Algebra 2/Trig: Solving Word Problems Directions for Sets A to E: Solve each problem while showing sufficient work. Sufficient work includes the following: 1) define your variable, 2) write an equation that will be used to solve the problem, 3) show work while solving your equation, and 4) include the correct label with your answer (if necessary). Problem Set A 1. Nine more than a number is 58. What is the number? 2. A number decreased by 8 is 31. What is the number? 3. A basketball player scored 56 points in a game. This was 12 points more than he had scored in an earlier game. What was his score on the earlier game? 4. Three members of the School Committee missed the last meeting. Twelve members were present at the meeting. How many members does the committee have? 5. If the temperature of the water in a beaker rises six degrees Celsius above what it is now, the water will be at the boiling point (100 degrees Celsius). What is the temperature of the water now? 6. If the latitude of New York is about 41 degrees North and the latitude of Rio de Janeiro is about 23 degrees South, what is the difference in latitude between the two cities? 7. A discount store chain hired 130 new employees during a year in which 27 employees retired and 59 left for other reasons. If there were 498 employees in the chain at the end of the year, how many were there at the beginning? 8. Rona paid $3.23 for 2 tubes of toothpaste. She paid the regular price of $1.79 for one tube, but bought the other one for less, because she used a cents-off coupon. What was the coupon worth? Problem Set B 1. Negative eight times a number is 376. What is the number? 2. One third of a number is 912. What is the number? 3. Daniel Erik paid $147 for six theater tickets. How much did each ticket cost? 4. The perimeter of a square lot is 156 m. How long is each side of the lot? 5. Twelve-year-old Lola is one fourth as old as her Uncle Hector. How old is Hector? 6. A 75-watt bulb consumes 0.075 kw * h (kilowatt hours) of energy when it burns for one hour. How long was the bulb left burning during a period when it consumed 3.3 kw * h of energy? 7. A police helicopter clocked an automobile for 10 seconds over a stretch of highway one-fifth of a mile long. At what rate was the automobile traveling in miles per a) second? b) hour? 8. For selling a home, an agent receives $6 for every one hundred dollars in the selling price. What was the selling price of a home for which the agent received $4725?
Problem Set C 1. The sum of two consecutive integers is 87. 2. The sum of four consecutive integers is 106. 3. The sum of four consecutive integers is 42. 4. The sum of three consecutive odd integers is 81. 5. The sum of three consecutive odd integers is 147. 6. The sum of four consecutive even integers is 100. 7. The greater of two consecutive even integers is six less than twice the smaller. 8. The smaller of two consecutive even integers is five more than one half of the greater. 9. The four Smith children were born at two-year intervals. The sum of their ages is 36. 10. A rectangle has a perimeter of 62 cm. The lengths in centimeters of its adjacent sides are consecutive integers. 11. There are four consecutive integers. Three times the greatest is 6 more than the sum of the other three. 12. There are three consecutive integers listed in natural order. The sum of the first and half the third is 13 less than the second. 13. Find four consecutive integers such that the sum of the two largest subtracted from twice the sum of the two smallest is 15. Problem Set D 1. The sum of 85 and twice a number is 237. Find the number. 2. Five times a number, decreased by 87, is 12. Find the number. 3. If you add 15 to the product of 4 and a number, you get 363. Find the number. 4. The perimeter of a rectangle is 326 and its length is 94. Find its width. 5. Together, a boat and outboard motor cost $1500. The boat cost five times as much as the motor. What was the cost of each? 6. When Joe Darrico retired from his parcel delivery service, his daughter Rose took over the business. By now, Rose has been running the service three years longer than her father did. If the business has been in operation for 27 years, how long did Joe run it? 7. Eighteen-year-old Manolo is 4 years older than twice as old as his sister, Julia. How old is Julia? 8. Find three consecutive integers whose sum is 81. 9. Find four consecutive odd integers whose sum is 112. 10. The lengths in meters of the sides of a triangle are consecutive even integers. The perimeter is 210 m. How long is each side? 11. Two numbers differ by 3. Four times the lesser diminished by three times the greater is 7. Find the numbers. 12. A magazine reported that in a survey of 100 teenagers, the number preferring broiled steak for dinner was 8 more than the number preferring beef stew. But the number preferring macaroni and cheese was 2 more than 4 times the number preferring beef stew. How many teenagers in the survey preferred beef stew? 13. Five centimeters are cut off one side of a square piece of paper and 8 cm are added to an adjacent side. The resulting rectangular piece of paper has a perimeter of 98 cm. What was the area of the original square?
