Proportions, Measurement Conversions, Scale, and Percents

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

How large is a meter? About the width 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? About a 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 about 1 gram from “Math Flash Measurement I” by M. and B. Yuskaitis

How heavy is a kilogram? A kitten weighs about 1 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

The letter C in Roman Numerals stands for 100 or “cent”. 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

Step 4 – Place the decimal point 2 places to the left in your answer. 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%

2) Find an equivalent fraction with 100 as the denominator. 60 6 10 ? 100 =

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%

Proportions, Measurement Conversions, Scale, and Percents