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208 Ratio and Proportion, A ratio which is a comparison of two numbers by division is the quotient. obtained when the first number is divided by the second nonzero number. Since a ratio is the quotient of two numbers divided in a definite order care. must be taken to write each ratio in its intended order For example the ratio of. 3 to 1 is written,1 as a fraction or 3 1 using a colon. while the ratio of 1 to 3 is written,3 as a fraction or 1 3 using a colon. In general the ratio of a to b can be expressed as. b or a b or a b, To find the ratio of two quantities both quantities must be expressed in the. same unit of measure before their quotient is determined For example to com. pare the value of a nickel and a penny we first convert the nickel to 5 pennies. and then find the ratio which is 51 or 5 1 Therefore a nickel is worth 5 times as. much as a penny The ratio has no unit of measure,Equivalent Ratios.

Since the ratio 51 is a fraction we can use the multiplication property of 1 to find. many equivalent ratios For example,1 5 51 3 22 5 10. 1 5 51 3 33 5 15,1 5 51 3 xx 5 5x, From the last example we see that 5x and lx represent two numbers whose ratio. In general if a b and x are numbers b 0 x 0 ax and bx represent two. numbers whose ratio is a b because,b 5 b 3 1 5 ba 3 xx 5 bx. Also since a ratio such as 16 is a,fraction we can divide the numerator and the. denominator of the fraction by the same nonzero number to find equivalent. ratios For example,16 5 24 4 2 12,16 4 2 5 8,16 5 24 4 4.

16 4 4 5 4,16 5 24 4 8 3,16 4 8 5 2, A ratio is expressed in simplest form when both terms of the ratio are whole. numbers and when there is no whole number other than 1 that is a factor of. both of these terms Therefore to express the ratio 24. 16 in simplest form we divide, both terms by 8 the largest integer that will divide both 24 and 16 Therefore 24. in simplest form is 32,Continued Ratio, Comparisons can also be made for three or more quanti. ties For example the length of a rectangular solid is 75. centimeters the width is 60 centimeters and the height 45 cm. is 45 centimeters The ratio of the length to the width is 60 cm 75 cm. 75 60 and the ratio of the width to the height is 60 45. We can write these two ratios in an abbreviated form as. the continued ratio 75 60 45, A continued ratio is a comparison of three or more quantities in a definite. order Here the ratio of the measures of the length width and height in that. order of the rectangular solid is 75 60 45 or in simplest form 5 4 3. In general the ratio of the numbers a b and c b 0 c 0 is a b c. An oil tank with a capacity of 200 gallons contains 50 gallons of oil. a Find the ratio of the number of gallons of oil in the tank to the capacity of. b What part of the tank is full,number of gallons of oil in tank 50.

a Ratio capacity of tank 5 200 5 14,b The tank is 4 full. Answers a 14 b 14 full,Compute the ratio of 6 4 ounces to 1 pound. Solution First express both quantities in the same unit of measure Use the fact that. 1 pound 16 ounces,6 4 ounces,1 pound 5 6 4 ounces 6 4 6 4 10 64 64 4 32 2. 16 ounces 5 16 5 16 3 10 5 160 5 160 4 32 5 5,210 Ratio and Proportion. Calculator On a calculator divide 6 4 ounces by 16 ounces. ENTER 6 4 16 ENTER,Change the decimal in the display to a fraction.

ENTER 2nd ANS MATH ENTER ENTER,Answer The ratio is 2 5. Express the ratio 134 to 112 in simplest form, Solution Since a ratio is the quotient obtained when the first number is divided by the. second divide 134 by 112,134 4 112 5 74 4 32 5 74 23 5 14 7. Answer The ratio in simplest form is 76 or 7 6,Writing About Mathematics. 1 Last week Melanie answered 24 out of 30 questions correctly on a test This week she. answered 20 out of 24 questions correctly On which test did Melanie have better results. Explain your answer, 2 Explain why the ratio 1 5 4 5 is not in simplest form.