Problem Set E 1. Find a number that is 96 greater than its opposite. 2. Find a number whose product with 9 is the same as its sum with 56 3. Three times a number, decreased by 8, is the same as twice the number, increased by 15. Find the number. 4. The sum of two numbers is 15. Three times one of the numbers is 11 less than five times the other. Find the numbers. 5. The greater of two consecutive integers is 15 more than twice the lesser. Find the integers. 6. Dina has 6 steel balls of equal mass. If she puts 5 of them in one pan of a beam balance and one ball along with a mass of 100 g in the other pan, the pans balance each other. What is the mass of each steel ball? 7. The lengths of the sides of a triangle are consecutive even integers. Find the length of the longest side if it is 14 units shorter than the perimeter. 8. Jean Ackyroyd s starting salary is $18,000, with semi-annual raises of $750. Sue Bathgate s starting salary is $16,200, with semi-annual raises of $900. After how many years will the two women be earning the same salary? 9. Mary has $6 more than Frank. Together Mary and Frank have $90 more than Joe, who has half as much money as Frank. How much does Frank have? 10. Find four consecutive multiples of 4 such that twice the sum of the least and greatest exceeds three times the least by 32. Problem Set F Directions: Copy and fill in each chart. Do not solve. 1. Dana is 2 years younger than Gary. Age now Age next year Gary x? Dana?? 2. Mark is half as old as Sue. Age now Age 3 years ago Sue x? Mark?? 3. Tim was twice as old as Tom last year. Age last year Age now Tom y? Tim?? 4. A mat is 4 cm longer than it is wide. A second mat is 1 cm wider and 2 cm longer than the first. Length Width First mat? z Second mat??
5. In a game played with white, blue, and red chips, one player had twice as many blue chips as white ones and 3 more red ones than blue ones. Each chip has the point value shown in the chart. Number Point value of each chip Total value of all chips of this kind White w 1 w Blue? 2? Red? 5? 6. The ages in years of three sisters are consecutive odd integers. Age now Age last year Age 10 years from now The youngest sister??? The middle sister m?? The oldest sister??? Problem Set G Directions for Sets G to I: For each problem you must 1) set-up a chart/table 2) write an equation that will be used to solve the situation, 3) show work while solving your equation, and 4) include the correct label with your answer (if necessary). You are only allowed to use one variable even if you want to use two, you aren t allowed. 1. Toby is 3 years older than Sarah. Last year their ages totaled 37. How old is each now? 2. Bob is 24 years younger than his father. In 2 years he will be half as old as his father. How old is each now? 3. Tom is 8 and his mother is 31. How long will it be before she is twice as old as he? 4. Bill Phipps is six times as old as his five-year-old niece. How long will it be before he is twice as old as she? 5. The length of a rectangle is 15 cm more than the width. A second rectangle whose perimeter is 72 cm is 5 cm wider but 2 cm shorter than the first rectangle. What are the dimensions of each rectangle? 6. Zelda Carter weighs 55 lb less than her husband. Their combined weight is 215 lb more than that of their daughter, who weighs half as much as her father. How much does Zelda Carter weigh? 7. Jack s age next year will be twice Jill s age last year. Their present ages total 45. How old is each now? 8. Al is three-fourths as old as Ann. Six years ago Al was half as old as Ann. How old is each now? 9. Susan is six years older than Ralph and Ralph is twice as old as Neil. Next year Susan s age will total the ages of Ralph and Neil. How old is each now? 10. Ruby is 6 years younger than Carlo, and the average of their ages is twice Ruby s age 5 years ago. How old are they? 11. George Washington was born eleven years before Thomas Jefferson. In 1770 Washington s age was three more than seven times the age of Jefferson in 1748. How old was each man in 1750? 12. In one match Marcia scored three times as many points as Dina. In the next match, Marcia scored 7 fewer points than she did in the first match, while Dina earned 9 more than she did in the first match. If they tied in the second match, what their scores in the first match? 13. Suppose that 9 servings of juice cost the same as 5 fruit cups. Suppose, also, that one fruit cup costs 50 cents more than one bowl of soup, while one bowl of soup costs 50 cents more than one serving of juice. What would be the cost of each item: serving of juice, fruit cup, bowl of soup? 14. The upper falls of Angel Falls, the highest waterfall on Earth, are 750 m higher than Niagara Falls. If each of the falls were 7 m lower, the upper Angel Falls would be sixteen times as high as Niagara Falls. How high is the upper Angel Falls? Niagara Falls?
