percent

189 is what percent of 84?

Using the percent proportion: Using the percent equation:

Step 1 Percent is unknown. Percent is unknown.

Step 2 Amount = Percent × Base

Step 3 Amount = 189 Amount = 189

Base = 84 Base = 84

n = percent number n = percent (as a decimal)

189 = n × 84

Step 4

Step 5 Answer: 225% 2.25% = 225% Answer

Step 6 Validate: Validate:

AmountBase

percent number100

=

18984 100= n

189 10084

84

8418900

84225

1

1× = ×

=

=

n

n

n

18984

84

8418984

2 25

1

1= ×

=

=

n

n

n.

the percent numberthe percent in decimal form

)84 18900

168

210

168

420

420

0

225

−

−

−

)84 189 00

168

210

168

420

420

0

2 25..

−

−

−

18984

189 100 225 8418 900 18 900

=

× = ×=

225100

, ,

?

?

225

84

900

18000

18900

×189 84189 2 25 84189 189

= ×= ×=

225%.

??

2 25

84

900

18000

189 00

.

.

×

If you choose the percent proportion methodology to solve a percent problem, you can estimate your answer as you would when solving any kind of proportion problem; and when you use the percent equation, you can estimate your answer to the fi nal multiplication or division step.

However, even before you decide which of the two methodologies to use, you can make a reasonable prediction about your answer as it relates to the two known quantities in the problem (its relative size).

How Estimation/Prediction Can Help


Solving for the amount or base when the other is given and the percent is known,

• When the known percent < 100%, the amount will be < the base (base > amount).

• When the known percent > 100%, the amount will be > the base. (base < amount).

Example: 10% of 60 is what? Example: How much is 250% of 40?

10% < 100% 250% > 100%

Predict: amount < 60 Predict: amount > 40

Actual answer: amount = 6 Actual answer: amount = 100 10% of 60 is 6 100 is 250% of 40

Example: 15 is 20% of what number? Example: 125% of what number is 50?

20% < 100% 125% > 100%

Predict: base > 15 Predict: base < 50

Actual answer: base = 75 Actual answer: base = 40 15 is 20% of 75 125% of 40 is 50

Solving for the percent when you know the amount and base,

• When the amount < the base, the percent will be < 100%.

• When the amount > the base, the percent will be > 100%.

Example: 30 is what percent of 120? Example: What percent of 8 is 20?

30 < 120, amount < base 20 > 8, amount > base

Predict: percent < 100% Predict: percent > 100%

Actual answer: percent = 25% Actual answer: percent = 250% 30 is 25% of 120 250% of 8 is 20

THINK THINK

THINK THINK

THINK THINK

P B P B

PA P A

B ABA


AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS

Issue Incorrect Process Resolution Correct Process Validation

Incorrectly identifying the amount and the base

What is 50% of 120?

50% of 120 is 240.

The amount is associated with “is,” and the base follows “of” in a percent problem.

What is 50% of 120?

Amount unknown = a

Base = 120

percent number = 50

Misinterpreting the answer when solving for the unknown percent

What percent of 300 is 12?

When the percent is the unknown and you solve by using the percent proportion, your solution is the percent number. Attach the percent sign to the percent number for the answer.

When you solve by using the percent equation, your solution is the decimal form of the percent. Convert to a percent for the answer.


120 50100

120 100

50

50

50240

2

1

1

1

b

b

b

=

× = ×

=a

a

a

12050

100

2

2

1202

60

1

2

1

1

=

× =

=

aaa

= ×= ×=

50 1200 50 12060

%.

60 0 5 12060 60= ×=

.

60120

50100

60 100 50 1206000 6000

=

× = ×=

?

?

?

12300 100

12 100

300

300

300

44 00

4 1 1

113

=

× = ×

==

n

n

n. 400%

AB

percent number100

= AB

percent number100

=

12300 100

12 100

300

300

300

4

4 1 1

113

=

× = ×

=

n

n

n4%

12 300

12300

300

3000 040 04

1

1

= ×

= ×

==

n

n

n.. 0.04%

12300

4100

12 100 4 3001200 1200

=

× = ×=

12 0 04 30012 12= ×=

.

OR

OR

12 300123000 040 04

= ×

=

==

n

n

n.. 4%

)300 12 00

12 00

0

0 04..

.

−

)300 12 00

12 00

0

0 04..

.

−

00 51

0

b

=

0001×

0

0 =

0404 .0

0

4 =0.0

OR

?

?

?


Issue Incorrect Process Resolution Correct Process Validation

Using the incorrect form of the given percent when solving for the unknown amount or base in the percent equation

What is 5.5% of 80?

A = Percent × Base

a = 5.5 × 80

a = 440

5.5% of 80 is 440

When solving for the unknown in the Amount = Percent x Base equation, you must use the decimal form of the given percent.

What is 5.5% of 80?

A = Percent × Base

a = 5.5% × 80

a = 0.055 × 80

a = 4.4

5.5% of 80 is 4.4

Incorrectly presenting a percent answer


Always attach the percent sign (%) to the answer when you solve for an unknown percent.


80

5 5

400

4000

440 0

× .

.

0 055

80

000

4400

4 400

.

.

×

4 4 0 055 804 4 4 4. .. .= ×=

4 480

5 5100

4 4 100 5 5 80440 440

. .

. .

=

× = ×=

?

OR

?

1768 100

17006825

=

=

=

n

n

n

Answer: 25

1768 100

17006825

=

=

=

n

n

n

Answer: 25%

1768

25100

17 100 25 681700 1700

=

× = ×=

?

?