Developing Skills, In 3 12 express each ratio in simplest form a as a fraction b using a colon. 3 36 to 12 4 48 to 24 5 40 to 25 6 12 to 3 7 5 to 4. 8 8 to 32 9 40 to 5 10 0 2 to 8 11 72 to 1 2 12 3c to 5c. 13 If the ratio of two numbers is 10 1 the larger number is how many times the smaller num. 14 If the ratio of two numbers is 8 1 the smaller number is what fractional part of the larger. In 15 19 express each ratio in simplest form, 15 34 to 41 16 118 to 38 17 1 2 to 2 4 18 0 75 to 0 25 19 6 to 0 25. In 20 31 express each ratio in simplest form, 20 80 m to 16 m 21 75 g to 100 g 22 36 cm to 72 cm. 23 54 g to 90 g 24 75 cm to 350 cm 25 8 ounces to 1 pound. 26 112 hr to 21 hr 27 3 in to 21 in 28 1 ft to 1 in. 29 1 yd to 1 ft 30 1 hr to 15 min 31 6 dollars to 50 cents. Applying Skills,32 A baseball team played 162 games and won 90. a What is the ratio of the number of games won to the number of games played. b For every nine games played how many games were won. 33 A student did six of ten problems correctly, a What is the ratio of the number right to the number wrong.

b For every two answers that were wrong how many answers were right. 34 A cake recipe calls for 114 cups of milk to 134 cups of flour Write in simplest form the ratio. of the number of cups of milk to the number of cups of flour in this recipe. 35 The perimeter of a rectangular garden is 30 feet and the width is 5 feet Find the ratio of. the length of the rectangle to its width in simplest form. 36 In a freshman class there are b boys and g girls Express the ratio of the number of boys to. the total number of pupils, 37 The length of a rectangular classroom is represented by 3x and its width by 2x Find the. ratio of the width of the classroom to its perimeter. 38 The ages of three teachers are 48 28 and 24 years Find in simplest form the continued. ratio of these ages from oldest to youngest, 39 A woodworker is fashioning a base for a trophy He starts with a block of wood whose. length is twice its width and whose height is one half its width Write in simplest form the. continued ratio of length to width to height, 40 Taya and Jed collect coins The ratio of the number of coins in their collections in some. order is 4 to 3 If Taya has 60 coins in her collection how many coins could Jed have. 212 Ratio and Proportion,6 2 USING A RATIO TO EXPRESS A RATE. When two quantities have the same unit of measure their ratio has no unit of. measure A rate like a ratio is a comparison of two quantities but the. quantities may have different units of measures and their ratio has a unit of. For example if a plane flies 1 920 kilometers in 3 hours its rate of speed is. a ratio that compares the distance traveled to the time that the plane was in. 1 920 kilometers,Rate distance,time 5 3 hours 5 640 kilometers.

1 hour 640 km h,The abbreviation km h is read kilometers per hour. A rate may be expressed in lowest terms when the numbers in its ratio are. whole numbers with no common factor other than 1 However a rate is most. frequently written as a ratio with 1 as its second term As shown in the example. above the second term may be omitted when it is 1 A rate that has a denomi. nator of 1 is called a unit rate A rate that identifies the cost of an item per unit. is called the unit price For example 0 15 per ounce or 3 79 per pound are unit. Kareem scored 175 points in seven basketball games Express in lowest terms. the average rate of the number of points Kareem scored per game. 175 points 25 points,Rate 7 games 5 1 game 25 points per game. Answer Kareem scored points at an average rate of 25 points per game. There are 5 grams of salt in 100 cubic centimeters of a solution of salt and water. Express in lowest terms the ratio of the number of grams of salt per cubic cen. timeters in the solution,3 5 3 5 g cm3,100 cm 20 cm 20. Answer The solution contains 20 grams or 0 05 grams of salt per cubic centimeter of. Using a Ratio to Express a Rate 213,Writing About Mathematics. 1 How is a rate a special kind of ratio, 2 How does the way in which a rate is usually expressed differ from a ratio in.

simplest form,Developing Skills,In 3 8 express each rate in lowest terms. 3 The ratio of 36 apples to 18 people 4 The ratio of 48 patients to 6 nurses. 5 The ratio of 1 50 to 3 liters 6 The ratio of 96 cents to 16 grams. 7 The ratio of 2 25 to 6 75 ounces 8 The ratio of 62 miles to 100 kilometers. Applying Skills, In 9 12 in each case find the average rate of speed expressed in miles per hour. 9 A vacationer traveled 230 miles in 4 hours, 10 A post office truck delivered mail on a 9 mile route in 2 hours. 11 A commuter drove 48 miles to work in 112 hours, 12 A race car driver traveled 31 miles in 15 minutes Use 15 minutes 14 hour. 13 If there are 240 tennis balls in 80 cans how many tennis balls are in each can. 14 If an 11 ounce can of shaving cream costs 88 cents what is the unit cost of the shaving. cream in the can, 15 In a supermarket the regular size of CleanRight cleanser contains 14 ounces and.

costs 49 cents The giant size of CleanRight cleanser which contains 20 ounces costs. a Find correct to the nearest tenth of a cent the cost per ounce for the regular can. b Find correct to the nearest tenth of a cent the cost per ounce for the giant can. c Which is the better buy, 16 Johanna and Al use computers for word processing Johanna can keyboard 920 words. in 20 minutes and Al can keyboard 1 290 words in 30 minutes Who is faster at entering. words on a keyboard, 17 Ronald runs 300 meters in 40 seconds Carlos runs 200 meters in 30 seconds Who is the. faster runner for short races,214 Ratio and Proportion. 6 3 VERBAL PROBLEMS INVOLVING RATIO, Any pair of numbers in the ratio 3 5 can be found by multiplying 3 and 5 by. the same nonzero number,3 3 9 3 7 21 3 0 3 0 9 3 x 3x.

