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.
- Magic Triangle Puzzles, Expanded [11/17/2012]
In the archives, a parent finds a discussion of magic triangles — but the puzzles assigned his daughter go beyond the counting numbers. Doctor Peterson walks through a range
of strategies, from guess-and-check to algebraic.
- Magic Triangle Sums [02/04/2003]
Find 6 consecutive odd numbers such that their magic triangle sum is
25. Find 6 consecutive numbers such their magic triangle sum is 4.
- Magic Wheel [03/20/2002]
Given a wheel with 16 spokes and a hub, enter the whole numbers 1-17 in
the hub and at the end of spokes. The sums of the 3 numbers along the
spokes must be equal.
- Making the Grade [04/13/2001]
After three tests, Amanda's average score is 88. What grade does she need
on her next test to score a four-test average of 90?
- Man Crossing a Bridge [09/27/2001]
A man is jogging across a bridge at 8 mph. When he is 3/8 of the way
- Mapping Functions in the Real World [3/20/1995]
What is the purpose of learning to map a function? What is it used for in
the real world?
- Marble Collection Fraction Problem [9/9/1996]
3/4 of Tim's marble collection is the same size as 3/5 of Danny's
- Math and Music: Harmonic Series [01/09/1998]
Could you supply me with information on how math relates to music?
- Math with Parentheses [3/10/1995]
Is there an easier way to do math with parenthesis?
- Matrix [4/11/1996]
What is a matrix?
- Maximum Number of Intersections of n Distinct Lines [10/07/1998]
Find a pattern for the maximum number of intersections of n lines, where
n is greater than or equal to 2.
- Meaning of Greater Than, Less Than Symbols [11/07/2002]
Does the meaning of less than and greater than signs depend on
- Meaning of Term in Algebra [09/16/2007]
Why is it that when you multiply or divide two numbers or variables
they become one term (such as 2*x becoming 2x), but when you add or
subtract two numbers or variables they are still two terms (such as 2
- Measuring the Length of a Moving Train [08/12/2003]
I was driving in the opposite direction from a moving train that had
many cars behind the locomotive. It took me approximately 2-3 minutes
from the time my car 'met' the front of the train until it passed me
going the opposite direction. My speed was about 40 miles an hour. Is
it possible to estimate the length of this moving train without
actually measuring when it is not moving?
- Minding the Gaps of Picket Fencing [03/17/2012]
Landscaping a fence leads to fractional pickets and frustration. After diagramming a
bird's eye view of a model fence line, Doctor Peterson applies some algebra to the key
insight: in a straight run of fencing, the spaces between the pickets outnumber the
pickets by one.
- Mixed Operations [04/08/1997]
When calculating 839-319+200/20x15, I got 540. But, should I have first
- Mixing Peanuts and Cashews [11/19/1999]
Peanuts sell for $3.00 per pound. Cashews sell for $6.00 per pound. How
many pounds of cashews should be mixed with 12 pounds of peanuts to
obtain a mixture that sells for $4.20 per pound?
- Mixture Problem [11/11/1997]
How much pure antifreeze should be added to 3 gallons of a 30 percent
antifreeze solution to get 65 percent antifreeze?
- Monetary Conversions [01/27/2001]
Given these monetary exchanges: 2 coconuts = 1 banana; 3 bananas = 2
mangos; 4 papaya = 1 coconut, what is the exchange for banana to papaya
and for papaya to mango?
- Monomial, Polynomials [11/14/1997]
What is a monomial?
- Monomials, Polynomials [03/02/2001]
Can you explain monomials and polynomials to me?
- More Variables Than Equations in a System [01/21/2002]
I have a problem that uses four variables but only two equations to
relate them. I don't seem to have enough information to figure out
the value of each variable or answer the question. Can you help?
- Mother's Age in Terms of Daughter's [06/27/2001]
The sum of the ages of a father, a mother, and a daughter is 73. When the
father is twice as old as the daughter, the sum of their two ages will be
- Mowing Half a Lawn [03/31/2002]
Chris and Lee are each responsible for mowing half of their 50-foot by
90-foot rectangular back yard...
- Mr. Brown's Walk [09/09/1997]
One day Mr. Brown took the 4:00 p.m. train, arrived at the station one
hour early, and started walking home... How long did Mr. Brown walk?
- A Mule and a Donkey [04/08/1999]
A famous Euclidean brainteaser.
- Multiplication and Division Equations [11/29/2001]
Solve for x: 17x = 85.
- Multiplying/Adding Fractions Gives Same Answer [03/01/2002]
Find two fractions which, when multiplied and added, give the same
- Multiplying a Fraction by a Variable [08/17/2006]
How do I find the product of something like (1/3) times 'a'?
- Multiplying and Dividing Fractions [8/18/1996]
Three hard problems, illustrated.
- Multiplying Binomials and Other Polynomials [04/16/2004]
I have a hard time multiplying binomials properly, such as (2x + 5)(x
+ 7). I do not know what order the problem should be solved in.
- Multiplying or Dividing by a Negative Number [04/08/1999]
When x(x-4) is less than 0...
- Multiplying Polynomials [2/10/1996]
How do you multiply polynomials?
- Names of Polynomials [05/22/1999]
Why is an equation having only two roots, one of which is raised to the
second power, called a quadratic equation?
- Natural Domain of a Function [03/12/1999]
Some inputs don't make sense for some functions.
- Negative Exponents [10/23/1996]
I have to simplify this expression with non-negative exponents: x^5.
- Negative Numbers Combined with Exponentials [03/09/2001]
Why in the order of operations is negation a multiplication done after
exponentiation, rather than as a part of the base? What about
- Negatives and Inequalities [09/10/1998]
Why when we multiply or divide an inequality by a negative number do we
reverse the inequality sign?
- Negatives in Absolute Value Expressions and Equations [10/15/2005]
How do I solve an equation like |x - 3| - 4 = 0? How do I evaluate
absolute value expressions with negatives, like |-7 - 3|?
- Negative Squared, or Squared Negative? [10/18/2002]
In -3^2, isn't -3 actually a number, and not -1*3? If so, then
shouldn't the value be -3*-3=9?