The next generation of problem solving
Seize the moment!
K3-4 through K-12 problems.
Concept-based problem solving.
Learning through problem solving.
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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.
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Fequently asked questions.
(Grade K-999).
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?
See the development of the solutions. 6/11/99
(Grade 1-12).
What is a cow without legs?
See the development of the solution. 4/13/99
(Grade 2-4).
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?
See the development of the solution. 11/23/98
(Grade 2-4).
Twenty-eight children are going on a picnic. Four
children can ride in each car. How many cars are needed?
See the development of the solution. 6/15/98
(Grade 2-4).
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?
See the development of the solution. 3/14/99
(Grade 2-8).
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?
See the development of the solution. 5/22/98
(Grade 3-5).
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?
See the development of the solution. 1/22/99
(Grade 3-5).
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?
See the development of the solution. 9/2/2002
(Grade 3-5).
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.
See the development of the solution. 3/6/200
(Grade 3-5).
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?
See the development of the solution. 5/21/98
(Grade 3-5). About Algebra
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?
See the development of the solution. 11/14/98
(Grade 3-4). IA
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?
See the development of the solution. 1/5/98
(Grade 3-5). IA
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?
See the development of the solution. 2/19/98
(Grade 3-5).
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?
See the development of the solution. 7/22/98
(Grade 3-5). IA
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?
See the development of the solution. 1/7/98
(Grade 3-5).
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?
See the development of the solution. 11/21/98
(Grade 3-5). IA
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?
See the development of the solution. 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?
See the development of the solution. 8/25/98
(Grade 3-5). IA
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?
See the development of the solution. 3/8/98
(Grade 3-5).
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?
See the development of the solution. 1/27/99
(Grade 3-6).
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.
See the development of the solution. 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.
See the development of the solution. 4/24/99
(Grade 3-6).
Grade 3-4
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?
Grade 5-6
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?
See the development of the solution. 2/19/00
(Grade 3-6).
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?
See the development of the solution. 2/26/99
(Grade 3-6).
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.
See the development of the solution. 10/19/98
(Grade 3-6).
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?
See the development of the solution. 10/21/98
(Grade 3-6). IA
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?
See the development of the solution. 4/11/98
(Grade 3-9).
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?
See the development of the solution. 12/7/98
See the development of the detailed (simpler) solution. 12/7/98
See the development of the elegant solution. 12/7/98
(Grade 4). IA
Lizzy and her brother Dizzy found 7 stinkbugs and 6 lice
on Sunday. How many insects did they find altogether?
See the development of the solution. 12/27/97
(Grade 4). IA
Suzy caught 16 flies in a week. The next week she caught only 9 flies. What was the difference?
See the development of the solution. 12/27/97
(Grade 4-5).
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?
See the development of the solution. 6/17/98
(Grade 4-5). IA
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?
See the development of the solution. 2/7/98
(Grade 4-6).
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?
See the development of the solution. 12/30/1999
(Grade 4-6).
A student averages 72 on 5 different tests. If the
lowest score is dropped, the average rises to 84.
What is the lowest score?
See the development of the solution. 2/24/99
(Grade 4-7).
There are 640 acres in a square mile. How many
square meters are there in one acre?
See the development of the solutions. 9/17/98
(Grade 4-7).
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?
See the development of the solutions. 7/2/99
(Grade 4-7).
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?
See the development of the solutions. 9/20/98
(Grade 4-7).
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?
See the development of the solutions. 8/9/98
(Grade 4-8).
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?
See the development of the solution. 4/8/99
(Grade 4-8).
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?
See the development of the solution. 12/17/99
(Grade 4-6).
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?
See the development of the solution. 8/26/98
(Grade 4-6).
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?
See the development of the solution. 8/26/98
(Grade 4-6).
Eight times a number plus 3 times the number
is the same as 20 more than 9 times the number.
Find the number.
See the development of the solution. 10/17/98
(Grade 4-6).
How many jelly beans fill a one liter bottle?. A Fermi problem.
See the development of the solution. 3/16/99
(Grade 4-7).
Solutions to a collection of simple linear equation.
See the development of the solutions. 12/7/99
(Grade 4-8).
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?
See the development of the solution. 4/29/98
(Grade 4-8).
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.
See the development of the solution. 3/16/99
(Grade 5). IA
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?
See the development of the solution. 12/17/97
(Grade 5-6).
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?
See the development of the solution. 6/19/98
(Grade 5-6). IA
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?
See the development of the solution. 3/12/98
(Grade 5-8).
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?
See the development of the solution. 8/9/98
(Grade 5-7).
