See also the
Dr. Math FAQ:
3D and higher
Browse Middle School Word Problems
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
- Two Planes and Not Enough Information [08/10/2004]
Two planes leave Chicago, one traveling due east and the other due
west. The first travels 100 miles per hour faster than the second.
How long will it be until they are 2000 miles apart?
- Two Quantities, Two Relations [04/04/2000]
Find two consecutive numbers whose squares, when subtracted, equal 43.
- Two Steps Forward, One Step Back [10/14/2003]
A salmon swims 30 feet upstream in 5 seconds, then rests for 2
seconds. If she moves 10 feet downstream during each rest, how long
does it take her to swim 1010 feet?
- Unitary Method Problem [02/01/1999]
How many runts are there in 6 cartons if each carton has 8 packages -
each of which has 3 boxes (each of which has 170 runts)?
- Units and Cylinder Volume [02/06/2003]
Find the volume and surface area of a cylindical storage tank with a
radius of 15 feet and a height of 30 feet.
- The Unknown Angle that Didn't Matter [10/01/2014]
A geometry word problem leads to the confounding result 120° equals
0. Doctor Ian takes a closer look at how to interpret this puzzle's statement —
and empty solution sets generally.
- Use One Weighing [02/06/2002]
There are 90 genuine coins and 10 fake coins in 10 stacks with 10 coins
in each stack... Using only one weighing, determine which stack contains
the fake coins.
- Using Guess and Check to Start an Algebra Problem [02/15/2008]
Laura is three times as old as Maria was when Laura was as old as
Maria is now. In two years Laura will be twice as old as Maria was two
years ago. Find their present ages. Dr. Ian shows that by first making
a guess and checking it you can then easily write the algebraic
equations needed to solve the problem.
- Using Improper Fractions in a Word Problem [12/6/1995]
Mr. Smith owns 1/4 of the stock of a business. Mrs. Smith owns 5/6 as
much as Mr. Smith. How much do they not own of the stock?
- Using Tables to Tame a Problem [09/19/2003]
Racing bikes are either silver or blue. Every silver bike is
a racing model. 1/2 of the blue bikes are racing models. 1/3 of the
bikes are racing models. There are 40 blue bikes and 30 silver
bikes. How many bicycles are neither blue nor silver?
- Using Units and Variables to Make Sense of Word Problems [05/24/2004]
I am having a hard time changing a word problem into a formula in
order to get the results. For example: Apples cost $1.50 per pound
and pears cost $2.30 per pound. A total of 8 pounds of apples and
pears were bought, and the price was $14.00. How many pounds of pears
- Using Variables to Solve Word Problems [05/10/1998]
Susan puts only dimes and quarters in her coin bank ...
- The Value of a Word [08/25/1998]
Think of a word that equals one dollar. The key is: a=.01, b=.02, c=.03,
- A Variable for Each Digit [11/28/2001]
A husband's age is the inverse of and greater than his wife's age... Some
examples are 82,28 or 54,45. Is there a way to solve this without simple
trial and error?
- A Velocity-Time Problem [02/16/1999]
If you drive 40 mph instead of 30 mph, you save 30 seconds per mile, but
if you make the same 10 mph increase from 70 mph to 80 mph, you only save
5.6 seconds per mile. Why?
- Visualizing Ratios to Scale Least Common Multiples [10/02/2010]
The ratio of Cotsworld sheep to Swaledales is 3:5, and the
ratio of Swaledales to Longwools is 7:4. What is the ratio
of Cotsworlds to Longwools?
- Visualizing Two Word Problems [03/12/1998]
Sandy has thrice as many beads as Wendy... Paul gives 20 of his marbles
- Watch Gaining Time [11/10/2002]
A watch is set accurately at midnight, but gains six minutes every
hour. If it stopped 30 minutes ago, and now snows 8:26 a.m., what's
the correct time now?