Problem Set H 1. Lex spent $8.40 for several pencils costing 20 cents each and some notebooks costing $1.20 each. He bought 7 more pencils than notebooks. How many notebooks did he buy? Number Price Cost Pencils??? Notebooks x?? 2. Fritz makes $4 an hour working after school and $5 an hour working on Saturdays. Last week he made $52.50 working a total of 12 hours. How many hours did he work on Saturday? Hours Wage Income Saturday s?? Weekdays??? 3. Della has some nickels, dimes, and quarters worth $6.10. She has twice as many dimes as quarters and 50 coins in all. How many of each kind of coin does she have? Number Coin Value Total Value Nickels??? Dimes??? Quarters q?? 4. Ruth purchases some one-cent stamps, some fifteen-cent stamps, and some twenty-cent stamps for $5. There are three times as many one-cent stamps as twenty-cent stamps, and 8 fewer twenty-cent stamps than fifteen-cent stamps. How many stamps in all does she buy? Number Value Total Value One-cent??? Fifteen-cent??? Twenty-cent t?? 5. Each of the 24 members of the Boosters Club bought either a pennant or a cap at the football game. The pennants were $1.25 each and the caps were $1.50 each. If the total bill was $34, how many people bought pennants? 6. Adult tickets for the senior class play were $4 each and student tickets were $2. A total of 1250 tickets worth $3400 were sold. How many student tickets were sold? 7. A plumber makes $4.50 an hour more than an apprentice. During an eight hour day, their earnings total $284. How much does each make per hour? 8. I have 40 coins, all dimes, nickels, and quarters, worth $4.05. I have seven more nickels than dimes. How many quarters do I have? 9. Lizzie bought several apples at 20 cents each, ate two of them, and sold the rest for 30 cents each. She made a profit of $2.20. How many did she buy? 10. Mercedes had a total of 37 nickels, dimes, and quarters. She had three fewer dimes than nickels and four more quarters than nickels. How much money did she have? 11. Gunnar bought 47 stamps. He bought five more ten-cent stamps than fifteen-cent stamps and six fewer twenty-cent stamps than fifteen-cent stamps. How much did he pay for the stamps? 12. Roger claims: I have $20 in quarters, half-dollars, and one-dollar bills. I have twice as many quarters as half-dollars, and half as many one-dollar bills as half-dollars. Explain why Roger must be wrong. 13. Can 46 pennies and nickels have a total value of a dollar? Justify your answer.
14. In a collection of coins worth $9.13, there are twice as many dimes as quarters, four more nickels than dimes, and twice as many pennies as nickels. How many of each kind of coin are in the collection? 15. Nadia has seven more nickels than dean has dimes. If Dean gives Nadia four of his dimes, then Dean will have the same value of money as Nadia. How much money do they have together? (Assume that Nadia has only nickels and Dean has only dimes.) Problem Set I 1. Maria can average 30 km/h on her 10-speed racing bike, but Cooper cannot go that fast on his 3- speed bike. When Maria finished a race from Boston to Worcester, 90 km away, Cooper was still 18 km from the finish line. What is Cooper s average speed? Maria??? Cooper x?? 2. Ann leaves Avon at noon biking to Batavia at 16 km/h. At 1 p.m. Bill leaves Batavia and bikes toward Avon at 20 km/h. If the distance between Avon and Batavia is 70 km, at what time do Ann and Bill meet? Ann? x? Bill??? 3. Tracy leaves for work at 8 a.m. traveling at 36 mph. Sarah finds that Tracy has forgotten her briefcase, so she leaves at 8:05 chasing after her at 50 mph. At what time will Sarah catch Tracy? Tracy? x? Sarah??? 4. Rip skates across a frozen lake at 10 mph and returns at 5 mph. If his total skating time is one hour, how wide is the lake? To? x? Back??? 5. Two jets leave St. Louis at 2 p.m., one traveling north at 850 km/h and the other south at 750 km/h. At what time will they be 4000 km apart? 6. Lisa can bike 6 km/h faster than Wendy. At noon, each girl leaves her house, traveling toward the other. If the distance between their houses is 20 km and they meet in 0.5 h, what is each girl s rate? 7. It takes a train 90 min longer to go from Farmington to Allentown at 60 km/h than it does to return at 80 km/h. What is the distance between these towns? 8. A car travels 418 km in 6 h. One hour of the trip is in the city, where its average speed is just half of what the car averaged on the turnpike for the rest of the trip. What was the car s average city speed? 9. Kelly rode her bike from home to the repair shop and then walked home. She spent 10 min riding, 5 min talking to the attendant, and 20 min walking. If Kelly s walking speed is 12 km/h less than her biking speed, find the distance from her house to the repair shop.
