## K3-4 through K-12 problems.Concept-based problem solving.Learning through problem solving.

 A extensive sequence of algebra problems will be developed in this file. These will be listed in the order of grade level from 3rd to 12th grade. The problems are from a variety of hard copy and electronic sources. The source is identified with each problem. The sources referenced provide additional problems and problem solving instruction. These pages serve to extending the value of these materials by explicitly using the problem solving process. Apply the reliable problem solving process shown in these examples to the word problems in any text or other teaching material and become an expert problem solver. The problems start at the 3-4 grade level. Studying the lower grade problems are of particular value to more advanced students. They provide an opportunity to correct unsatisfactory problem solving habits. Such study is exceptionally rewarding to college students. The problems provide a path for gentle introduction of algebraic concepts. Problem based learning provides the initiative for learning new things through the need to do so. The answers can be obtained by methods known to be effective problem-solving tools, such as using manipulatives, guess and check, forming tables, looking for patterns, acting out, and so forth. In this way, problem solving using algebra can be introduced in a natural way at appropriate levels.

A medley of simple problems. All you need to know about problem solving.

Sara has 15 apples and 12 oranges. How many pieces of fruit does she have?

Sara has 15 apples and 3 times as many oranges. How many pieces of fruit does she have?

Two consecutive numbers have a sum of 91. What are the numbers?

Two numbers have a sum of 87. The larger of the numbers is twice the smaller. What are the numbers?
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6/11/99

What is a cow without legs?
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4/13/99
It is said that a picture is worth a thousand words. It is also claimed that words are a dime a dozen. Assuming these propositions are true, what is the dollar value of a picture?
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11/23/98
Twenty-eight children are going on a picnic. Four children can ride in each car. How many cars are needed?
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6/15/98
The castle kitchen servants brought in 4 pies left over from the feast. 12 pies were eaten at the feast. Queen Mab took 2 home with her. How many pies did the servants bring into the feast at the beginning?
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3/14/99
My towing rope was cut in half
and half was thrown away.
The other half was cut again
one third along the way.
The longer part (ten metres length)
is what I use today.
But how long was my towing rope
before this cutting fray?
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5/22/98
The total fare for 2 adults and 3 children on the Tilt-n-Spew ride is \$14.00. If a child's fare is one half of an adult's fare, what is the adult fare?
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1/22/99
A man is 30 years older than his youngest son. In 17 years he will be twice his son's age. How old is the son?
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9/2/2002
A piece of string is 40 centimeters long. It is cut into three pieces. The longest piece is 3 times as long as the middle-sized piece and the shortest piece is 23 centimeters shorter than the longest piece. Find the lengths of the three pieces.
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3/6/200
Convenience Store Math. The cost of 2 cans of caffeine-rich cola and 3 temporary tattoos is \$4.60. If the cost of 2 tattoos is \$2.20, what is the cost of 2 cans of caffeine-rich cola?
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5/21/98
A street vendor sells two types of newspapers, one for 25 cents and the other for 40 cents. If in one day she sold 100 newspapers and took in exactly 28 dollars, how many of the 25-cent newspapers did she sell?
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11/14/98
One neeb and one noob together cost 3 cents. One neeb and one nub together cost 4 cents. One nub and one noob together cost 5 cents. How much does a neeb cost by itself?
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1/5/98
You put 1/10 of your weekly allowance in the bank for college. If have \$5.00 dollars in the bank, how much allowance have you earned?
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2/19/98
A painted wooden cube, such as a child's block, is cut into twenty seven equal pieces. First the saw takes two parallel and vertical cuts through the cube, dividing it into equal thirds; then it takes two additional vertical cuts at 90 degrees to the first ones, dividing the cube into equal ninths. Finally, it takes two parallel and horizontal cuts through the cube, dividing it into twenty seven cubes. How many of these small cubes are painted on three sides? On two sides? On one side? How many cubes are unpainted?
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7/22/98
There are eight Christmas trees in the lobby of the Cheryl Hotel. Each tree has seven dozen lights. How many lights is that in all?
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1/7/98
A round wheel with a radius of 37 cm rolls at a constant speed of 3 revolutions per second. How far does the axle of the wheel move in 8 seconds?
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11/21/98
Donna has twice as many coins as Cathy. When Donna gives Cathy 2 coins, they each have the same number. How many coins do they each have?
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2/16/98
(Grade 3-5). Bill is twice as old as Juan. The sum of their ages three years ago was 45 years. How old are they now?
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8/25/98
The agricultural school garden is rectangular in shape and measures 20' x 45'. The students plant beans in 2/3 of the garden. One half of that contains lima beans. How many square feet of the garden are planted in lima beans?
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3/8/98
Doris spent 2/3 of her savings on a used car, and she spent 1/4 of her remaining savings on vintage fuzzy dice. If the dice cost her \$250, how much did her original savings total?
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1/27/99
Wendell and Terry both rounded the number 3.4682. Wendell says that he rounded the number up.Terry says that he rounded the number down. To what place value might the number have been rounded by Wendell? By Terry? Explain.
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4/24/99
(Grade 3-6). An improved solution of the previous problem.
Note the change in the order of the steps.

