Study Guide and Intervention

Transcription

1 Study Guide and Intervention You can use two-step equations to represent situations in which you start with a given amount and then increase it at a certain rate. Example PRINTING: A laser printer prints pages per minute. Liza refilled the paper tray after it had printed 2 pages. In how many more minutes will there be a total of 245 pages printed? EXPLORE You know the number of pages printed and the total number of pages to be printed. You need to find the number of minutes required to print the remaining pages. PLAN Let m = the number of minutes. Write and solve an equation. The remaining pages to print is m. remaining pages pages printed total pages m SOLVE m m m 153 m 15 3 m 17 EXAMINE The remaining 153 pages will print in 17 minutes. Since = 2, the answer is correct. Exercises 1. METEOROLOGY During one day in 118, the temperature in Granville, North Dakota, began at 33 and rose for 12 hours. The high temperature was about 51. About how many degrees per hour did the temperature rise? 33 12d 51; about 7 degrees 2. SAVINGS John has $825 in his savings account. He has decided to deposit $65 per month until he has a total of $1800. In how many months will this occur? 65m ; 15 months 3. SKYDIVING A skydiver jumps from an airplane at an altitude of 12,000 feet. After 42 seconds, she reaches 4608 feet and opens her parachute. What was her average velocity during her descent? 12,000 42f 4608; 176 feet per second 4. FLOODING The water level of a creek has risen 4 inches above its flood stage. If it continues to rise steadily at 2 inches per hour, how long will it take for the creek to be 12 inches above its flood stage? 4 2h 12; 4 hours Glencoe/McGraw-Hill 132 Glencoe Pre-Algebra

2 Skills Practice 1. Eleven less than 5 times a number is 24. 5n 11 24; 7 2. The quotient of a number and increased by 10 is 11. n 10 11; 3. Five less than the product of 3 and a number is 2. 3n 5 2; 1 Lesson 4. Fifteen more than twice a number is n 23; 1 5. The difference between 5 times a number and 4 is 16. 5n 4 16; 4 6. Nine more than 8 times a number is 7. 8n 7; 2 7. The difference between 12 and ten times a number is n 28; 4 8. Seven more than three times a number is 52. 3n 7 52; 15. Eleven less than five times a number is 1. 5n 11 1; Thirteen more than four times a number is 1. 4n 13 1; Seven less than twice a number is 43. 2n 7 43; SHOPPING The total cost of a suit and 4 ties is $22. The suit cost $200. Each tie cost the same amount. Find the cost of one tie. 4x ; $ AGES Mary s sister is 7 years older than Mary. Their combined ages add up to 35. How old is Mary? 2x 7 35; 14 Glencoe/McGraw-Hill 133 Glencoe Pre-Algebra

3 Practice 1. Eight less than 7 times a number is 2. 7n 8 2; 3 2. Twenty more than twice a number is n 52; The difference between three times a number and 11 is 10. 3n 11 10; 7 4. One more than the difference between 18 and seven times a number is n ; 4 5. Eight times a number plus 6 less than twice the number is 34. 8n 2n 6 34; more than the product of a number and 17 is n 26 42; 4 7. Twelve less than the quotient of a number and 8 is 1. n 12 1; ANIMAL TRAINING Last summer, Gary trained 32 more dogs than Zina. Together they trained 126 dogs. How many dogs did Gary train? g (g 32) 126; 7 dogs. SALES Julius sold five times as many computers as Sam sold last year. In total, they sold 78 computers. How many computers did Julius sell? 5s s 78; 65 computers 10. TRACK In one season, Ana ran 18 races. This was four fewer races than twice the number of races Kelly ran. How many races did Kelly run? 18 2r 4; 11 races 11. BASEBALL André hit four more home runs than twice the number of home runs Larry hit. Together they hit 10 home runs. How many home runs did André hit? h 2h 4 10; 8 home runs Glencoe/McGraw-Hill 134 Glencoe Pre-Algebra

