Wordproblems. 1. Problem solving

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1 Wordproblems 1. Problem solving Many problems can be translated into algebraic equations. When problems are solved using algebra, we follow these steps: Step 1: Read the problem. Step 2: Decide on the unknown quantity and allocate a variable. Step 3: Decide which operations are involved. Step 4: Translate the problem into an equation. Step 5: Solve the equation by isolating the variable. Step 6: Check that your solution does satisfy the original problem. Step 7: Write your answer in sentence form. Remember, there is usually no variable in the original problem. Example 1 When a number is trebled and subtracted from 7, the result is 11. Find the number. Let x be the number, so 3x is the number trebled. 7 3x is this number subtracted from 7. So, 7 3x = 11 3x = 18 x = 6 So, the number is 6. Check: = 7 18 = 11 Example 2 Sarah s age is one third her father s age. In 13 years time her age will be a half of her father s age. How old is Sarah now? Let Sarah s present age be x years, so her father s present age is 3x years. Table of ages: Now 13 years time Sarah x x+13 Father 3x 3x + 13 So, 3x+13 = 2(x+13) 3x+13 = 2x+26 3x 2x = x = 13 Sarah s present age is 13 years. 1

2 2 Wordproblems-3 o ESO Exercises - Set A 1. When three times a certain number is subtracted from 15 the result is 6. Find the number. 2. Five times a certain number, minus 5, is equal to 7 more than three times the number. What is the number? 3. Three times the result of subtracting a certain number from 7 gives the same answer as adding eleven to the number. Find the number. 4. I think of a number. If I divide the sum of 6 and the number by 3, the result is 4 more than one quarter of the number. Find the number. 5. The sum of two numbers is 15. When one of these numbers is added to three times the other, the result is 27. What are the numbers? 6. What number must be added to both the numerator and denominator of the fraction 2 5 to get the fraction 7 8? 7. What number must be subtracted from both the numerator and denominator of the fraction 3 4 to get the fraction 1 3? 8. Eli is now one quarter of his father s age. In 5 years time his age will be one third of his father s age. How old is Eli now? 9. When Maria was born, her mother was 24 years old. At present, Maria s age is 20 % of her mother s age. How old is Maria now? 10. Five years ago, Jacob was one sixth of the age of his brother. In three years time his age doubled will match his brother s age. How old is Jacob now? 2. Money and investment problems Problems involving money can be made easier to understand by constructing a table and placing the given information into it. Example 3 Britney has only 2-cent and 5-cent stamps. Their total value is $1,78, and there are two more 5-cent stamps than there are 2-cent stamps. How many 2-cent stamps are there? If there are x 2-cent stamps then there are (x+2) 5-cent stamps. Type Number Value 2-cent x 2x cents 5-cent x+2 5(x+2) cents 2x + 5(x + 2) = 178 2x+5x+10 = 178 7x = 168 x = 24 (equating values in cents) So, there are 24 2-cents stamps.

3 Wordproblems-3 o ESO 3 Exercises - Set B 1. Michaela has 5-cent and 10-cent stamps with a total value of 5,75e. If she has 5 more 10-cent stamps than 5-cent stamps, how many of each stamp does she have? 2. The school tuck-shop has milk in 600 ml and 1 litre cartons. If there are 54 cartons and 40 ml of milk in total, how many 600 ml cartons are there? 3. Aaron has a collection of American coins. He has three times as many 10 cent coins as 25 cent coins, and he has some 5 cent coins as well. If he has 88 coins with a total value $11,40, how many of each type does he have? 4. Tickets to a football match cost 8e, 15eor 20e each. The number of 15etickets sold was double the number of 8etickets sold more 20e tickets were sold than 15e tickets. If the gate receipts totalled e, how many of each type of ticket were sold? 5. Kelly blends coffee. She mixes brand A costing $6 per kilogram with brand B costing $8 per kilogram. How many kilograms of each brand does she have to mix to make 50 kg of coffee costing her $7,20 per kg? 6. Su Li has 13 kg of almonds costing $5 per kilogram. How many kilograms of cashews costing $12 per kg should be added to get a mixture of the two nut types costing $7,45 per kg? 3. Motion problems Motion problems are problems concerned with speed, distance travelled, and time taken. These variables are related by the formulae: speed = distance time distance = speed time time = distance speed Speed is usually measured in either kilometres per hour (denoted km/h or km h 1 ) or metres per second (denoted m/s or m s 1 ). Example 4 A car travels for 2 hours at a certain speed and then 3 hours more at a speed 10 km h 1 faster than this. If the entire distance travelled is 455 km, find the car s speed in the first two hours of travel. Let the speed in the first 2 hours be s km h 1. Speed (km h 1 ) Time (h) Distance (km) First section s 2 2s Second section (s + 10) 3 3(s + 10) Total 455 So, 2s+3(s+10) = 455 2s+3s+30 = 455 5s = 425 s = 85 The car s speed in the first two hours was 85 km h 1.