Wendell and Terry both rounded the number 3.4682. Wendell says that he rounded the number up.Terry says that he rounded the number down. To what place value might the number have been rounded by Wendell? By Terry? Explain.
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4/24/99
Donna has twice as many coins as Cathy. When Donna gives Cathy 2 coins, they each have the same number. How many coins do they each have?

Karen is 3 times as old as Allan. In 6 years from now, Karen will be two times as old as Allan. How old will Karen be 6 years from now?
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2/19/00
Mary looked out of her farmhouse window and saw a group of pigeons and donkeys passing by. She counted all the legs of the pigeons and donkeys and found that the total number of legs add up to 66. How many of each kind of animals (pigeons and donkeys) passed by her window if the total number of animals is 24?
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2/26/99
Your class goes on a trip to see a matinee of Cirque du Soleil. The following costs are for the entire class: the bus is \$645, trip insurance is \$99, lunch is \$97, and admission at the child's matinee price is \$580. Several extension problem are given. An excellent set of problems for group work.
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10/19/98
In Pizza Tossing 101, Lily earned an 82 on the first test, another 82 on the second test, a 76 on the third test, and a 92 on the fourth test. What score does she need to get on the fifth test to end up with a five-test mean score of 86?
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10/21/98
In London there were three gangs operating on August 11, 1891. Holmes knew from some inside information that his equal in crime, the clever Moriarty, led a gang with five members. At the same time the treacherous Smerzi headed a gang with seven members and Gilda Z, the trickiest of them all, a gang with eight members.

From certain information from Scotland Yard, it was known that originally none of the gangs was large enough to pull off the Great Train Robbery. The must have added another organiztion that was twice the size of the original gang. Altogether, twenty-one members were involved in the robbery.

Which organization was added to the original organization?
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4/11/98

Three solutions are shown. These provide and opportunity to examine what is meant by "different ways". Compare the solutions. A link to an arithmetic solution to the problem is also provided for those who have not yet developed algebraic skills.

5 years ago, Jay was seven times older than Mary. In five years, Mary will be half as old as Jay (or Jay will be twice as old as Mary). How old is each now?
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12/7/98
See the development of the detailed (simpler) solution. 12/7/98
See the development of the elegant solution. 12/7/98

Lizzy and her brother Dizzy found 7 stinkbugs and 6 lice on Sunday. How many insects did they find altogether?
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12/27/97
Suzy caught 16 flies in a week. The next week she caught only 9 flies. What was the difference?
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12/27/97
A certain stock begins the week trading at 87 1/2 per share. If the average (arithmetic mean) gain for the next four days is 1/2, by how much should the price of the stock increase during Friday so that the total gain for the stock during the entire five days is 5 percent?
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6/17/98
It was claimed that the shepherd was the shepherd of 2000 sheep. The shepherd exclaimed, "I am not the shepherd of two thousand sheep!" Pointing to his flock, he added, "If I had that many sheep plus another flock as large as that, then again half as many as I have out there, I would be the shepherd of two thousand sheep." How many sheep were in the shepherd's flock?
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2/7/98
Before the market opens on Monday, a stock is priced at \$25. If its price decreases \$4 on Monday, increases \$6 on Tuesday, and then decreases \$2 on Wednesday, what is the final price of the stock on Wednesday?
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12/30/1999
A student averages 72 on 5 different tests. If the lowest score is dropped, the average rises to 84. What is the lowest score?
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2/24/99
There are 640 acres in a square mile. How many square meters are there in one acre?
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9/17/98
By increasing her usual speed by 25 kilometers/hr a bus driver decreases the time on a 25K trip by ten minutes. What is her usual speed?
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7/2/99
An Island has no currency; it instead has the following exchange rate:

50 bananas = 20 coconuts
30 coconuts = 12 fish
100 fish = 1 hammock.

How many bananas equal 1 hammock?
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9/20/98
Examples of unit conversion. Algebraic and Factor-Label processes.

1. A car is traveling 65 miles per hour. How many feet does the car travel in one second?

2. The density of water is one gram per cubic centimeter. What is the density of water in pounds per liter?
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8/9/98
If 5 times the first number plus three times the second number equals 47, and 10 times the first number minus 4 times the second number equals 54, what are the numbers?
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4/8/99
If the ratio of Howard Stern fans to non-fans in a group is 4 to 1, what fraction of the people in the group are non-fans?
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12/17/99
Jenny wanted to purchase 2 dozen pencils and a pen. Those items cost \$8.45 and she did not have enough money. So she decided to purchase 8 fewer pencils and paid \$6.05. How much was a pen?
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8/26/98
Bob, Jim and Cathy each have some money. The sum of Bob's and Jim's money is \$18.00. The sum of Jim's and Cathy's money is \$21.00. The sum of Bob's and Cathy's money is \$23.00. How much money does each person have?
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8/26/98
Eight times a number plus 3 times the number is the same as 20 more than 9 times the number. Find the number.
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10/17/98
How many jelly beans fill a one liter bottle?. A Fermi problem.
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3/16/99
Solutions to a collection of simple linear equation.
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12/7/99
Mark reads at an average rate of 30 pages per hour, while Mindy reads at an average rate of 40 pages per hour. If Mark starts reading a novel at 4:30 PM, and Mindy begins reading the same novel at 5:20 PM, at what time will they be reading the same page?
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4/29/98
Adam is 3 times as old as Cynthia and Fred is 16 yrs. younger than Adam. One year ago, Adam's age was twice the sum of Chyntia's & Fred's age. Find their present age.
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3/16/99
In the first year of production a play sells 1572 tickets, in its second year it sells 1753 tickets, in its third year it sells 152 less than in its second year. How many tickets are sold in 3 years?
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12/17/97
You've heard that the coldest temperature ever recorded has been -90 degrees Centigrade at the Soviet installation in Antarctica a few years ago. How much is this in degrees Fahrenheit?
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6/19/98
Daisy is driving from Maryville to a country fair in Yorkville. After driving for two hours at an average speed of 70 kilometers per hour, she still has 80 kilometers left to travel. What is the distance between the two towns, in kilometers?
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3/12/98
Nicole is cruising down the lodge in her '96 Lexus Coupe at a speed of 162 ft/s. How fast is she traveling in miles/hour? Will she get a ticket if the speed limit is 65 miles/hour?
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8/9/98
Paul makes \$25.00 a week less than the sum of what Fred and Carl together make. Carl's weekly income would be triple Steven's if he made \$50.00 more a week. Paul makes \$285.00 a week and Steven makes \$75.00 a week. How much does Fred makE?.
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9/8/98
Harry and Jon plan to spend the afternoon at the fair. After paying the entrance price of \$5.00 each they entered the fair ground. Jon looked around and saw that the Dragon ride was \$3.50 and the Loopo was \$2.75. In addition, there were 3 activities he wanted to do which cost \$1.50 each. Jon guessed that snacks and drinks would cost \$3.50. Jon could see that he did not have enough money. Jon then borrowed \$6.00 from Harry. They noted that after Harry gave Jon \$6.00 Harry still had \$12.00 more than Jon. How much more money did Harry have than Jon had before Harry gave Jon \$6.00?
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9/8/98
A cube has a volume of 8 cm3. The cube is deformed such that the length of each side doubles. What is the volume of this new cube?
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3/21/1999
The radius of both circles shown is 1 meter. One diagram shows a square circumscribed and the other shows a square inscribed on identical circles. Find the sum of the shaded areas of these two figures.
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9/23/98
Three friends, returning from a movie on Friday the 13th, stopped to eat at a restaurant. After dinner, they paid their bill and noticed a bowl of mints at the front counter. Sean took 1/3 of the mints, but returned four because he had a momentary pang of guilt. Faizah then took 1/4 of what was left but returned three for similar reasons. Eugene then took half of the remainder but threw two that looked like they had been slobbered on back into the bowl. (He felt no pangs of guilt - he just didn't want slobbered-on mints.) The bowl had only 17 mints left when the raid was over. How many mints were originally in the bowl?
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9/13/98
Late one night a burglar somehow got into one of the vaults in Fort Knox and started out with a big sack of gold coins. No one really knows how much he stole. At any rate, on his way out, he was stopped by one of the guards, who caught him "holding the bag" so to speak. Fortunately for him, the burglar was able to talk his way out of trouble by offering the guard half the money he had taken with a bonus of \$2,000 thrown in. Just as he was walking away, praising his good luck at having gotten free, he was stopped by a second guard. It took the same bribe, half of all the money he had left, with \$2,000 thrown in, to get by the second guard. Just as he was about to leave, you guessed it, he was stopped by yet a third guard who let him go only after receiving half of all the burglar had left, with \$2,000 thrown in.