5 3 15 5 7 35 5 0 3 1 5 5 x 5x,3 5 9 15 3 5 21 35 3 5 0 9 1 5 3 5 3x 5x. Thus for any nonzero number x 3x 5x 3 5, In general when we know the ratio of two or more numbers we can use the. terms of the ratio and a nonzero variable x to express the numbers Any two. numbers in the ratio a b can be written as ax and bx where x is a nonzero real. The perimeter of a triangle is 60 feet If the sides are in the ratio 3 4 5 find the. length of each side of the triangle,Solution Let 3x the length of the first side. 4x the length of the second side,5x the length of the third side. The perimeter of the triangle is 60 feet Check,3x 4x 5x 60 15 20 25 3 4 5.

12x 60 15 20 25 60, Answer The lengths of the sides are 15 feet 20 feet and 25 feet. Two numbers have the ratio 2 3 The larger is 30 more than 12 of the smaller. Find the numbers,Solution Let 2x the smaller number. 3x the larger number,Verbal Problems Involving Ratio 215. The larger number is 30 more than 12 of Check, the smaller number The ratio 30 45 in lowest terms. 3x 12 2x 1 30,One half of the smaller number,3x x 30 30 is 15 The larger number 45.

is 30 more than 15,2x 2 15 30,3x 3 15 45,Answer The numbers are 30 and 45. Writing About Mathematics, 1 Two numbers in the ratio 2 3 can be written as 2x and 3x Explain why x cannot equal zero. 2 The ratio of the length of a rectangle to its width is 7 4 Pete said that the ratio of the. length to the perimeter is 7 11 Do you agree with Pete Explain why or why not. Developing Skills, 3 Two numbers are in the ratio 4 3 Their sum is 70 Find the numbers. 4 Find two numbers whose sum is 160 and that have the ratio 5 3. 5 Two numbers have the ratio 7 5 Their difference is 12 Find the numbers. 6 Find two numbers whose ratio is 4 1 and whose difference is 36. 7 The lengths of the sides of a triangle are in the ratio of 6 6 5 The perimeter of the triangle. is 34 centimeters Find the length of each side of the triangle. 8 The perimeter of a triangle is 48 centimeters The lengths of the sides are in the ratio 3 4 5. Find the length of each side, 9 The perimeter of a rectangle is 360 centimeters If the ratio of its length to its width is 11 4. find the dimensions of the rectangle, 10 The sum of the measures of two angles is 90 The ratio of the measures of the angles is 2 3.

Find the measure of each angle, 11 The sum of the measures of two angles is 180 The ratio of the measures of the angles is. 4 5 Find the measure of each angle,216 Ratio and Proportion. 12 The ratio of the measures of the three angles of a triangle is 2 2 5 Find the measures of. each angle, 13 In a triangle two sides have the same length The ratio of each of these sides to the third. side is 5 3 If the perimeter of the triangle is 65 inches find the length of each side of the. 14 Two positive numbers are in the ratio 3 7 The larger exceeds the smaller by 12 Find the. 15 Two numbers are in the ratio 3 5 If 9 is added to their sum the result is 41 Find the. Applying Skills, 16 A piece of wire 32 centimeters in length is divided into two parts that are in the ratio 3 5. Find the length of each part, 17 The ratio of the number of boys in a school to the number of girls is 11 to 10 If there are.

525 pupils in the school how many of them are boys. 18 The ratio of Carl s money to Donald s money is 7 3 If Carl gives Donald 20 the two then. have equal amounts Find the original amount that each one had. 19 In a basketball free throw shooting contest the points made by Sam and Wilbur were. in the ratio 7 9 Wilbur made 6 more points than Sam Find the number of points made. 7 5 cm 60 cm On a calculator divide 6 4 ounces by 16 ounces ENTER 6 4 16 DISPLAY Change the decimal in the display to a fraction ENTER DISPLAY Answer The ratio is 2 5 EXAMPLE 3 Express the ratio to in simplest form Solution Since a ratio is the quotient obtained when the first number is divided by the second divide by Answer The ratio in simplest form is or 7 6 Writing About

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