GEARING UP EXAMPLES. 4 to 3 4:3

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1 GEARING UP EXAMPLES B 2 Teeth A 8 Teeth DEFINITION - RATIO As gear A revolves times, it will cause gear B to revolve times. Hence, we say that gear ratio of A to B is to. In mathematics, a ratio is a comparison of two numbers by division. The gear ratio above can be epressed in the following ways: to : A ratio is commonly epressed as a fraction in simplest form! Eample : What is the ratio of gear A to gear B if gear A has 0 teeth to 6 teeth (in fraction form)? The ratio is 0 or. 6 Eample 2: What is the ratio of 20 inches to feet? The units should be the same! feet 8 inches The ratio is 20in or 8in 2.

$The gear ratio above can be epressed in the following ways: to : A ratio is commonly epressed as a fraction in simplest form!$

2 Eample : What is the gear ratio, as a fraction, for 0 teeth to 2 teeth? 0 6 The ratio is 6. 2 Eample : What is the ratio, as a fraction, for 6 inches to feet? The units must be the same! feet 6 inches The ratio is 6in 6in 9. DEFINITION PROPORTION a b An equation of the form that states that two ratios are equal is called a proportion. Every proportion consists of four terms. c d First a b Second c d Third Fourth The first and fourth terms, a and d, are called the etremes. The second and third terms, b and c, are called the means. MEANS-EXTREMES PROPERTY OF PROPORTIONS: In a proportion, the product of the etremes is equal to the product of the means. a c If, then ad bc. b d You can use this property to solve some equations. The equations must be in the form of a proportion. 2

c d First a b Second c d Third Fourth The first and fourth terms, a and d, are called the etremes. The second and third terms, b and c, are called the means.

3 Eample : Solve: (8) 2 Means-Etremes Property 8 2 Eample 6: Solve: Means-Etremes Property (0) ( + ) 0 + ()

4 Eample : Write a proportion for the following word problem, then use the - step approach to problem solving to solve the problem. A gear of 96 teeth required a matching gear of 6 teeth. At that rate, how many teeth would be needed on gear B if gear A had 2 teeth? (Round answer to nearest whole number of teeth. Eplore: Let teeth required for gear B Plan: Write a proportion for the problem Solve: /2 or 9. A gear of 0 teeth is required. Eamine: Gear B is /6 as large as gear A. Is (/6)(2) 0 (approimately)? (YES)

(Round answer to nearest whole number of teeth. Eplore: Let teeth required for gear B Plan: Write a proportion for the problem.

5 Eample 8: Write a proportion for the following word problem, then use the -step approach to problem solving to solve the problem. In the diagram at the right, gear A connects to the differential at gear B. There are 2 teeth on gear A and teeth on Gear B. It has been decided that the gear ratio needs to be increased to a to 2 (A:B) ratio. If gear B remains the same, how many teeth must gear A have? A B Eplore: Let teeth on gear A after change Plan: Write a proportion for the problem. 2 Solve: A new gear would require teeth. Eamine: Gear B must retain teeth. Gear A must be in the ratio of :2 times larger. Is (/2)? (YES)

It has been decided that the gear ratio needs to be increased to a to 2 (A:B) ratio. If gear B remains the same, how many teeth must gear A have?

6 Name: Date: Class: GEARING UP WORKSHEET Write each ratio as a fraction in simplest form:. gears to gears 2. gears to gears. 2 feet to 6 feet. 6 cm to cm. 2 ounces to 6 ounces 6. km to km. 8 feet to 28 inches 8. pounds to 00 ounces 9. 6 cm to 0 mm 0. 2 mm to 90 mm. minutes to 2 hours 2. hours to 2 minutes Show steps to solve each proportion

8 feet to 28 inches 8. pounds to 00 ounces 9. 6 cm to 0 mm 0. 2 mm to 90 mm.

7 GEARING UP WORKSHEET For each problem use the -step approach to problem solving: a. Eplore Define a variable b. Plan Write an equation c. Solve Solve the equation and answer the problem d. Eamine Check to see if the answer makes sense B A 2. If gear A turns times each time gear B turns 2 times, then how many times will Gear A turn if Gear B turns 0 times? 22. If gear A turns times each time gear B turns /2 times, then how many times will Gear B turn if Gear A turns 2 times? 2. Stewart earns $9 in days. At that rate, how many days will it take him to earn $8? 2. Shawn used 2 gallons of gasoline in traveling 0 miles. How much gasoline will be used in traveling 62 miles? 2. One gear has 6 teeth. The ratio of this gear to a second gear is to. How many teeth does the second gear have?