)68 1700

136

340

340

0

25

−

−

=

440

Perce= P

.5

r 25

?


PPREPARATION REPARATION IINVENTORYNVENTORY

Before proceeding, you should have an understanding of each of the following:

the terminology and notation associated with translating the language of a percent problem to equation form, including the percent proportion

how to identify the known and unknown quantities in a percent problem

how to set up the unknown and the known quantities in an equation

how to validate your answer to a percent problem

505

ACTIVITY Solving Percent Problems

PPERFORMANCE ERFORMANCE CCRITERIARITERIA

• Setting up the equation for a percent problem correctly:

– correct identifi cation of the unknown– appropriate substitution of the known quantities into the equation

• Solving the percent equation or the percent proportion correctly– correct presentation of the answer– validation of the answer

CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS

1. What are the four parts of a percent proportion?•

•

•

•

2. What are the three parts of a percent equation?•

•

•

3. What are the key words or phrases that will identify the given parts and the unknown part in a percent problem?

Amount

Base=

percent number

100 or Amount = Percent Bas× ee

Section 5.2


4. When using the percent equations to solve for the amount or base, what form of the percent must you use to compute the answer?

5. How can you predict the relative size of your answer to a percent problem?

6. How do you validate your answer to a percent problem?

7. What is the difference in the two Methodologies for Solving Percent Problems when the unknown is the percent?


TTIPS FOR IPS FOR SSUCCESSUCCESS

DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING

1. Identify the base, amount, and percent for the following problems. Use a variable for the unknown. The fi rst one is given for you.

Problem Base Amount Percent

a) 12 is 15% of what number? n 12 15%

b) What percent of 25 is 8?

c) Find 2.69% of 32.

d) 95 is what percent of 300?

e) 20% of what number is 8.8?

f) 0.25% of 440 is what number?

g) What is 12.5% of 250?

h) What percent of 95 is 323?

i) 0.3% of what number is 90?

j) 0.4 is what percent of 125?

• Read the problem carefully. Underline the key words and symbols in the problem, such as “is” (which signals the amount), “of” (which signals the base), and the percent sign.

• When solving for an unknown percent, be attentive to whether you are solving for the

percent number (when using the percent proportion , ) or the

decimal form of the percent (when using the percent equation, Amount = Percent × Base)

• Before solving, predict the size of the unknown quantity relative to the known quantities—

amount < base means the percent < 100%; and percent < 100% means amount < base

amount > base means the percent > 100%; and percent > 100% means amount > base

• A way to recall the set-up of the percent proportion is

• Reduce, when possible, to simplify the calculation(s).

"is"

"of "=

percent number

100

Amount

Base=

percent number

100


2. Predict the relative size of the answer for each of the following percent problems. Without solving, fi ll in each blank with the “less than” < or “greater than” > symbol.

a) 25% of 200 is ______ 200.

b) 130 is what percent of 50? The percent is ______ 100%

c) 150% of what number is 24? The base number is ______ 24.

d) 104% of 40 is ______ 40.

e) 62 is 20% of what number? The base number is ______ 62.

f) 1.6% of 50 is ______ 50.

g) What percent of 35 is 15.75? The percent is ______ 100%.

3. Use the Percent Proportion Methodology or the Percent Equation Methodology to solve each of the following percent problems.

Problem Worked Solution Validation

a) 12 is 15% of what number?

b) 8 is what percent of 25?

c) Find 2.69% of 32.

MENTAL MENTAL MATHMATH



d) To the nearest hundredth percent, what percent of 300 is 95?

e) 8.8 is 20% of what number?

f) What number is 0.25% of 440?

g) 12.5% of 250 is what number?



h) What percent of 95 is 323?

i) 0.3% of what number is 90?

j) 0.4 is what percent of 125?


TEAM EXERCISESTEAM EXERCISES

1. Tony’s monthly income, after deductions, is $1800. Rent accounts for 34% of his income. Internet, cable, and phone combined account for 9%, 15% goes for food, and car expenses (including his loan) total 14.5%. What percent remains to cover his other expenses?

2. One situation might generate several kinds of percent questions. Answer each of the following. Validate your answers.

a) Suppose there are nineteen women and eleven men in a creative writing class. What percent represents the number of women in the class? That is, “19 is what percent of 30?”

b) If 40% of the class earned an “A” grade on the mid-term project, how many students earned an “A”? That is, “What is 40% of 30?”

c) If this quarter’s enrollment of 30 students is up 25% from last quarter, how many students were enrolled last quarter? That is, “30 is 125% of what number?”


In the second column, identify the error(s) you fi nd in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column.

Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

1) Find 1½% of 30. Did not use the decimal form of 1½% in the percent equation.

A = P × B

2) 36.6 is 2% of what number.

3) 0.2% of 31,500 is how much?

IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS

112

1 5 0 015% . % .= =

A or

= ×=

0 015 300 45 45

.. .

.

.

01530

0000450

0 450

× Answer:0.45

AB

percent number=

100

. .

. .

4530

1 5100

45 100 1 5 3045 45

=

× = ×=

?

?


Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

4) What percent of 45 is 1.62?

5) 288 is what percent of 64?


ADDITIONAL EXERCISESADDITIONAL EXERCISES

Solve the following percent problems and validate your answer for each.

1. 130 is what percent of 50?

2. 62 is 20% of what number?

3. What is 1.6% of 50?

4. What percent of 35 is 15.75?

5. Find 0.7% of 520.

6. 14.4 is what percent of 90?

7. 80 is 0.4% of what number?

8. How much is 140% of 25?

9. 5% of what number is 7?

10. To the nearest hundredth percent, what percent of 68 is 12?

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