TRANSLATING WORD PROBLEMS

KEYWORDS

|Addition |increased by |
| |more than |
| |combined, together |
| |total of |
| |sum |
| |added to |
|Subtraction |decreased by |
| |minus, less |
| |difference between/of |
| |less than, fewer than |
|Multiplication |of times, multiplied by |
| |product of |
| |increased/decreased by a factor of(this type can involve both |
| |addition or subtraction and multiplication!) |
|Division |per, a |
| |out of |
| |ratio of, quotient of |
| |percent(divide by 100) |
|Equals |is, are, was, were, will be |
| |gives, yields |
| |sold for |

Application Examples: 1. Translate the “the sum of 8 and y” into an algebraic expression 2. Translate the “4 less than x” into an algebraic expression 3. Translate the “x multiplied by 13” into an algebraic expression 4. Translate the “the quotient of x and 3” into an algebraic expression 5. Translate the “the difference of 5 and y” into an algebraic expression 6. Translate the “the ratio of 9 more than x to x” into an algebraic expression 7. Translate the “nine less than the total of a number and 2” into an algebraic expression, and simplify. 8. The length of a football field is 30 yards more than its width. Express the length of the field in terms of its width w. 9. Twenty gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the small container in terms of the amount g poured into the larger container.

AGE WORD PROBLEMS 1. In January of the year 2000, my husband John was eleven times as old as my son William. In January of 2012, he will be three times as old as my son. How old was my son in January of 2000? 2. In three more years, Miguel’s grandfather will be six times as old as Miguel was last year. When Miguel’s present age is added to his grandfather’s present age, the total is 68. How old is each one now? 3. 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? 4. “Here lies Diophantus,” the wonder behold…. Through art algebraic, the stone tells how old: “God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by this science of numbers for four years, he ended his life.”

Find Diophantus’ age at death.

GEOMETRIC WORD PROBLEMS 1. Three times the width of a certain rectangle exceeds twice its length by three inches, and four times its length is twelve more than its perimeter. Find the dimensions of the rectangle. 2. Suppose a water tank in the shape of right circular cylinder is thirty feet long and eight feet in diameter. How much sheet metal was used in its construction? 3. A piece of wire 42 cm long is bent into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle. 4. A circular swimming pool with a diameter of 28 feet has a deck of uniform width built around it. If the area of the deck is 60(pi) square feet, find its width. 5. If the height of a triangle is five inches less than the length of its base, and if the area of the triangle is 52 square inches, find the base and the height. 6. If the sum of the sides of a right triangle is 49 inches and the hypotenuse is 41 inches, find the two sides. 7. The smallest angle of a triangle is two-thirds of the middle angle, and the middle angle is three-sevenths of the largest angle. Find all three angle measures. 8. The rectangle is 8 feet long and 6 feet wide. If each dimension is increased by the same number of feet, the area of the new rectangle formed is 32 square feet more than the area of the original rectangle. By how many feet was each dimension increased? 9. You worked for a fencing company. A customer called, wanting to fence in his 1,320 square-foot garden. He ordered 148 feet of fencing, but you forgot to ask him the length and width of the garden (these dimensions will determine some of the details of the order, so you need the information). You don’t want the customer to think that you are incompetent, so you figure out the length and width from the information the customer has already given you. What are the dimensions? 10. You need to make a pizza box. You know that the box needs to be two inches deep, it needs to be a square, and the web site you found said that the box needs to have a volume of 512 cubic inches. After cursing the occasional near-uselessness of information that you find on the Internet, you start calculating the dimensions you will need. You have a large piece of cardboard, but you do not have enough cardboard to make a mistake and try again, so you will have to get right it the first time. You will be forming a box by cutting a large square, and then cutting out the two-inch squares from the corners that will allow you to fold up the edges to make a two-inch-deep box. What should be the dimensions of the large square? (Ignore the top of the box: you will just make another open box, slightly larger, turn it upside down, and slip it over the first box to make the “top”.) 11. Find the largest rectangular area that you can enclose, assuming that you have 128 meters of fencing. What is the (geometric) significance of the dimensions of this largest possible enclosure?

“COIN” WORD PROBLEMS

1. Your uncle walks in, jingling the coins in his pocket. He grins at you and tells you that you can have the coins if you can figure out how many of each kind of coin he is carrying. You are not interested until he tells you that he’s been collecting those gold-tone dollar coins. The twenty-six coins in his pocket are dollars and quarters, and they add up to seventeen dollars. How many of each coin does he have? 2. 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? 3. 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?

DISTANCE WORD PROBLEMS 1. A 555-mile, 5-hour plane trip was flown at two speeds. For the first of the trip, the average speed was 105 mph. Then the tailwind picked up, and the remainder of the trip was flown at an average speed of 115 mph. For how long did the plane fly at each speed? 2. An executive drove from home at an average speed of 30 mph to an airport where a helicopter was waiting. The executive boarded the helicopter and flew to the corporate offices at an average speed of 60 mph. The total distance was 150 miles; the entire trip took three hours. Find the distance from the airport to the corporate offices. 3. A car and a bus set out at 2 p.m. from the same point, headed in the same direction. The average speed of the car is 30 mph slower than twice the speed of the bus. In two hours, the car is 20 miles ahead of the bus. Find the rate of the car. 4. A passenger train leaves a train depot 2 hours after the freight train leaves 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. 5. Two cyclists started 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? 6. A boat travels for three hours with a current of 3 mph and 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? 7. With the wind, an airplane travels 1120 miles in seven hours. Against the wind, however, it takes eight hours. Find the rate of the plane in still air and the velocity of the wind. 8. A spike is hammered into a trail rail. You are standing at the other end of the trail. 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 the sound travels through air at 1100 feet per second and through steel at 16,500 feet per second. How far away is that spike?

