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:
- Coin Combinations Quickened by Process of Elimination [11/04/2013]
A student seeks a method for efficiently attacking a puzzle that involves partitioning
sums of coins. Doctor Greenie suggests he reverse his thinking and work backwards.
- Combining Rates of Work [08/30/1998]
It takes me 3 hours to paint a house. It takes you 5 hours to paint a
house. How long will it take for both of us to paint a house?
- Combining Rates of Work [10/27/1998]
Jones takes 12 hours to complete a task. Marco arrives, and they finish
in 2 hours. How long would Marco have needed to do the job alone?
- Combining Rates of Work: From Unit Rates to a General Model [12/23/2013]
A student solves a problem about combined rates of work, but struggles to
understand "units of work per man-hour" and the other ratios it entails. Doctor Ian
approaches the problem first by simplifying it into unit rates, then distinguishes the
direct proportions from the inverse ones en route to a general template that would
model the combination of any work rates.
- Combining Rates of Work, Revealing Constants of Word Problems [11/13/2011]
An adult wonders how to logically reason through problems that entail
combining rates of work. Moved to re-examine the mechanics he learned
in high school, Doctor Jerry introduces a constant tacitly assumed in
the wording of such problems.
- Commissions [03/19/2002]
Mildred receives a 5% commission on her sales of exercise equipment and a
6% commission on her sales of weight training equipment...
- Comparing Prices [09/29/2003]
Which of these is the better buy: a 6-ounce can of tuna that sells for
$1.59, or a package of three 3-ounce cans for $2.19?
- Comparing Running Rates [01/20/2005]
Jack and Jane ran the 100 yard dash. When Jane won, Jack was 10 yards
behind her. They raced again, but this time Jane started 10 yards
behind the starting line. If they both ran at the same rate as in the
first race, who won the second race?
- Comparison of On-line Rates [5/20/1996]
Jack is interested in two on-line companies, Easy Comp and Destruct
- Complicated Age Problem [02/04/2002]
Eight years ago Mary was half as old as Jane will be when Jane is one
year older than Tim will be at that time when Mary will be five times as
old as Tim will be two years from now...
- Concept of Farmer Crossing a River [04/30/2002]
What concept does the farmer, fox, and chicken problem use, and how
does it relate to the sheriffs and outlaws problem, husband and wives
problem, and missionaries and cannibals problem?
- A Container Holds 5/8 Gallon... [11/05/1997]
If the container is 4/5 full and then 25 percent of the liquid is lost
due to evaporation, how much liquid is left in the container?
- Converting Miles/Hour to Feet [08/14/1997]
If my father is traveling in his car at 45 mph, how long will it take him
to travel 100 feet?
- Converting to an Equation [06/13/1999]
How can I convert "the square of a number is 96 more than 10 times the
number" into an equation?
- Cooking at a Set Rate [6/13/1996]
If a 10 pound turkey takes four hours to cook, how long will it take a 14
pound turkey to cook?
- Cost of a Ruler [08/01/1997]
Alice and Betty want to buy the same kind of ruler but Alice is 22 cents
short and Betty is 3 cents short...
- Cost of Pencils [6/26/1996]
Drawing pencils cost ... and coloured pencils cost... Two dozen assorted
pencils cost $2.16. How many coloured pencils are there?
- Cost, Overhead, Selling Price, and Profit [06/24/2008]
Larry's Lamp Shop buys lamps for $18 each. If Larry's overhead is
$7.29 per lamp and he makes a profit of $3.38 per lamp, what is the
selling price of each lamp?
- Counting Students [10/06/1998]
In a class, 3/4 of the students are less than 16 years old, 2/3 are less
than 15 years old, .... How many students are in the class?
- A Crabby Word Problem [2/13/1996]
A crab is trying to swim to its home, 35 meters out in the ocean. It
takes the crab one minute to swim ten meters; then a wave washes it back
five meters, where it rests for one minute before trying again. How long
will it take the crab to get home?
- Crossing a River: Cannibals and Missionaries [10/15/2001]
Three missionaries and three cannibals wish to cross a river using a
single boat that will hold only two people. If at any time the cannibals
outnumber the missionaries, those missionaries will be eaten...
- Crossing the Desert [05/22/2001]
A truck gets one mile per gallon, and can hold 400 gallons at a time. How
much is the minimum amount to cross a 1000-mile desert?
- Crossing the Desert [08/19/1998]
One person can carry a 4-day supply of food and water for a trip across a
desert that takes 6 days to cross...
- Cutting a Pipe [06/24/2003]
You have a pipe that is 21 1/8' long. How many 2' segments can be made
of this pipe if 1/8' waste is created from each cut?
- Cutting Equal Strips [09/08/2005]
Given five 4 ft by 8 ft rectangular sheets of plywood, how many strips
that are 7 1/2 inches wide and 8 feet long could you cut? Each saw cut
eliminates 1/16 inch of the plywood as sawdust, and you are not able
to glue any strips together.
- Cycling Motion Problem [6/26/1996]
A cyclist travels from A to B over a high pass...
- Cycling Uphill [7/8/1996]
I don't know how to handle the unknown speed at which the cyclist travels
- Dart Data [02/29/2012]
A mother and daughter struggle to enumerate possible scores from throwing four
darts at three scoring rings. Doctor Peterson suggests another way to tabulate the
tosses: methodically organize the data into groups of four outcomes rather than
treating each dart individually.
- Decreasing by Degrees [01/07/2016]
A teacher wonders about her choice of context for a problem designed
to elicit subtraction of a negative number. Doctor Peterson provides
troubleshooting linguistic, mathematical, and instructional.
- Delivering Papers [11/15/1999]
How many papers will Juni deliver if Uri delivers 5 newspapers in the
same time that Juni delivers 4 newspapers, and they have a total of 54
papers to deliver?
- Density Equation Family [03/27/2002]
I need to know how to solve for mass using volume and density.
- Dependent and Independent Variables [12/26/2001]
Identify the dependent and independent variable: time it takes to travel
50 miles and the speed of the vehicle.
- Depth of a Tank [09/04/1997]
A tank 100' long and 10' wide holds 15,000 cu. ft. of water...
- Determining Which Number to Divide by Which [10/04/2004]
Julia plans to use a recipe that yields 15 pounds of fudge. She
intends to wrap the fudge she makes in 3/4 pound boxes and give one
box to each of her friends and relatives for gifts. How many gifts
will she have? Should I divide 15 by 3/4 or 3/4 by 15?
- The Difference of x and y... [06/05/2003]
The difference of a number and its square is 42. Is the equation x^2 -
x = 42 or x - x^2 = 42?
- Different Combinations of Coins [2/18/1996]
Francisco has some change in his pocket. He gives his friends three clues
to see if they can guess how much he has: the coins equal exactly one
dollar; he has no more than 100 coins; he has at least one coin. What
combination of coins can Francisco have in his pocket?
- Digital Riddle [10/04/2016]
A non-native English speaker struggles to parse the clues in a riddle. After unpacking the verbiage step by step, Doctor Peterson offers general suggestions for approaching such place value puzzles.
- Diophantus' Lifespan [05/14/1997]
Diophantus' youth lasted 1/6 of his life. He grew a beard after 1/12 more
of his life. After 1/7 more of his life, he married. Five years later he
had a son. How many years did Diophantus live?
- Direct and Indirect Variation [07/09/1998]
Can you explain the difference between direct and indirect variation? How
would you interpret them in a word problem?
- Direct and Inverse Variation [04/25/2001]
An example of each type of problem.