10. Because Chip can run 1 m/s faster than Dean, he is able to run across a field in 20 s, a full 2.5 s quicker than Dean. How fast can Chip run? 11. Laura can run at 8 m/s and Mary at 7.5 m/s. On a racetrack, Mary is given a 25 m head start and the race ends in a tie. How long is the track? 12. At 1:30 p.m., George left Exeter for Portsmouth, cycling at 20 km/h. At 2:00 p.m., Luke left Portsmouth for Exeter, cycling at 24 km/h. If the distance from Exeter to Portsmouth is 32 km, at what time did the boys meet? 13. At 2:00 p.m., a small plane had been flying 1 h when a change of wind direction doubled its average ground speed. If the entire trip of 860 km took 2.5 h, how far did the plane travel in the first hour? 14. A ship must average 22 knots (nautical miles per hour) to make its 10-hour run on schedule. During the first four hours, bad weather caused it to reduce its average speed to 16 knots. What should its average speed be for the rest of the trip to maintain its schedule? Problem Set J Suppose you could paddle a canoe at the rate of 3 mph in still water. If you paddle downstream where the current is 1 mph you will be traveling at 4mph. If you paddle upstream with the same 1 mph current you will be traveling at 2 mph. Use this idea to answer the following problems. 1. Suppose you can paddle a canoe in still water at 5 km/hr and you are going to travel down the river and back. The river current is 1 km/hr, and you have only 2.5 hours for your trip, how far can you go? Downstream? x? Upstream? 2.5 - x? 2. Sam can paddle in still water at 6 mph. The Saco River is flowing at 2 mph. How far down stream can Sam travel if he must return in two hours. Down? x? Back??? 3. A powerboat has a four-hour supply of gasoline. How far can this boat travel from the marina if the rate out against the current is 40 km/h and the rate back in with the current is 60 km/h? From Marina? x? To Marina??? 4. A jet can travel the 6000 km distance from Washington DC and London in 6 h with the wind. The return trip against the same wind takes 7.5 h. Find the rate of the jet in still air. 5. A plane flies 3750 km in 3 h. On the return trip, the plane is flying against the wind and the trip takes 5 h. Find the rate of the plane in still air and the speed of the wind. 6. An airplane takes 3 h to fly 1200 km against the wind. The return trip takes 2 h. What would the speed of the plane be if there were no wind? What is the wind speed? 7. A boat can go 20 km downsteam in 2 h. The return trip takes 5 h. What would the speed of the boat be if there were no current? What is the speed of the current? 8. The 4200 km trip from New York to San Francisco takes 6 h flying against the wind but only 5 h returning. Find the speed of the plane in still air and the wind speed. 9. The 1080 km trip from Madrid to Paris takes 2 h flying against the wind and 1.5 h flying with the wind. Find the speed of the plane in still air and the wind speed.
