Proportions, Measurement Conversions, Scale, and Percents

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Proportions, Measurement Conversions, Scale, and Percents. by Lauren McCluskey . Credits. “Prentice Hall Mathematics: Algebra I” “Changing Percents” by D. Fisher “Percent I” by Monica and Bob Yuskaitis “Percent II” by Monica and Bob Yuskaitis

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### Proportions, Measurement Conversions, Scale, and Percents

by Lauren McCluskey

Credits
• “Prentice Hall Mathematics: Algebra I”
• “Changing Percents” by D. Fisher
• “Percent I” by Monica and Bob Yuskaitis
• “Percent II” by Monica and Bob Yuskaitis
• “Percent Formula Word Problems” by Rush Strong
• “Math Flash Measurement I” by Monica and Bob Yuskaitis
Ratios, Rates, and Proportions:
• “A ratio is a comparison of two numbers by division.”
• A rate is a ratio which compares two different units, such as 20 pages /per 10 minutes.
• “A unit rate is a rate with a denominator of 1.” An example of this is miles / per hour.

from Prentice Hall Algebra I

Try It!
• Find the unit rate:

1) \$57 / 6 hr.

2) \$2 / 5 lb.

5) A 10-ounce bottle of shampoo costs \$2.40. What is the cost per ounce?

from Prentice Hall, Algebra I

Proportions:
• “A proportion is an equation that states that two ratios are equal.”
• “The products ad and bc are the cross products of the proportion a/b = c/d.”

Example: 3/12 = x / 24

from Prentice Hall Algebra I

Multi-step Proportions:

Now you try it!

X+ 37

4 8

Use cross products:

4 * 7 = 8(x + 3)

28 = 8x + 24

-24 -24

4= 8x

8 8

x = 1/2

=

7

12

a – 6

5

=

from Prentice Hall Algebra I

7

12

a – 6

5

=

• 7(a – 6) = 5 * 7
• 7a – 42 = 35
• +42 +42
• 7a = 77
• 7
• a= 11
Proportions can be used when:
• Solving Unit Rate problems
• Converting Measurements
• Indirect Measurements via Similar Figures
• Converting between Scale and the actual object/ distance
• Solving Percent problems
Try It! (Unit Rates:)

30) A canary’s heart beats 200 times in 12 seconds. How many times does it beat in 1 hour?

from Prentice Hall Algebra I

Proportions:

31) “Suppose you traveled 66 km in 1.25 hours. Moving at the same speed, how many km would you cover in 2 hours?”

from Prentice Hall Algebra I

Measurement Conversions:

52) “The peregrine falcon has a record diving speed of 168 miles per hour. Write this speed in feet per second.”

from Prentice Hall Algebra I

How large is a millimeter?

The width

of a pin

from “Math Flash Measurement I” by M. and B. Yuskaitis

How large is a centimeter?

The width

of the top

from “Math Flash Measurement I” by M. and B. Yuskaitis

How large is a meter?

of one & 1/2 doors

1 meter

from “Math Flash Measurement I” by M. and B. Yuskaitis

How large is a kilometer?

Whitmore

A little over

1/2 of a

mile

1 kilometer

Walter

White

from “Math Flash Measurement I” by M. and B. Yuskaitis

How large is a milliliter?

drop of

liquid

from “Math Flash Measurement I” by M. and B. Yuskaitis

How large is a liter?

Half of a

large pop

bottle

1

liter

from “Math Flash Measurement I” by M. and B. Yuskaitis

How heavy is a gram?

A paper clip

weighs

gram

from “Math Flash Measurement I” by M. and B. Yuskaitis

How heavy is a kilogram?

A kitten

weighs

kilogram

from “Math Flash Measurement I” by M. and B. Yuskaitis

Measurement Conversions:
• 12m = _________km
• 12m = _________mm
• 12m = _________cm
• 48 in. = _________ft.
• 48 in. = _________ yd.
• 48 in. = _________ mile
Similar Figures:
• “Similar figures have the same shape but not necessarily the same size.
• In similar triangles, corresponding angels are congruent and corresponding sides are in proportion.”

from Prentice Hall Algebra I

What is the missing measure?

x

15cm

8m

8m

5cm

3cm

20cm

4cm

4m

12m

X

Scale:
• “A scale drawing is an enlarged or reduced drawing that is similar to an actual object or place.
• The ratio of a distance in the drawing to the corresponding actual distance is the scale of the drawing.”

from Prentice Hall Algebra I

Scale

22) A blueprint scale is 1 in. : 9 ft. On the plan, the room measures 2.5 in. by 3 in. What are the actual dimensions of the room?

from Prentice Hall Algebra I

Proportions and Percent Equations:

is% OR: part

of 100 whole

n2080=w 8020

100 25 100 25 100 z

=

=

=

Find the percent

Find the part

Find the whole

Understanding Percents:
• Percent can be defined as

“of one hundred.”

of

100

from “Percent I” by M. and B.Yuskaitis

“Cent” comes from the Latin

and means 100.