If gear A turns times each time gear B turns /2 times, then how many times will Gear B turn if Gear A turns 2 times? 2. Stewart earns $9 in days.

8 GEARING UP WORKSHEET KEY Write each ratio as a fraction in simplest form: gears to gears 2. gears to gears. 2 feet to 6 feet. 6 cm to cm. 2 ounces to 6 ounces 6. km to km. 8 feet to 28 inches 8. pounds to 00 ounces 9. 6 cm to 0 mm 0. 2 mm to 90 mm. minutes to 2 hours 2. hours to 2 minutes Show steps to solve each proportion: (8) 2 6 () 9 8

km to km. 8 feet to 28 inches 8. pounds to 00 ounces 9. 6 cm to 0 mm 0. 2 mm to 90 mm.

9 GEARING UP WORKSHEET KEY (0) ( + 2)() () ( ) () / (6) ( + 2) / ( ) 0( + ) () ( 8) 2 9 / 9

9 / 8. + 2 6 + 2 6 (6) ( + 2) 8 + 0 8 / 9.

10 GEARING UP WORKSHEET KEY For each problem use the -step approach to problem solving: a. Eplore Define a variable b. Plan Write an equation c. Solve Solve the equation and answer the problem d. Eamine Check to see if the answer makes sense B A 20. If gear A turns times each time gear B turns 2 times, then how many times will Gear A turn if Gear B turns 0 times? Let be the number times gear A will turn. 2 0 (0) If gear B turns 0 times then this is times larger. Is 2 five times larger than? (YES) 2. If gear A turns times each time gear B turns /2 times, then how many times will Gear B turn if Gear A turns 2 times? Let be the number of times gear B turns. 2 2 (2)( /2) 8 2 / If gear A turns 2 times then this is a little less than twice as many turns. Is 2 times ( /2 ) approimately 2 /? (YES) 22. Stewart earns $9 in days. At that rate, how many days will it take him to earn $8? Let be number of days to earn $ $8 is times greater than $9. Then it should take times longer to earn the needed money. Is () 20? (YES) 0

Let be the number times gear A will turn. 2 0 (0) 2 0 2 2 If gear B turns 0 times then this is times larger. Is 2 five times larger than? (YES) 2.

11 GEARING UP WORKSHEET KEY 2. Shawn used 2 gallons of gasoline in traveling 0 miles. How much gasoline will be used in traveling 62 miles? Let be the amount of gasoline used (62) miles is a little less than /2 times larger than 0 miles. Is approimately /2 times larger than 2? (YES) 2. One gear has 6 teeth. The ratio of this gear to a second gear is to. How many teeth does the second gear have? Let be the number of teeth the second gear has. 6 (6) 08 2 Is the ratio of 6 teeth to 2 teeth the same as to? (YES)

2 0 62 2(62) 0 0 0 62 miles is a little less than /2 times larger than 0 miles. Is approimately /2 times larger than 2? (YES) 2.

12 Student Name: Date: GEARING UP CHECKLIST. On each problem, did the student write each ratio as a fraction in simplest form? a. All twelve (0 points) b. Ten or more of the twelve (2 points) c. Eight or more of the twelve (20 points) d. Si or more of the twelve ( points) e. Four or more of the twelve (0 points) f. Two or more of the twelve ( points) 2. On each problem, did the student show steps to solve each proportion correctly? a. All eight (0 points) b. Seven of the eight (2 points) c. Si of the eight (20 points) d. Five of the eight ( points) e. Four of the eight (0 points) f. Three of the eight ( points). On each problem, did the student use the -step approach to problem solving? a. All five (0 points) b. Four of the five (2 points) c. Three of the five (20 points) d. Two of the five ( points) e. One of the five (0 points) Total Number of Points A B C D F 8 points and above 2 points and above 6 points and above points and above points and below Any score below C needs remediation! 2

On each problem, did the student show steps to solve each proportion correctly? a. All eight (0 points) b. Seven of the eight (2 points) c. Si of the eight (20 points) d.

Test 4 Sample Problem Solutions, 27.58 = 27 47 100, 7 5, 1 6. 5 = 14 10 = 1.4. Moving the decimal two spots to the left gives

Test 4 Sample Problem Solutions Convert from a decimal to a fraction: 0.023, 27.58, 0.777... For the first two we have 0.023 = 23 58, 27.58 = 27 1000 100. For the last, if we set x = 0.777..., then 10x