By the time the burglar left the front gate of Fort Knox, he had mixed emotions. After all, he did leave with \$9,000 more than he had when he arrived and he escaped a free man. But as he thought of all the money he had left behind with the guards, he wept. Oh, by the way, you now know enough to calculate how much he had taken in the first place.
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4/9/98

Solve the following equation.

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12/20/97
One hundred bushels of corn are to be divided among 100 persons. Men get 3 bushels each. Women get 2 bushels each. Children get 1/2 bushel each. How will the bushels be distributed?
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3/20/98
Doris spent 2/3 of her savings on a used car, and she spent 1/4 of her remaining savings on vintage fuzzy dice. If the dice cost her \$250, how much did her original savings total?
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1/27/98
A bacteria splits in half after 20 minutes, so that after 20 minutes there are 2 bacteria, and after 40 minutes there are 4 bacteria. How many bacteria will there be after 1 hour and 20 minutes? After 2 hours?
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If the area of a circle is 144pi square units, find its circumference.
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12/31/97
A drug store parking lot has space for 1000 cars. 2/5 of the spaces are for compact cars. On Tuesday, there were 200 compact cars and some standard size cars in the parking lot. The parking lot was 3/4 full. How many standard size cars were in the parking lot?
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A number is divided by 2 and the result multiplied by 1/3. The result is then squared and 1 is add. The result of these operations is 50. What is the number?
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1/30/98
A car travels from A to B at a speed of 40 mph then returns from B to A at a speed of 60 mph. What is the average speed for the round trip?
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11/7/99
In the right triangle ABC, CD is an altitude. The circles centered P and Q are inscribed in triangles ACD and BCD respectively. For AC=15 and BC=20 compute PQ.
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8/31/01
Two railway trains, one 400 feet long and the other 200 feet long, run on parallel rails, each at its own constant speed. When they move in opposite directions, the trains pass each other in 5 seconds. When they move in the same direction, the faster train takes 15 seconds to pass the slower train. How many MPH is each train moving?
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11/7/99
A person travels 18 MPH going to a lodge and 12 MPH coming back. The total round trip took 6 hours. How far is the lodge from his house?
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NEW 9/8/2001
A train approaches a tunnel AB. Inside the tunnel is a cat located at a point that is 3/8 of the distance AB measured from the entrance A. When the train whistles the cat moves. If the cat moves to the entrance of the tunnel, A, the train catches the cat exactly at the entrance. If the cat moves to the exit, B, the train catches the cat at exactly the exit. Determine how many times the speed of the train is greater than the speed of the cat.
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3/29/99
On a trip, Debbie spent as much time on the train as she did on the bus. The train averaged 50 miles per hour and the bus averaged 35 miles per hour. If she went 105 miles more on the train, how far did she go altogether?
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10/29/2000
Convert 39800 rods/hour to furlongs per fortnight.
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5/1/98
(Grade 6-10). You are driving along a highway at constant speed of 55 MPH. You observe a car one-half mile behind you. The car is moving fast and zooms by you exactly one minute later. How fast is the car traveling?
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9/28/98
(Grade 6-11). Leon, who is always in a hurry, walked up an escalator, while it was moving, at the rate of one step per second and reached the top in 20 steps. The next day he climbed two steps per second (skipping none), also while it was moving, and reached the top in thirty-two steps. If the escalator had been stopped, how many steps did the escalator have from the bottom to the top?
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9/20/98