4 Answers (Lessons 3-5 and ) 3-5 Enrichment Al-Khowarizmi: The Father of Algebra The title Father of Algebra should be awarded to the Arabian mathematician Al-Khowarizmi. In the ninth century, he wrote a work entitled Hisab al-jabr-w al muqubalah, meaning the science of restoring and canceling. In this work, he gave a clear and complete explanation of how to solve an equation by performing the same operation on both sides of the equation. In the thirteenth century, Al-Khowarizmi s work was translated into Latin, the language of educated people in Europe, which launched algebra into the Western world. It is from the title of his work that we get the word algebra. Here is an example of how Al-Khowarizmi solved an equation like x To solve this equation, 5 must be added, or restored, to each side. Thus, x , or x Another example of restoring can be used to solve x To solve this equation, 2 5 must be restored to each side. This is the same as subtracting 5 from each side. Thus, x , or x 5 5. Here is an example of solving an equation by canceling (or dividing). To solve 3x 5, each side must be canceled by a factor of 3. Thus, } 3 3 x } 5 } }, or x Solve each equation and label it with an R for restoring or with a C for canceling. 1. x y x 5 12 x 30; R y 5; R x 3; C 4. 5y y x 5 18 y 3; C y 11; R x 6; C 7. x y y 5 21 x 1; R y 5; C y 7; C 10. Make up your own equations to solve. Three should use restoring and three canceling. See students work. Glencoe/McGraw-Hill 131 Glencoe Pre-Algebra Study Guide and Intervention You can use two-step equations to represent situations in which you start with a given amount and then increase it at a certain rate. Example PRINTING: A laser printer prints pages per minute. Liza refilled the paper tray after it had printed 2 pages. In how many more minutes will there be a total of 245 pages printed? EXPLORE You know the number of pages printed and the total number of pages to be printed. You need to find the number of minutes required to print the remaining pages. PLAN Let m = the number of minutes. Write and solve an equation. The remaining pages to print is m. remaining pages 1 pages printed 5 total pages m SOLVE m m m m 5 } 15 3 } m 5 17 EXAMINE The remaining 153 pages will print in 17 minutes. Since = 2, the answer is correct. Exercises 1. METEOROLOGY During one day in 118, the temperature in Granville, North Dakota, began at 2 33 and rose for 12 hours. The high temperature was about 51. About how many degrees per hour did the temperature rise? 33 12d 51; about 7 degrees 2. SAVINGS John has $825 in his savings account. He has decided to deposit $65 per month until he has a total of $1800. In how many months will this occur? 65m ; 15 months 3. SKYDIVING A skydiver jumps from an airplane at an altitude of 12,000 feet. After 42 seconds, she reaches 4608 feet and opens her parachute. What was her average velocity during her descent? 12,000 42f 4608; 176 feet per second 4. FLOODING The water level of a creek has risen 4 inches above its flood stage. If it continues to rise steadily at 2 inches per hour, how long will it take for the creek to be 12 inches above its flood stage? 4 2h 12; 4 hours Glencoe/McGraw-Hill 132 Glencoe Pre-Algebra Lesson 3-5 Glencoe/McGraw-Hill A14 Glencoe Pre-Algebra

5 Answers (Lesson ) Skills Practice 1. Eleven less than 5 times a number is 24. 5n 11 24; 7 2. The quotient of a number and 2 increased by 10 is 11. } n } 10 11; 3. Five less than the product of 2 3 and a number is n 5 2; 1 4. Fifteen more than twice a number is n 23; 1 5. The difference between 5 times a number and 4 is 16. 5n 4 16; 4 6. Nine more than 2 8 times a number is n 7; 2 7. The difference between 12 and ten times a number is n 28; 4 8. Seven more than three times a number is 52. 3n 7 52; 15. Eleven less than five times a number is 1. 5n 11 1; Thirteen more than four times a number is n 13 1; Seven less than twice a number is 43. 2n 7 43; SHOPPING The total cost of a suit and 4 ties is $22. The suit cost $200. Each tie cost the same amount. Find the cost of one tie. 4x ; $ AGES Mary s sister is 7 years older than Mary. Their combined ages add up to 35. How old is Mary? 2x 7 35; 14 Glencoe/McGraw-Hill 133 Glencoe Pre-Algebra Practice 1. Eight less than 7 times a number is n 8 2; 3 2. Twenty more than twice a number is n 52; The difference between three times a number and 11 is 10. 3n 11 10; 7 4. One more than the difference between 18 and seven times a number is n ; 4 5. Eight times a number plus 6 less than twice the number is 34. 8n 2n 6 34; more than the product of a number and 17 is n 26 42; 4 7. Twelve less than the quotient of a number and 8 is 2 1. } n } 12 1; ANIMAL TRAINING Last summer, Gary trained 32 more dogs than Zina. Together they trained 126 dogs. How many dogs did Gary train? g (g 32) 126; 7 dogs. SALES Julius sold five times as many computers as Sam sold last year. In total, they sold 78 computers. How many computers did Julius sell? 5s s 78; 65 computers 10. TRACK In one season, Ana ran 18 races. This was four fewer races than twice the number of races Kelly ran. How many races did Kelly run? 18 2r 4; 11 races 11. BASEBALL André hit four more home runs than twice the number of home runs Larry hit. Together they hit 10 home runs. How many home runs did André hit? h 2h 4 10; 8 home runs Glencoe/McGraw-Hill 134 Glencoe Pre-Algebra Lesson Answers Glencoe/McGraw-Hill A15 Glencoe Pre-Algebra