NAME ________________________________________ DATE ______________ PERIOD _____ Ratios (pages 380–383) You can compare two quantities by using a ratio. A common way to express a ratio is as a fraction in simplest form. If the two quantities you are comparing have different units of measure, this kind of ratio is called a rate. A rate is in the form of a unit rate when the denominator is 1. A rate is a ratio of two measurements that have different units. To write a Writing a Rate ratio as a unit rate, divide the numerator and denominator by the same and a Unit Rate number to rewrite the ratio as a fraction with a denominator of 1. A Write the ratio 5 sixth-graders out of 15 B Express the ratio 15 pencils for $5 as a students in three different ways. Express unit rate. How many pencils can you buy this ratio as a fraction in simplest form. for $1? 15 pencils As a fraction ᎏ5 Write the ratio as a fraction. ᎏᎏ 15 $5 As a ratio 5:15 To rewrite the fraction with a denominator of 1, In words 5 to 15 divide numerator and denominator by 5. Another way is in the problem: 5 out of 15. 15 pencils ᎏ1ᎏ5 pencils ᎏᎏ ϭ ᎏᎏ5 or 3 pencils for $1 $5 $1 ᎏ5 in simplest form is ᎏ1 . 15 3 Try These Together 1. Write the ratio 7 sodas out of 20 are 2. Express the ratio $14.50 for 5 rides as a sugar free in three different ways. rate. What is the cost for one ride? HINT: Write the numbers in the same order as HINT: Divide numerator and denominator by 5. they appear in the problem. Write each ratio as a fraction in simplest form. 3. 4 out of 16 papers are typed 4. 5 out of 10 horses are white 5. 7 blue bicycles out of 21 bicycles 6. 4 watermelons out of 10 melons Write each ratio as a unit rate. 7. $1.50 for 3 bottles of juice 8. 5 bracelets for $25.00 B 3. C 4. C A B 5. C 6. B A B 7. A 8. 9. Standardized Test Practice If milk costs $5.50 for 2 gallons, how much does it cost per gallon? A $11.00 B $10.50 C $2.75 D $3.50 C 9. bracelet per $5.00 8. 5 3 2 4 rides 5 20 $0.50 per bottle of juice of bottle per $0.50 3. 4. 5.ride 6.per 7. $2.90 or 2. 20; to 7 20, 7: 1. Answers: ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ 2 1 1 1 $14.50 7 © Glencoe/McGraw-Hill Parent and Student Study Guide 69 Mathematics: Applications and Concepts, Course 1 NAME ________________________________________ DATE ______________ PERIOD _____ Solving Proportions (pages 386–389) A proportion is an equation that shows that two ratios are equivalent. The general form of a proportion is ᎏa ϭ ᎏc , where neither b nor d is equal to b d zero. The cross products of a proportion are ad and bc. Property of The cross products of a proportion are equal. Proportions If ᎏa ϭ ᎏc , then ad ϭ bc. b d A 2 y Use cross products to find out whether B Solve the proportion ᎏ ϭ ᎏ for y. this pair of ratios forms a proportion. 7 21 3 9 Write the cross products. ᎏ, ᎏ ϫ ϭ ϫ 4 12 2 21 7 y 42 ϭ 7y Does ᎏ3 ϭ ᎏ9 ? Are the cross products equal? 4 12 ᎏ42 ϭ ᎏ7y Divide each side of the equation ϫ ϭ ϫ ϭ 7 7 Does 3 12 4 9? Yes, because 36 36. by 7. 3 9 6 ϭ y ᎏ ϭ ᎏ is a proportion because the cross 4 12 The solution is 6. products are equal. Try These Together 1. Use cross products to determine 2. Solve the proportion ᎏ3 ϭ ᎏ4 . p 20 whether this pair of ratios forms a HINT: Set the cross products equal to each proportion. ᎏ0.5 , ᎏ0.4 other and solve for p. 2 1.6 Determine whether each pair of ratios forms a proportion. 3. ᎏ1 , ᎏ5 4. ᎏ4 , ᎏ2 5. ᎏ4 , ᎏ1 6. ᎏ8 , ᎏ2 2 10 8 4 5 8 13 5 Solve each proportion. 7. ᎏ3 ϭ ᎏx 8. ᎏ4 ϭ ᎏ2 9. ᎏ9 ϭ ᎏ6 10. ᎏd ϭ ᎏ2 6 2 w 11 12 z 25 10 B 3. C 4. C A B 5. C 6. B A B 7. A 8. 11. Standardized Test Practice The home economics class is making a casserole. They need 3 eggs for 1 casserole. How many eggs do they need for 4 casseroles? A 9 B 12 C 15 D 10 B 11. 5 10. 8 9. 22 8. 1 7. no 6. no 5. yes 4. yes 3. 