4 4 Wordproblems-3 o ESO Exercises - Set C 1. Joe can run twice as fast as Pete. They start at the same point and run in opposite directions for 40 minutes. The distance between them is now 16 km. How fast does Joe run? 2. A car leaves a country town at 60 km per hour. Two hours later, a second car leaves the town; it catches the first car after 5 more hours. Find the speed of the second car. 3. A boy cycles from his house to a friend s house at 20 km h 1 and home again at 25 km h 1. If his round trip takes 9 10 of an hour, how far it is to his friend s house? 4. A motor cyclist makes a trip of 500 km. If he had increased his speed by 10 km h 1,he could have covered 600 km in the same time. What was his original speed? 5. Normally I drive to work at 60 km h 1. If I drive at 72 km h 1 I cut 8 minutes off my time for the trip. What distance do I travel? Revision problems (1) 1. One-half of Heather s age two years from now plus one-third of her age three years ago is twenty years. How old is she now? 2. A piece of 16-gauge copper wire 42 cm long is bent into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle. 3. A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there? 4. A wallet contains the same number of pennies, nickels, and dimes. The coins total $1.44. How many of each type of coin does the wallet contain? 5. An aircraft carrier made a trip to Guam and back. The trip there took three hours and the trip back took four hours. It averaged 6 km/h on the return trip. Find the average speed of the trip there. 6. A passenger plane made a trip to Las Vegas and back. On the trip there it flew 432 mph and on the return trip it went 480 mph. How long did the trip there take if the return trip took nine hours? 7. A cattle train left Miami and traveled toward New York. 14 hours later a diesel train left traveling at 45 km/h in a effort to catch up to the cattle train. After traveling for four hours the diesel train finally caught up. What was the cattle train s average speed? 8. Jose left the White House and drove toward the recycling plant at an average speed of 40 km/h. Rob left some time later driving in the same direction at an average speed of 48 km/h. After driving five hours Rob caught up with Jose. How long did Jose drive before Rob caught up? 9. A passenger train leaves the train depot 2 hours after a freight train left the same depot. The freight train is traveling 20 mph slower than the passenger train. Find the rate of each train, if the passenger train overtakes the freight train in three hours. 10. Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet?

5 Wordproblems-3 o ESO 5 Revision problems (2) 1. A boat travels for three hours with a current of 3 mph and then returns the same distance against the current in four hours. What is the boat s speed in calm water? How far did the boat travel one way? 2. A spike is hammered into a train rail. You are standing at the other end of the rail. You hear the sound of the hammer strike both through the air and through the rail itself. These sounds arrive at your point six seconds apart. You know that sounds travels through air at 1100 feet per second and through steel at feet per second. How far away is that spike? 3. The sum of two consecutive integers is 15. Find the numbers. 4. A golf shop pays its wholesaler $40 for a certain club, and then sells it to a golfer for $75. What is the markup rate? 5. A shoe store uses a 40 % markup on cost. Find the cost of a pair of shoes that sells for $ An item is marked down 15 %; the sale price is $127,46. What was the original price? 7. 2 m 3 of soil containing 35 % sand was mixed into 6 m 3 of soil containing 15 % sand. What is the sand content of the mixture? 8. 5 fl. oz. of a 2 % alcohol solution was mixed with 11 fl. oz. of a 66 % alcohol solution. Find the concentration of the new mixture lbs. of mixed nuts containing 55 % peanuts were mixed with 6 lbs. of another kind of mixed nuts that contain 40 % peanuts. What percent of the new mixture is peanuts? 10. Working alone, Ryan can dig a 10 ft by 10 ft hole in five hours. Castel can dig the same hole in six hours. How long would it take them if they worked together? 11. It takes Trevon ten hours to clean an attic. Cody can clean the same attic in seven hours. Find how long it would take them if they worked together. 12. Working together, Paul and Daniel can pick forty bushels of apples in 4,95 hours. Had he done it alone it would have taken Daniel 9 hours. Find how long it would take Paul to do it alone.

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