Understanding Ratio, Proportion, and Percent Chapter Thirteen

RATIOS & PROPORTIONS A ratio is a comparison of two numbers by division. To write ratios, use the word to, a colon, or a fraction bar. EXAMPLE #1: John read 3 books in 4 days. Write the ratio of books to days. Words: 3 to 4 Colon: 3:4 Fraction: 3/4

RATIOS & PROPORTIONS A proportion is a statement that two ratios are equal. EXAMPLE #2: Is a proportion? Find the cross products. 30 = 30 TRUE If TRUE, the statement is a proportion.

RATIOS & PROPORTIONS 20 = 21 FALSE EXAMPLE #3: Is a proportion? Find the cross products. 20 = 21 FALSE If FALSE, the statement is NOT a proportion.

RATIOS & PROPORTIONS EXAMPLE #4: Solve the proportion: Find the cross products. Divide both sides by 10. 10

RATIOS & PROPORTIONS Find the missing numbers to make the following proportions. 10 10 9 9

Equal ratios Beginning activities should include number patterns and repeated addition In a basketball game, Amy made 3 of 5 free throws and Diane made 4 of 7 attempts. Which performance was better? 3 5 7 21 35 x = Answer: AMY 4 7 5 20 35 x =

Cross Products Travel On A car travels 212 km in 4 hours. At this rate of speed, how far would it travel in 7 hours? 212 km 4 hrs. x km 7 hrs. = 4x = 1484 4 4 X = 371 km

Cross Products Travel On 8 people 6 onions 12 people x onions = A recipe for soup for 8 people calls for 6 onions. How many onions should be used to make enough soup for 12 people? 8 people 6 onions 12 people x onions = 8x = 72 8 8 X = 9 onions

SCALE DRAWINGS A scale drawing is a drawing of an object with dimensions proportional to those of the actual object. To find lengths from scale drawings, set up and solve proportions. Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree?

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? Mary’s height Tree’s height Mary’s shadow Tree’s shadow = x 5.5 2 7

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? 5.5 x 2 7 = Mary’s height Tree’s height Mary’s shadow Tree’s shadow = x 5.5 2 7

5.5 x 2 7 = x 7 ( 5.5 ) = 2 x 38.5 = 2 x x = 19.25 7 The height of the tree is 19.25 feet

SCALE DRAWINGS USING PROPORTIONS EXAMPLE#1: ¼ in = 5 ft is a scale for a rock band’s concert stage. A. The scale length of the drum stand is 1 in. What is the actual length of the drum stand? Cross multiply. Divide by .25 .25 .25

SCALE DRAWINGS USING PROPORTIONS EXAMPLE#2: ¼ in = 5 ft is a scale for a rock band’s concert stage. B. The actual length of the stage is 60 ft. What is the scale length of the stage? Cross multiply. Divide by 5 5 5

Percent Meaning and notation Number sense Term percent means “parts per hundred” Part-to-whole ratio that has 100 as its second term Can be thought of as fractions with denominators of 100 Number sense Introductory activities should help children to develop visual images and a quantitative feel for numbers expressed as percents Concrete Level – Base-ten blocks Pictorial Level – Squares on a 10-by-10 grid

Percent (cont’d) Fraction and decimal equivalents Decimals as percents involves finding an equivalent decimal in hundredths. Fractions as percents can be determined by finding an equivalent fraction with a denominator of 100. Another method – first write the fraction in decimal form and then multiply the number by 100. Percents as fractions with a denominator of 100 can be done by using the definition of percent.

Finding a Percent of a Number Everyday situations include: Sales tax Discounts Commission Interest Early examples should include percents that have simple fraction or decimal equivalents 10% of 70 30% of 80 50% of 14 1% of 324 25% of 40 200% of 52

Percents Most real-life computations will be carried out on a calculator which means instruction should focus on Estimating answers Justifying the reasonableness of the results What’s the estimate of the following problems? 52% of 65 17% of 543 61% of 46 9.5% of 628

Other Procedures for Solving Percent Problems The proportion method Of the sixth-grade children in a school, 21 usually walk to school. This represents 30% of the sixth grade children. How many sixth graders are there? 21 x 30 100 = 30x = 2100 x = 70

Other Procedures for Solving Percent Problems (cont’d) The equation method 50% of _____ = 32 .5x = 32 x = 64 6 is 10% of ______ 6 = .10x 60 = x 78 is 120% of some number. What do you know about the number? 78 = 1.20x 65 = x

Solving Equations Containing Percents Most percent problems are word problems and deal with data. Percents are used to describe relationships or compare a part to a whole. Sloths may seen lazy, but their extremely slow movement helps make them almost invisible to predators. Sloths sleep an average of 16.5 hours per day. What percent of the day do they sleep? Solution

What percent of 24 is 16.5. n · 24 = 16.5 n = 0.6875 n = 68.75% Proportional method Part Part Whole Whole Equation Method What percent of 24 is 16.5. n · 24 = 16.5 n = 0.6875 n = 68.75%