MIXTURE WORD PROBLEMS 1. Your school is holding a “family holiday” event this weekend. Students have been pre-selling tickets to the event; adult tickets are $5.00, and child tickets (for six-year-old and under) are $2.50. From past experience, you expect about 13,000 people to attend the event. But this is the first year in which tickets prices have been reduced for the younger children, so you really don’t know how many child tickets and how many adult tickets you can expect to sell. Your boss wants you to estimate the expected ticket revenue. You decide to use the information from the pre-sold tickets to estimate the ratio of adults to children, and figure the expected revenue from this information. You consult with your student ticket-sellers, and discover that they have not been keeping track of how many child tickets they have sold. The tickets are identical, until the ticket-seller punches a hole in the ticket, indicating that it is a child ticket. But they don’t remember how many holes they have punched. They only know that they have sold 548 tickets for $2460. How much revenue from of each of child and adult tickets can you expect? 2. Suppose you work in a lab. You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% solution. How many liters of 10% solution and 30% solution should you use? 3. How many liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution? 4. How many ounces of pure water must be added to 50 ounces of a 15% saline solution to make a saline solution that is 10% salt? 5. Find the selling price per pound of a coffee mixture made from 8 pounds of coffee that sells for $9.20 per pound and 12 pounds of coffee that costs $5.50 per pound. 6. How many pounds of lima beans that cost $0.90 per pound must be mixed with 16 pounds of corn that costs $0.50 per pound to make a mixture of vegetables that costs $0.65 per pound? 7. Two hundred liters of a punch that contains 35% fruit juice is mixed with 300 liters of another punch. The resulting fruit punch is 20% fruit juice. Find the percent of fruit juice in the 300 liters of punch. 8. Ten grams of sugar are added to a 40-g serving of a breakfast cereal that is 30% sugar. What is the percent concentration of sugar in the resulting mixture?

NUMBER WORD PROBLEMS 1. The sum of two consecutive integers is 15. Find the numbers. 2. The product of two consecutive negative even integers is 24. Find the numbers. 3. Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers?

WORK WORD PROBLEMS 1. Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house? 2. One pipe can fill a pool 1.25 times faster than a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used? 3. When the tub faucet is on full, it can fill the tub to overflowing in 20 minutes (we’ll ignore the existence of the overflow drain). The drain can empty the tub in 15 minutes. Your four-year-old has managed to turn the faucet on full, and the drain was closed. Just as the tub starts to overflow, you run in and discover the mess. You grab the faucet handle, and it comes off in your hand. You yank the drain open, and run for towels to clean up the overflow. How long will it take for the tub to empty, with the faucet still on but the drain now open/ 4. Two guys were working on your car. One can complete the given job in six hours, but the new guy takes eight hours. They worked together for two hours, but then the first guy left to help another mechanic on a different job. How long will it take the new guy to finish your car? 5. Working alone, Maria can complete a task in 100 minutes. Shaniqua can complete the same task in two hours. They work together when Liu, the new employee, joins and begins helping. They finish the task 20 minutes later. How long would it take Liu to complete the task alone? 6. Working together, Bill and Tom painted a fence in 8 hours. Tom painted the fence by himself. The year before, Bill painted it by himself, but took 12 hours less than Tom took. How long did Bill and Tom take, when each was painting alone? 7. Ben takes 2 hours to wash 500 dishes, and Frank takes 3 hours to wash 450 dishes. How long till they take, working together, to wash 1000 dishes? 8. If six men can do a job in fourteen days, how many would it take to do the job in twenty-one days?

MARKUP-MARKDOWN PROBLEMS 1. A computer software retailer used a markup rate of 40%. Find the selling price of a computer game that cost the retailer $25. 2. A golf shop pays its wholesaler $40 for a certain club, and then sells it for $75. What is the markup rate? 3. A shoe store uses a 40% markup on cost. Find the cost of a pair of shoes that sells for $63. 4. An item originally priced at $55 is marked 25% off. What is the sale price? 5. An item that regularly sells for $425 is marked down to $318.75. What is the discount rate? 6. An item is marked down 15%; the sale price is $127.46. What was the original price?

INCREASE-DECREASE PROBLEMS 1. Growing up, you lived in a tiny country village. When you left for college, the population was 840. You recently heard that the population has grown by 5%. What is the present population? 2. Your friend diets and goes from 125 pounds to 110 pounds. What was her percentage weight loss? 3. Your boss says that his wife has put an 18 x 51 foot garden in along the whole back end of their backyard. He says that this has reduced the lawn area by 24%. You know that properties in that part of town tend to be long and skinny. What are the total dimensions of his backyard? What are the dimensions of the remaining lawn area?