15 2. yes 1. Answers: © Glencoe/McGraw-Hill Parent and Student Study Guide 70 Mathematics: Applications and Concepts, Course 1 NAME ________________________________________ DATE ______________ PERIOD _____ Scale Drawings and Models (pages 391–393) A scale drawing is exactly the same shape as an object, but the drawing may be larger or smaller than the real object. The scale written on the drawing or model gives the ratio that compares the Reading a measurements on the drawing to the actual measurements of the object. Scale Drawing Use the scale of the drawing for one of the ratios and the known and unknown measurements for the other ratio. Write a proportion and solve it for the unknown measurement. A A model car has a scale of 1:16. A B The doorway of an actual house window on the model measures ᎏ1 of a measures 3 ft wide. How wide will the 32 doorway in a model house be if the scale meter. What will this same window is 1 ft ϭ 2 in.? measure on the real car? 1 ft 3 ft ᎏ1ᎏ meter ᎏ ϭ ᎏ so 1 x w ϭ 6 or w ϭ 6. 1 ᎏᎏ32 2 in. w ᎏ ϭ ᐉ Write a proportion. 16 The model doorway will be 6 inches wide. ᐉ ϭ 16΂ᎏ1 meter΃ Find cross products. 32 ᐉ ϭ ᎏ1 meter Solve. 2 The actual window measures ᎏ1 meter. 2 Try These Together 1. The scale of a map is 1 inch ϭ 25 miles. 2. A line on a scale drawing of a building The distance on the map between two measures 15 inches. The same length on cities is 7 inches. How many miles apart the actual building is 5 yards. What is are they? the scale of the drawing in simplest form? HINT: Write a proportion using ᎏ1 as one ratio. HINT: One ratio is ᎏ15 and the other is ᎏx inches . 25 5 1 yard 3. Transportation The oldest monorail system in the world is in Wuppertal, Germany. Its track is 8.5 miles long. If you wanted to build a model of the track that has a scale of 1 inch ϭ 0.5 miles, how long would the model track be? B 3. C 4. C A B 5. C 6. B A B 7. A 8. 4. Standardized Test Practice Mavis and Reese want to rearrange the furniture in their living room. Before they move the furniture, they make a model. The scale for the model is 1 inch ϭ 2 feet. If their sofa is actually 6 feet long, how long is the model of the sofa? A 3 inches B 4 inches C 3 feet D 4 feet A 4. inches 17 3. yard 1 to inches 3 2. miles 175 1. Answers: © Glencoe/McGraw-Hill Parent and Student Study Guide 71 Mathematics: Applications and Concepts, Course 1 NAME ________________________________________ DATE ______________ PERIOD _____ Modeling Percents (pages 395–397) A percent is a ratio that compares a number to 100. Since percent means out of one hundred, you can use a 10 ϫ 10 grid to model percents. A Model 20% B Model 36% 20% means 20 out of 36% means 36 out 100. So, shade 20 of of 100. So, shade 36 20% 36% the 100 squares. of the 100 squares. Model each percent. 1. 8% 2. 45% 3. 17% 4. 63% 5. 55% 6. 90% Identify each percent modeled. 7. 8. 9. 10. 11. 12. 13. At the school cafeteria, 65% of the students drink soda. Make a model to show 65%. 14. Use a model to show which is smaller, 83% or 77%. B 3. C 4. C A B 5. C 6. B A B 7. A 8. 15. Standardized Test Practice In the eighth grade class, 16% exercise 3 to 4 hours per week, 23% exercise 2 to 3 hours per week, 25% exercise 1 to 2 hours per week, 36% exercise 0 to 1 hours per week. What amount of time spent exercising has the least percentage of students? A 3 to 4 hours B 2 to 3 hours C 1 to 2 hours D 0 to 1 hours A 15. smaller. 77% is 77% 14. Key. Answer See 13–14. 18% 12. 6% 11. 72% 10. 32% 9. 85% 8. 25% 7. Key Answer See 1–6 Answers: © Glencoe/McGraw-Hill Parent and Student Study Guide 72 Mathematics: Applications and Concepts, Course 1 NAME ________________________________________ DATE ______________ PERIOD _____ Percents and Fractions (pages 400–403) A percent is a ratio that compares a number to 100. Writing To write a percent as a fraction, follow these steps.