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?.
See the development of the solution. 9/8/98
(Grade 5-7).
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?
See the development of the solution. 9/8/98
(Grade 5-7).
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?
See the development of the solution. 3/21/1999
(Grade 5-7).
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.
See the development of the solution. 9/23/98
(Grade 5-7).
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?
See the development of the solution. 9/13/98
(Grade 5-12). IA
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.
See the development of the solution. 4/9/98
(Grade 6). IA
Solve the following equation.
See the development of the solution. 12/20/97
(Grade 6). IA
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?
See the development of the solution. 3/20/98
(Grade 6). IA
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?
See the development of the solution. 1/27/98
(Grade 6). IA
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?
See the development of the solution.
(Grade 6). IA
If the area of a circle is 144pi square units,
find its circumference.
See the development of the solution. 12/31/97
(Grade 6). IA
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?
See the development of the solution.
(Grade 6). IA
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?
See the development of the solution. 1/30/98
(Grade 6-9).
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?
See the development of the solution. 11/7/99
(Grade 6-9).
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.
See the development of the solution. 8/31/01
(Grade 6-9).
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?
See the development of the solution. 11/7/99
(Grade 6-9).
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?
See the development of the solution. NEW 9/8/2001
(Grade 6-9).
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.
See the development of the solution. 3/29/99
(Grade 6-9).
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?
See the development of the solution. 10/29/2000
(Grade 6-9).
Convert 39800 rods/hour to furlongs per fortnight.
See the development of the solution. 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?
See the development of the solution. 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?
See the development of the solution. 9/20/98
(Grade 6-12).
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.
See the development of the solution. 8/15/98
(Grade 7). IA
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?
See the development of the solution.
(Grade 7). IA
There are two numbers whose sum is 50. Three times
the first is 5 more than twice the second. What are
the numbers?
See the development of the solution. 3/8/98
(Grade 7). IA
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?
See the development of the solution. 12/30/97
(Grade 7). IA
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?
See the development of the solution. 1/10/98
(Grade 7). IA
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?
See the development of the solution.
(Grade 7).
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?
See the development of the solution. 3/19/99
(Grade 7-8).
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?
See the development of the solution. 8/9/98
(Grade 7-11).
Find an equation of the line containing (5,3) and parallel to the line 5x-3y=4
See the development of the solution. 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?
See the development of the solution. 10/29/98
(Grade 8).IA
The first number is four more than two times the second number. The sum of the two numbers is ninety seven.
Find the numbers.
See the development of the solution. 12/23/97
(Grade 8). IA
See the development of the solution. 12/30/97
(Grade 8). IA
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?
See the development of the solution. 1/2/98
(Grade 8).IA
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?
See the development of the solution.
(Grade 8).IA
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?
See the development of the solution. 12/25/97
(Grade 8-10).
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.
See the development of the solution. 7/27/99
(Grade 8-10).
Find the solution set-
square root of 2x-3 = 2 times the square root of 3x -2
See the development of the solution. 3/6/2000
(Grade 8-11).
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.
See the development of the solution. 9/20/98
(Grade 9). IA
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?
See the development of the solution. 3/28/98
(Grade 9-11).
An equilateral triangle with each side = x cm is inscribed
in a circle. Find the radius r of the circle in terms of x.
See the development of the solution. 3/4/2000
(Grade 9-12).
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?
See the development of the solution.
5/15/98
(Grade 9). IA
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.
See the development of the solution.
2/14/98
(Grade 9). IA
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?
See the development of the solution.
1/15/98
(Grade 9). IA
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)
See the development of the solution.
(Grade 9). IA
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.
See the development of the solution. 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. |
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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?
See the development of the solution. 10/2/98
(Grade 9-12).
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?
See the development of the solution. 8/31/98
(Grade 9-12).
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.
See the development of the solution. 2/4/99
(Grade 9-12).
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.
See the development of the solution. 2/4/99
(Grade 9-12).
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.
See the development of the solution. 8/31/98
(Grade 9-12).
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.
See the development of the solution. 5/18/98
(Grade 10-12).
Determine the roots of z3 + 6z2 - 4z - 24 = 0
See the development of the solution.
(Grade 10-12).
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?
See the development of the solution.
(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). |
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See the development of the solution. 12/31/97
General
All grades.
Examples of conversion of units.
See the examples.
All grades.
Use Quick Math.
All grades.
Use MathServe.
For a comprehensive organization of problem solving instruction see:
Planning for Problem Solving Instruction ... Integrated and Developmental
To be continued
Copyright. Howard C. McAllister, 1997-2003.
