lesson 1: ratios and price per unit this lesson covers the

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Unit 4: Ratios and Proportions

LESSON 1: RATIOS AND PRICE PER UNIT This lesson covers the following information:

• Understanding ratios • Calculating price per unit, miles per gallon, and miles per hour

Highlights include the following:

• Ratios are a way to compare two quantities used to show a relationship or pattern. • Rations can be written in the following manner.

§ 3:4 3 to 4 34

• Ratios written as fractions must be reduced to lowest terms. • A ratio must contain two numbers. It can never be a mixed numeral or a whole number. • A ratio can be shown in charts. • In order to fund the unit price, you will need to find the ratio of the price to units. • Price per unit can be written:

§ price: unit price to unit priceunit

• Use division to find unit prices. § Another frequently used ratio is miles per hour. Known as rate or speed.

Reflection Ratio problems ask the question, “What is the ratio of ____ to ___?” Whatever is asked for first (the first blank) is always the first or top number in the answer. Ratio word problems sometimes describe the two items being compared in a different order, but the question to be answered will always tell you the order to be used --- first item first or on top. Notes:



Cryptogram Solve the puzzle.

Practice Problems Solve the following problems.

1. There are twelve members in a group. 5 of the members are girls. What is the ratio of girls to boys? ____________________

2. A bag of candy has 10 apple flavored fruit snacks and 8 cherry flavored snacks. What is the ratio of cherry flavored snacks to apple-flavored snacks? ____________________

3. A youth group went to a concert. They needed 50 tickets. When they arrived at the concert, there were 4 front row seats available and 46 tickets for the back row. What was the ratio of back row tickets to front row tickets? ____________________

4. Jackie is a hostess at a restaurant. She is making sure each waitress has the same amount of customers in an evening. There are 5 full tables and 3 empty tables in the back room. What is the ratio of full tables to tables in the back room? ____________________



5. At a union meeting, 25 employees voted yes to a proposed strike. 70 employees voted not to strike. What is the ratio of number of employees to number of employees who voted not to strike? ____________________

6. George is shopping for art pencils. He can buy 10 pencils for $4.00 or 5 pencils for $2.75. Which is the better buy? ____________________

7. Which is the best bargain? ____________________ a. 1 gallon of milk for $2.15 b. 2 gallons of milk for $4.10 c. 3 gallons of milk for $5.40 d. 10 gallons of milk for $17.00

8. A family is planning on driving across country. They have 1,500 miles to drive in order to reach their destination. They used 50 gallons of gas. How many miles per gallon did they get? ____________________

9. You have 7 nickels worth $0.35 and 15 dimes worth $1.50. What is the ratio of nickels to dimes? ____________________

10. What is the ratio of hands in humans to paws in dogs? ____________________



LESSON 2: RATIOS AND PROPORTIONS This lesson covers the following information:

• Identifying a proportion • Understanding true proportions


• A proportion is a statement that says two equal are ratios. • Proportions compare two sets of items. • Proportions can be true or false. • The simplest way to check whether two fractions (a proportion) are equal is to cross-multiply. If

the two products are equal, then the proportion is true. Reflection Learning to cross-multiply to prove that a proportion is true is an important step before learning to set up and solve proportions. Notes:



Letter Tiles Unscramble the tiles to solve the phrase.

Practice Problems Solve the following problems.

1. Identify if the proportion is true or false. 318

= 20105

____________________

2. Identify if the proportion is true or false. 56= 1018

____________________




= 7890

____________________


= 2234

____________________


____________________


____________________


= 22121

____________________


= 38

____________________


____________________


=22121

____________________



LESSON 3: FINDING THE UNKNOWN TERM IN A PROPORTION This lesson covers the following information:

• Finding the unknown term in a proportion. Highlights include the following:

• A proportion is a statement saying that two ratios are equal to each other. • To determine if the proportion is true, cross-multiply the two ratios. If the products are the same,

the proportion is true. • These are the two steps to solving a proportion with an unknown term (or number):

§ Cross multiply. If you have a missing number in a proportion, you'll only be able to cross multiply in one direction. Multiply diagonally, not horizontally.

§ Divide the product from step one by the third, unused number in the proportion. Reflection A proportion is a statement saying that two ratios are equal to each other. A proportion problem with a missing

number looks like 23= ?6

.