10 A motorboat travels 36 km downstream in the same amount of time that it takes to go 24 km upstream. If the current is flowing at 3 km/hr, what is the rate of the boat in still water? 11. The rate of the current in the Susquehanna River is 4 km/h. If a canoeist can paddle 5 km downstream in the same amount of time that she can paddle 1 km upstream, how fast can she paddle in still water? 12. The steamboat River Queen travels at the rate of 30 km/h in still water. If it can travel 45 km upstream in the same amount of time that it takes to go 63 km downstream, what is the rate of the current? 13. An airplane whose speed in still air is 760 km/h can travel 200 km with the wind in the same amount of time that it takes to fly 1800 km against the wind. What is the wind speed? 14. The Farrells can row 10km/h in still water. They are on a river whose current is 4 km/h. How far upstream can they go if they want to be back at their starting point in one hour? 15. An airplane flies 3000 mi in 4 h, but takes 5 h to make the return trip. What is the plane s speed in still air? 16. A small aircraft flies 400 km from Ardmore to Mackin and then returns. The plane travels 275 km/h in still air, and the wind speed is 25 km/h. Find the total time the trip took. 17. Sam and Janet have rented a canoe. How far upstream can they paddle if their rate in still water is 5 km/h, the rate of the current is 3 km/h, and they must return to their starting point in 3 h? 18. A boat can travel 45 km upstream in 3.6 h. The return trip takes 2 h. Find the rate of the boat in still water and the speed of the current. Problem Set K Directions: These are the review problems. By now you should know what work is expected. If you don t know what work is required, you need to learn the requirements quickly. 1. Find a number whose product with 7 is the same as its sum with 24. 2. Anna is 4 years older than Tina. Last year their ages totaled 38. How old are they now? 3. Rosa is 18 years old and Arthur is 24 years old. How many years ago was Arthur three times as old as Rosa? 4. Helen s age next year will be three times Dan s age last year. Their present ages total 32. How old is each now? 5. Marie paid 80 cents for some packages of cards. She sold all but 18 packages for $2 each, making a profit of $48. How many packages of cards did she buy? 6. Pam has some nickels, dimes, and quarters worth $6.75. There are four times as many nickels as dimes, and five more quarters than dimes. How many of each kind of coin does she have? 7. Hal makes $4 an hour working after school and $6 an hour working on Saturdays. Last week he made $88 working a total of 18 hours. How many hours did he work on Saturday? 8. Adult tickets for a concert were $5 each and student tickets were $2 each. A total of 980 tickets, worth $3460, were sold. How many adult tickets were sold? 9. Sara Jessup earns $6.00 an hour more than her assistant. During an 8-hour day they earn $240. How much does each earn per hour? 10. A bicyclist rode up a mountain road at 12 km/h and then back down at 30 km/h. If the round trip took 3.5 h, how long did the ride up the mountain take? 11. A car leaves a town traveling at 72 km/h. An hour later, a second car leaves the same town, on the same road, traveling at 90 km/h. In how many hours will the second car overtake the first? 12. Two cars start at the same time from the same point and travel in opposite directions. One car travels 10 mph faster than the other. In 3 hours they are 300 miles apart. Find the rate of each car.
13. Two airplanes leave Chicago at 12 noon, one traveling west at 575 km/h and the other east at 625 km/h. At what time will they be 3000 km apart? 14. A motorist driving 90 km/h can go from Midland to Shiloh in two hours less time than a train that averages 75 km/h. How far apart are the two towns? 15. Betty Sue has a bunch of coins. The number of nickels, dimes, and quarters are consecutive even integers, in that order. She has a total of $2.80. How many of each coin does she have? 16. Jimmy Ray has two jobs. Holding traffic signs and an assistant fry cook. He makes 8 bucks an hour holding signs and 5 bucks an hour as a cook. He has to pay a lot pf rent sp he works 49 hours a week. If he made $276.50 last week, how many hours did he work at each job? 17. Jimmy Ray and Billy Joe go bass fishin at Moses Lake. On the trip over they average 60 mph. Coming home however, they only average 45 mph due to the weight of the fish they caught and it took an extra hour. How long did it take them to get home? 18. Betty Sue is going to meet Jimmy Ray for the fish fry. Each forgets who is going to pick up who. Betty Sue leaves her house at 2:00 p.m. traveling 15 mph going towards Jimmy Ray s house. Jimmy Ray leaves his house traveling toward Betty Sue s house at 2:20 p.m. traveling at 12 mph. Their houses are 35 miles apart. At what time will Betty Sue pass Jimmy Ray, and state Geez, that sure looks like Jimmy Ray? 19. Jimmy Ray just purchased his new bass boat using all five of his VISA cards. The boat and the motor cost $15,000. The boat cost 4 times as much as the motor. How much was the boat? 20. Betty Sue, Billy Joe, and Jimmy Ray get jobs at the local golf course as greenskeepers. Billy Joe can mow and fertilize 1.5 greens per hour. Jimmy Ray can mow and fertilize 1.25 greens per hour. Betty Sue being a little sharper can do 2.5 greens per hour. Betty Sue works two hours less than her cousins, Billy Joe and Jimmy Ray. If they worked until they got 16 greens done, how long did each person work?