See also the
Dr. Math FAQ:
order of operations
3D and higher
Browse Middle School Algebra
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Direct and indirect variation.
Inequalities and negative numbers.
Positive/negative integer rules.
Solving simple linear equations.
- First Year Algebra [6/19/1996]
What should a person have learned after one year of Algebra I? I want to
be prepared for Geometry next year.
- Fitting a Picture to a Frame [11/14/1996]
I have a picture frame that needs to have twice as much border on the
side of the picture as on the top. If the photo is half the area of the
frame, how wide should the borders be?
- Five Bags of Nuts [04/17/2002]
Given 100 nuts in five paper bags, in the first and second bags there
are altogether fifty-two nuts; in the second and third there are
forty-three; in the third and fourth, thirty-four; in the fourth and
fifth, thirty. How many nuts are there in each bag?
- Flipping for Profit [01/19/2014]
A young entrepreneur struggles to reverse-engineer the mark-up price that
guarantees his take-home profit. Starting with his own steps as a framework, Doctor
Peterson introduces variables and applies algebra to model the scenario more
- Formulas and Missing Quantities [10/09/2004]
Variations on the question "Could you drink enough water in a lifetime to empty a pool?" lead to a discussion of using formulas to find missing quantities.
- Formulas for the Equation of a Line [09/28/1998]
Can you explain the meaning behind the point-slope formula? What about
other formulas for the equation of a line?
- Four Methods of Solving Classic Chicken and Egg Problems [02/24/2009]
If 6 cats can kill 6 rats in 6 minutes, how many cats will it take to
kill 100 rats in 50 minutes? A look at four different solution methods
for this sort of problem.
- The Fourth Dimension [8/24/1995]
What is the fourth dimension mathematically?
- Fraction: Algebra without the Equations [8/23/1996]
I have come up with a way to solve fraction problems like "2/3 of ? =
66", but I have no idea why it works.
- Fraction Equation Problem [5/10/1995]
Let 1/a = a + 1/b = b + 1/c = c + 1/d = 2. What is the value of abcd?
- Fraction Equivalents of Decimals that Repeat ... Mostly [05/04/2010]
How do you find the fraction equivalent of a repeating decimals that also includes
digits that do not repeat? Doctor Greenie tweaks a student's formal algebraic
approach, introduces another wrinkle grounded in the base ten number system, then
shares two algorithms based on his work with high school students who take part in
- Fraction or Decimal Form? [07/30/2002]
Is it better to write 1/2(6x-3(2x-1)) or 0.5(6x-3(2x-1)), and why?
- Fractions and Cross Multiplication [07/14/1998]
When do you cross-multiply in solving fractions?
- Fractions and Lowest Terms [07/19/2001]
Simplifying expressions: how do you finish cancelling fractions, and then
what do you multiply? What is the lowest term?
- From Representing Two Bags of Marbles, a Way to Solve All Kinds of Problems [04/03/2011]
A student struggles with an algebra word problem. Doctor Ian breaks it down into
manageable chunks, stepping through how to represent the question's relationships
algebraically and then check his interpretation for reasonableness.
- From Word Problems to Equations [01/28/1999]
If the sides of a square are lengthened by 3m, the area becomes 81m
squared. Find the length of a side of the original square.
- Functions and Equations [02/10/2001]
When is an equation also a function? How do you recognize a function?
- A Game in Three Rounds [04/19/1999]
After three rounds each of the girls has 8 chips. How many chips did they
have at the start?
- Geometrically Completing the Square [08/07/1997]
What are the steps for geometrically completing the square?
- Getting the Job Done [12/17/1996]
Two typists share a job and each does half of the work. The second typist
begins working one hour after the first...
- Given Perimeters, Find Sides [8/5/1996]
A triangle and a rectangle have equal perimeters... find the sides.
- Grade Averages [11/16/1995]
If Johnny has an 82% average for the 1st 9wks and a 75% average for the
2nd 9wks, what grade would he have to get on the final to receive an 80%
semester grade? The grades are weighted as follows: each 9wks 40% and the
semester test 20%.
- Graphing Intercepts [11/06/1997]
How would I find the intercept for a problem such as 3x-2y = 12?
- Graphing Slope Intercept [02/13/1999]
When given the slope intercept form, how do you know whether you rise or
drop when looking at the slope?
- Grid Patterns [09/10/99]
Finding a formula for the difference between the diagonal products of the
numbers in a 2x2 square within a 3x5 rectangle of numbers, and
generalizing it for any rectangle of numbers.
- Guess and Check Tables, Algebra [10/12/2002]
Find the mass of Hector's math, science, and Spanish textbooks.
- Hands of a Clock [10/10/1997]
How many times do the hour and minute hands cross in a 12-hour period of
- Heads and Legs [07/25/2001]
Joe counts 48 heads and 134 legs among the chickens and dogs on his farm.
How many dogs and how many chickens does he have?
- A Hen and a Half [09/23/1998]
If a hen and a half lays an egg and a half in a day and a half, how many
eggs will five hens lay in 6 days?
- Hot Air Balloons [12/25/1996]
A hot air balloon covered 2100 miles in 7 days. If it covered 50 miles
more each day than the day before, how many miles did it cover each day?
- How Does the Trick Work? [06/20/2003]
From time to time I've run across math tricks that ask me to give a
number, then take me through a series of other questions and by the
time I work my way to the end, the trick tells me my original number.
- How Does the Volume Change? [03/04/1999]
We have been doing a project where we change one side of an object and
see how it affects the volume...
- How Long is Duke the Dog? [09/04/2001]
Duke the dog's head is 6 inches long, his tail is as long as his head
plus half the length of his body, and his body is as long as the head and
the tail combined.
- How Long is the Train? [05/21/1999]
Two trains are traveling in opposite directions along parallel tracks,
train A at 72kph and train B at 54kph...
- How Many Apples? [09/19/2002]
How many apples did John get from his boss?
- How Many Coins Did Each Originally Have? [10/22/2001]
Natalie has some nickels, Dirk has some dimes, and Quincy has some
quarters. Dirk has five more dimes than Quincy has quarters... How many
coins did each originally have?
- How Many Factor Pairs Does a Given Number Have? [03/19/2004]
I'm wondering if there is a formula or algorithm that will tell me
exactly how many pairs of factors a number has?
- How Many Games Were Tied? [06/07/1999]
How many games are tied if each team receives 3 points for a win, 1 point
for a tie, and no points for a loss, and after 4 rounds the accumulated
points are: Lions 22, Tigers 19, Mounties 14, Royals 12?
- How Many Mice, Cats, and Dogs? [01/13/2003]
You must spend $100 to buy 100 pets, choosing at least one of each
pet. The pets and their prices are: mice @ $0.25 each, cats @ $1.00
each, and dogs @ $15.00 each. How many mice, cats, and dogs must you
- How Many of Each Ticket Were Sold? [09/06/2001]
Five hundred tickets were sold for a play, for $8 at the lower level and
$6 at the upper level, totaling $3600.