You are writing a short adventure story for your English class. In your story, two submarines on a secret mission need to arrive at a place in the middle of the Atlantic ocean at the same time. They start out at the same time from positions equally distant from the rendezvous point. They travel at different velocities but both go in a straight line. The first submarine travels at an average velocity of 20 km/hr for the first 500 km, 40 km/hr for the next 500 km, 30 km/hr for the next 500 km and 50 km/hr for the final 500 km. In the plot, the second submarine is required to travel at a constant velocity, so the captain needs to determine the magnitude of that velocity.
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8/15/98
A graphic arts company can print posters for \$4 each with a daily overhead of \$600. If they sell the posters at \$5.20 each, then how many posters must they sell to have a profit of greater than 10 per cent above their daily cost?
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There are two numbers whose sum is 50. Three times the first is 5 more than twice the second. What are the numbers?
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3/8/98
Sandy held a garage sale during which she charged a dime for everything, but accepted a nickel if the buyer bargained well. At the end of the day she realized she had sold all 12 items and raked in a grand total of 95 cents. She had only dimes and nickels. How many of each did she have?
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12/30/97
Ramona can walk at 2 miles per hour going up a mountain. Going down the same trail, she can walk at 6 miles per hour. If she spends no time at the top, what will be Ramona's average speed for the whole hike?
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1/10/98
Fred and Frank want to know their grandmother's age. She tells them that the digits of their age make up her age. She also tells them that their ages plus her age total 83. What is the age of the grandmother?
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Three ducks and two ducklings weigh 32 kg. Four ducks and three ducklings weigh 44kg. All ducks weigh the same and all ducklings weigh the same. What is the weight of two ducks and one duckling?
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3/19/99
Jessica is planning to drive to Sun Valley, Idaho from her home in Portland, Oregon for a ski vacation. She is concerned that her car's cooling system may not have enough antifreeze for the colder temperatures in Sun Valley. Her car has a 20% solution of antifreeze that protects her car's 8 liter system down to 15o F. However, temperatures in Sun Valley can be as low as -17o F.
How much of the 20% antifreeze should Jessica drain from her car's cooling system and replace with pure antifreeze to protect her auto's cooling system down to -17o F?
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8/9/98
(Grade 7-11). Find an equation of the line containing (5,3) and parallel to the line 5x-3y=4
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10/29/99
(Grade 7-11). Leon, who is always in a hurry, walked up an escalator, while it was moving, at the rate of one step per second and reached the top in 20 steps. The next day he climbed two steps per second (skipping none), also while it was moving, and reached the top in thirty-two steps. If the escalator had been stopped, how many steps did the escalator have from the bottom to the top?
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10/29/98
The first number is four more than two times the second number. The sum of the two numbers is ninety seven. Find the numbers.
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12/23/97

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12/30/97
A man purchases some 2-cent and some 15-cent stamps at the same time. He pays \$1.56 for all the stamps. There are 10 more 2-cent stamps than 15-cent stamps. How many of each kind did he buy?
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1/2/98
Chip said to Dale, "If you give me one acorn, then we will have an equal number of acorns." Dale replied with delight, "If you give me one acorn, then I will have double the number you have!" What was the total number of acorns they had in their trees? How many did Chip have and how many did Dale have?
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The length of a rectangle is four times as long as its width. If the area is 100 m2 what is the length of the rectangle?
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12/25/97