Many words have come from the root cent such as century, centimeter,

centipede, & cent.

from “Percent I” by M. and B.Yuskaitis

CCC means three

hundred.

from “Percent I” by M. and B.Yuskaitis

Percent and Money
• We write our change from a dollar in hundredths.

If you understand

money, learning

percents is a breeze.

from “Percent I” by M. and B.Yuskaitis

25¢

25¢

25¢

25¢

Comparing Money & Percents
• \$ .25 is ¼ of a dollar
• 25% also means ¼

\$1.00

=

from “Percent I” by M. and B.Yuskaitis

How to Find the Percent of a Whole Number
• The first thing to remember is “of” means multiply in mathematics.

x

of

=

from “Percent I” by M. and B.Yuskaitis

How to Find the Percent of a Whole Number
• Step 1 - When you see a percent problem you know when you read “of” in the problem you multiply.

x

25% of 200

from “Percent II” by M. and B.Yuskaitis

Step 2 – Change your percent to a decimal and then move it two places to the left.

.

.

25% x 200

from “Percent I” by M. and B.Yuskaitis

Step 3 – Multiply just like a regular decimal multiplication problem.

200

x .

25

1000

+400

5000

from “Percent II” by M. and B.Yuskaitis

200

x .

25

1000

+400

.

5000

from “Percent II” by M. and B.Yuskaitis

Percent Problems:

There are 3 types of percent problems:

1) What is ____% of ____?

2) What % of ____ is ____?

3) _____ is ___% of what #?

Problem 1

Brittany Berrier became a famous

skater. She won 85% of her

meets. If she had 250 meets in

2000, how many did she win?

• from ”Percent Formula Word Problems” by R. Strong
What is 85% of 250?

250

● 0.85

1250

20000

21250

So 212.50 is 85% of 250.

Problem 2

Matt Debord worked as a produce manager for Walmart. If 35 people bought green peppers and this was 28% of the

total customers, how many

customers did he have?

• from ”Percent Formula Word Problems” by R. Strong
35 = 28 So what is x? x 100

Use cross products:

3500 = 28x

28 28

X = 125 customers

Problem 3

Brett Mull became a famous D.J.

He played a total of 175 C.D’s in

January. If he played 35 classical

C.D.’s, what percent of CD’s were

Classical?

• adapted from ”Percent Formula Word Problems” by R. Strong
35 = x175 100

Use cross products:

3500 = 175x

175 175

X = 20 %

(or 1/5 of the CD’s played were classical)

Changing Fractions to Equivalent Percents:

There are 3 ways to change a fraction to an equivalent percent:

1) Divide the denominator into the numerator, then change the decimal to a percent.

=

0.60

6÷10

0.60 * 100

=

60%

3) Draw an illustration

using a 100 grid.

1/10

1/10

1/10

1/10

1/10

1/10

6/10= 60%

Try it!

1) What is 20% as a fraction in simplest form?

2) What is 0.6 as a percent?

3) What is 3/5 as a decimal?

20

20 ÷ 20

1

=

=

100

100 ÷ 20

5

What is 20% as a fraction in simplest form?

20/100?

1/5

1/5

1/5

1/5

1/5

20/100 = ?

adapted from a slide by D. Fisher

Rewrite 0.6 as a percent.

6/10= ?%

6÷ 10

= 0.6

adapted from a slide by D. Fisher

Remember: Multiply by 100 when changing a decimal to a percent because percent means “out of 100”.

0.6 * 100

= 60

So…

0.6

= 60%

What is 3/5 as a decimal?

1) Divide then multiply…

3 ÷ 5 = 0.6

0.6 * 100 = 60

So 3/5 = 60%

OR

2) Find an equivalent fraction 3/5 = ? /100

5 * 20 = 100

Do the same to the numerator:

3 * 20 = 60

So 3/5 = 60/100 which equals 60%

3) Make an Illustration:

1/5

1/5

1/5

= 60

out of

100

or

60%

Percent of Change:
• “Percent of change is the ratio:

amount of change

original amount expressed as a percent.

Try it!

“Suppose you increase the strength in your elbow joint from 90 foot-pounds to 135 foot-pounds. Find the percent of increase to the nearest percent.”

adapted from Prentice Hall Algebra I

90 to 135:

135- 90 = 45 change

45 ÷ 135 = 0.33

or 33 1/3 % increase.

Maximum and Minimum Areas:
• “The greatest possible error in measurement is one half of that measuring unit.”
• Find the maximum and minimum areas for a room that is 13 ft. by 7ft.

from Prentice Hall Algebra I

13 ft could be 12.5 or 13.5while 7 ft could be 6.5 or 7.5
• 12.5 * 6.5 = 81.25 ft2

(minimum)

• 13.5 * 7.5 = 101.25 ft2 (maximum)