- Water Main Section Lengths [01/09/2003]
A water main is being laid using pipe that comes in either 18-foot or
20-foot sections. The water main will require 14 fewer sections of 20-
foot pipe than if 18-foot sections are used. Find the total length of
the water main.
- Weighted Averages [11/30/2001]
An advertisement for an orange drink claims that the drink contains 10%
orange juice. How much pure orange juice would have to be added to 5
quarts of the drink to obtain a mixture containing 40% orange juice?
- Weight of Clown, Trapeze Artist [9/9/1996]
The clown weighs 60 lbs. more than the trapeze artist; the trapeze artist
weighs 2/3 as much as the clown. How much does each weigh?
- What is the Circumference of the Reservoir? [04/09/2003]
John and Tina start running around a round reservoir in opposite
directions, meeting for the first time after John has traveled 100
yards, and again 60 yards before Tina has completed her first lap.
- What is the Number? [3/29/1995]
You know that a certain 5-digit number is a power of 7 and that its ones
digit also is 7. What is the number?
- What is the Speed of the Truck? [3/20/1995]
A storm is 50 miles offshore and its path is perpendicular to a straight
shoreline. It is approaching the shore at a rate of 4mph....
- What is the Value of K? [01/24/2003]
For positive integer walues on N, define N! to be the sum of integers
1 to N inclusive.
- What's His Street Address? How Many Neighbors Does He Have? [09/29/2010]
A student struggles with a word problem that asks for specific sums of counting
numbers. Three different doctors weigh in with increasingly sophisticated and
comprehensive problem-solving approaches: programming spreadsheet formulas;
applying combinatorics; and invoking quadratic Diophantine and Pell equations.
- What Time Did Ajay Start? [08/06/2003]
Three persons, Ajay, Bhanu, and Chandu, plan to travel from P and reach
Q at 4.00 PM...
- What Time Does She Get to Work? [04/30/2002]
Lillian drives to work at an average speed of 45 miles per hour. If
she travels 30 miles to work and leaves at 8:00 A.M., what time does
she get to work?
- When Was King George Born? [08/23/2002]
King George died 180 years after King Alfred was born. Their combined
age when they died was 73 years. King Alfred died in 315 A.D. In what
year was King George born?
- Where would the Two Trains Meet? [6/5/1995]
If a train left Baltimore at 6:00 pm and travels at 19 m.p.h. and another
train left Philadelphia travelling at 85 mph in the opposite direction,
where would they meet?
- Who Gets the Job? [12/11/2001]
Each job candidate can see the other two candidates' black or red dots
but not his own. Whoever can figure out the color of his own dot gets the
- Who Picked the Most? [09/12/1998]
Arrange the names of the people in the order of the number of peaches
that each picked, starting with the person who picked the most.
- Who Will Win the Race? [11/26/2002]
Who will win the race and by how many meters?
- The Width of a Walkway Surrounding a Garden [04/04/1998]
A walkway surrounds a 24x30 garden and is 1/4th the area of the garden.
How wide is the walkway?
- Winking Lights [09/12/1998]
A red light winks every 6 seconds, a white one every 10 seconds and a
green one every 14 seconds. How often do they wink together?...
- With How Much Money Did Each Start the Game? [04/24/2002]
Ferne, Donna, and Susan have just finished playing three games...
After three rounds, each has $24.00. How much money did each have at
the start of the game?
- Word Problem Using Whole Numbers [5/25/1996]
If someone went to the store and bought 20 fish for $45...
- Working Backward [11/30/1999]
A woman sells half of her eggs and half an egg to Peter. Later, she sells
half her eggs again and half an egg to John, and so on with more people.
Finally the woman is left with only one egg. How many eggs did she start
- Working Backward [01/22/2001]
Ann, Barry, Chip, Doug, Evelyn, and Fred went to lunch to celebrate
Evelyn's and Fred's birthdays... How much should each person pay?
- Writing Equations for Word Problems [09/01/1998]
Can you help me convert the following word problems to equations and
solve them? An oil painting is 16 years older than a watercolor ....