Notes:




Practice Problems

1. 79= 1418

____________________

a. True b. False

2. 815

= 2430

____________________

a. True b. False

3. 2125

= 35

____________________

a. True b. False

4. 1330

= 3990

____________________

a. True b. False



5. 2752

= 108218

____________________

a. True b. False

6. 416

= 16

____________________

a. True b. False

7. 617

= 1234

____________________

a. True b. False

8. 89= 1627

____________________

a. True b. False

9. 3045

= 69

____________________

a. True b. False

10. 1,5002,000

= 35

____________________

a. True b. False



LESSON 4: PROBLEM SOLVING WITH PROPORTIONS This lesson covers the following information:

• Setting up proportions with unknown terms • Using charts set up proportions


• Pattern 1 The first ratio includes the numbers from one situation; the second ratio includes the numbers from the second, related situation, in the same order as the first situation.

• Pattern 2 In this second pattern, the first fraction compares boys from the one situation to the boys in the other situation. The second fraction compares the girls from the corresponding situations.

• There are 4 steps to follow to set up and solve a proportion problem: § Set up the first ratio with the information you are given § Set up the second ratio in exactly the same order as the first § Cross-multiply the 2 diagonal numbers. § Divide the answer from #3 by the 3rd number. This is the answer. Check by cross-multiplying

both sets of numbers. Reflection There are two different patterns for correctly writing proportions for story problems. Be aware that if you do not write the corresponding parts vertically or horizontally from each other, (if you write corresponding parts diagonally from each other), you will end up with a proportion with a different, wrong, cross product, and thus a wrong answer. Notes:



Word Search Find all the words in the list. Words can be found in any direction.

CORRESPONDING CROSS PRODUCT DIAGONALLY DIFFERENT PATTERNS PROPORTIONS TWO VERTICALLY



Practice Problems

1. Sally is making sugar cookies and the recipe calls for a ratio of 2 cups of sugar to 3 sticks of butter. If you are using 6 sticks of butter, how much sugar will you need? ____________________

2. Coils Incorporated makes car parts. If a line worker can make 240 coils in 5 minutes, how many coils can they make in 20 minutes? ____________________

3. Stephen has a new job as a receptionist. He can type 120 words in 90 seconds. How long will it take him to type 450 words? ____________________

4. Sarah and Janice are planning a wedding. The ratio of soda to fruit juice is 5:8. If they have 45 parts of soda, how much fruit juice will they need? ____________________

5. Carly wants to paint her bedroom a very specific color of purple by mixing 3 pints red paint to 4 pints blue. In order to have enough paint to cover her room, she needs 8 pints of blue paint. How many pints of red paint will she need? ____________________

6. If 3 feet is equal to 1 yard, how many yards are in 27 feet? ____________________

7. Property tax for a home that is valued at $45,000.00 is $800.00. What is the tax for a home that is valued at $72,000.00? ____________________

8. Grace wants to have a lemonade stand and make a profit of $25.00. To make her lemonade, she uses 4 lemons to 3 pints of water. How many lemons would she need if she had 24 lemons? ____________________

9. A shrub that is 3 feet tall casts a shadow that is 7 feet long. How long is the shadow of a tree that is 14 feet tall? ____________________

10. Kathy reads a book for 60 minutes every night. She tries to read 50 pages at a time. How many hours would it take to read a 300-page book? ____________________



LESSON 5: PROPORTIONS AND GEOMETRY This lesson covers the following information:

• Defining similar triangles and similar figures • Finding unknown lengths of sides for pairs of similar figures using proportions


• Two figures that are similar have the same shape, but not the same size. • Mathematicians define similar triangles this way:

§ The corresponding (matching) angles are congruent (or equal in measure). AND

§ The corresponding sides are proportional. This means they will make a true proportion. • Proportions to help us determine the length of a missing side of similar figures. • Since we know the rectangles are similar, their sides must be proportional.

Reflection Two figures that are similar have the same shape, but not the same size. Proportion problems can be used to identify the length of missing side. Notes:




Practice Problems

1. A picture is 3 inches tall and 5 inches long. Coryn wants to enlarge the picture so the length is 15 inches long. How tall will the picture have to be? ____________________

2. Ken is building two shelves. The smaller shelf has a length of 24 inches and a width of 6 inches. He would like to make a second shelf that has the same proportions as the first. If the second shelf has a width of 8 inches, how long does it have to be? ____________________

3. A scale model of a cruise ship is 100 inches long and 15 inches wide. If the actual ship is 900 feet long, how wide is it? ____________________

4. The United States Flag has an official ratio of length to width of 1:1.9. If a flag is 10 feet high, how wide should it be? ____________________

5. Bill’s garden is 8 feet long by 12 feet wide He wants to make a smaller garden this year and make it 4 feet long. How wide should it be? ____________________

lesson 1: ratios and price per unit this lesson covers the

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