A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, three miles north of the north gate, and it can just be seen from a point nine miles east of the south gate. The problem is to find the diameter of the wall that surrounds the city.
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7/27/99

Find the solution set- square root of 2x-3 = 2 times the square root of 3x -2
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3/6/2000

To the nearest degree, find the angle formed by intersection of a diagonal on the face of a unit cube with the diagonal of the cube drawn from the same vertex.
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9/20/98
On an insect collecting project Lizzy caught mudwasps, Izzy caught waterbugs and Dizzy caught flies. It was observed that three times the number of mudwasps that Lizzy caught less 14 was equal to the difference between the number of flies Dizzy caught and the number of waterbugs Izzy caught. In addition, it was observed that the sum of the mudwasps caught by Izzy and the waterbugs caught by Lizzy was 10 less than three times the flies caught by Dizzy. On further examination of the collection it was seen that three times the number of waterbugs caught by Izzy plus the difference between the number of mudwasps caught by Lizzy and the number of flies caught by Dizzy was 16. How many mudwasps did Lizzy collect?
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3/28/98
An equilateral triangle with each side = x cm is inscribed in a circle. Find the radius r of the circle in terms of x.
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3/4/2000
When Lee was thrice as old as Kevin,
His sister Kate was twenty-seven.
When Kevin was half as old as Kate,
Then brother Lee was thirty-eight.
Their ages add to one forty three.
How old are Kevin, Kate and Lee?
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5/15/98
The area of a rectangle is 360 m2. If its length is increased by 10m and its width is decreased by 6m, then its area does not change. Find the perimeter of the original rectangle.
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2/14/98
The average of 10 students' test scores was 69. Three students scored 80, 60 and 40, respectively. Three students scored 90 and two students scored 50. The other two students received the same score. What were their scores?
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1/15/98
Alex, Fred and Thomas run at constant rates. In a race of 1000m, Alex finished 200m ahead of Fred and 400m ahead of Thomas. When Fred finished, how far was he ahead of Thomas? (in m)
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The mass of the Great Pyramid is 557t greater than that of the Leaning Tower of Pisa. Stone Henge has a mass of 2695t, which is 95t less than the Leaning Tower of Pisa. There once was a Greater Pyramid which had a mass twice that of the Great Pyramid. What was the mass of the Greater Pyramid.
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12/25/97

 (Grade 9-12). Determine the length of the diagonal of the parallelogram shown if a = 20 cm, b = 12 cm and h = 10 cm.

See the development of the solution.  8/31/98
A 100-kg astronaut in space throws a 10-kg rock with a velocity of 25 m/s. How is the astronaut affected?
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10/2/98
You wish to temporarily mount a wire to serve as the antenna of your transmitter in your laboratory. It is necessary that this be straight and as long as possible. You recall that when you had the floor covered required exactly 200 square yards of carpet and that the long side of the room was 20 yards. You also know that the height of the room is 12 feet. You go to the store room to get the wire but find that it can only be supplied in lengths that are an integral multiple of 1 meter. What length of wire do you take back to the lab?
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8/31/98
An urn contains three white and four black balls. One of the balls is drawn out of the urn. Find the probability that the ball is white.
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2/4/99
An urn contains three white and four black balls. We take out a ball and put it in a drawer without looking at it. After that we take out a second ball. Find the probability that this ball is white.
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2/4/99
Given two intersecting straight lines and a point P marked on one of them, show how to construct a circle that is tangent to both lines including point P.
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8/31/98
A farmer has a circular field, radius R metres, in which he has a goat tethered to one edge by a length of chain, L metres long. If the goat is able to graze exactly half of the available area, find an expression for L in terms of R.
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5/18/98
Determine the roots of z3 + 6z2 - 4z - 24 = 0
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A piece of straight, level rail whose length is one mile in winter increases in length by one foot in summer. If the ends were firmly anchored in winter, we are to assume that the rail will become an arc of a circle in summer. How far will the center point move relative to its winter position?
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 (Grade 11). IA C is the center of the circle and F is a point on the circle such that BCDF is a 2cm by 3cm rectangle. What is the area of the shaded region? (in cm2).

See the development of the solution. 12/31/97

# General

Examples of conversion of units.
See the examples.

Use Quick Math.