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.
- Precedence of Unary Operators [09/01/99]
The PEMDAS rule for order of operations ignores unary operators. Can you
explain the proper precedence for them, and give an example showing how
not knowing the rule can cause an incorrect result?
- Price Plus 1/2 Price [12/05/2001]
A puzzle book costs $5.00 plus one-half of its price. How much does the
puzzle book cost?
- Probate Puzzle [12/15/2003]
A will specifies that every son should receive three times as much as
a daughter and every daughter should get two times as much as their
mother. What is the mother's share?
- Problem about Rain [09/20/2000]
During a recent span of time, 11 days had some rain... How many days had
no rain at all?
- Problem Solving and Mental Process [10/03/2002]
Two numbers add up to 19. The difference between the two numbers is 3.
What are the numbers?
- Proof That All Numbers Are Equal? [12/06/2005]
I saw a proof that all numbers are equal, which makes math pointless.
There must be something wrong with the proof. Can you tell me what?
- Quadratic Equation [6/22/1996]
What is a quadratic equation used for?
- Quadratic Equation [8/28/1996]
What is the formula for the quadratic equation?
- Quadratic Equation [04/08/1997]
The sum of two numbers is 15 and their product is 54. Find the numbers...
- Quadratic Equations [3/19/1996]
How are quadratic equations used in the real world?
- Quadratic Equations [6/25/1996]
Solving quadratic equations using the zero-factor property, the square
root property, and the quadratic formula.
- Quadratic Formula [5/11/1995]
Could you give me the quadratic formula?
- Quadratic Formula, Distributive Property [06/02/1997]
Who invented the quadratic formula?
- Quadratic Function: Definition [8/6/1996]
What is a "quadratic function"?
- Quadrilateral Patterns [03/06/1997]
What is the formula that gives the number of quadrilaterals within a
square grid when you increase the square grid by one unit on each side?
- Questions about Zero - Undefined or Indeterminate? [8/16/1995]
Is 0,0,0,... a geometric sequence? What is the value of 0/0? What is the
value of 0^0 (^ means exponent)? Can it be considered to be the limit of
a^0 as a approaches 0?
- Questions Answered Correctly [06/11/2003]
There are 30 multiple-choice questions in a contest. Five marks are
awarded for each correct answer, and three marks are deducted for each
incorrect answer. No marks are awarded for questions left unanswered.
If a student scores 78 marks, what is the greatest possible number of
questions answered correctly?
- Raising Sides of an Equation to the 0 Power [12/04/1997]
If you raise both sides of an equation to the 0 power, will both sides
then be equal to one?
- Rates of Paddling [04/25/2000]
I can travel 12 miles downstream in six hours less than it takes me to
travel the same distance upstream... How fast is this river's current?
- Rationalizing the Denominator [05/27/1997]
Why do we have to rationalize the denominator?
- Rational Number and its Reciprocal [03/14/2002]
A rational number greater than one and its reciprocal have a sum of 2
1/6. What is this number? Express your answer as an improper fraction in
- Rational Root Theorem [08/27/1998]
List all possible rational zeros of the each function, then determine the
rational zeros: f(x) = x^3 - 4x^2 + x + 2.
- Ratio of Anti-Freeze to Water [02/05/2003]
A barrel is filled with 45 gallons of anti-freeze. 9 gallons are
removed and the barrel is refilled with water. 9 gallons of the
mixture are removed and again, the barrel is refilled with water. What
is the final ratio of anti-freeze to water in the barrel?
- Ratio of Ratios [06/05/2002]
In a community, 3/5 of the men are married to 2/3 of the women. What
fraction of the residents are married?
- The Ratio of Water to Wine [10/22/1995]
A cask is filled with 45 gallons of wine. Nine gallons are removed and
the cask is refilled with water. The nine gallons of the mixture are
removed and the cask is again refilled with water. What is the ratio of
water to wine in the final mixture?
- Ratios and Mixtures [05/10/2001]
Sue needs to mix up a potion for her client... in what ratio must Sue mix
the honey and turpentine?
- Rearranging Variables [10/01/1998]
Why when you use a variable in a problem does the answer sometimes still
have a variable, so you can't get an actual number answer?
- Rectangles: Area and Perimeter the Same [4/15/1996]
There are two different rectangles whose sides are integers and whose
area and perimeter are the same. Find the dimensions.
- Relationship Between Commutative and Associative Property? [06/30/2002]
Are both the commutative and associative properties necessary to prove
a + b + c = c + b + a?
- Relations versus Functions [10/27/1998]
What is the difference between a relation and a function? What about the
domain and range of a function?
- Restrictions on Variables in Division [8/4/1996]
What are the restrictions on the variables in (y^3+8)/(y^2+3y+2)?
- Reversing the Inequality, Explained to a Student with Scale and Number Line Analogies [09/21/1998]
Why does the inequality sign reverse when you multiply by a negative
- Reversing the Inequality, Explained to a Teacher with an Antigravity Metaphor [03/09/1999]
Why do we reverse the inequality symbol when dividing/multiplying by a
- A Right Triangle of Points [01/14/1999]
Determine the values of x that would make the points (x,0), (-2,1), and
(3,4) the vertices of a right triangle.
- Rooting out the Reason for a Switch in Sign [01/13/2015]
Solving a radical inequality in one variable, a teacher struggles to make sense of two
steps that differ slightly but lead to opposite results. Doctor Peterson helps her spot
the error in her reasoning.
- Rule of Three [06/15/2002]
Please explain to me briefly (and with an example) the concept of the
rule of three.
- Rules of Exponent Manipulation [1/22/1996]
f sub r equals the reciprocal of ...
- Running Rates, Line Segments, and Basic Algebra [6/26/1996]
(1) Steven ran a 12-mile race at an average speed of 8 miles per hour. If
Adam ran the same race... (2) Given a line made of three segments, which
statements are true...? (3) If 3 more than x is 2 more than y, what is x
in terms of y?
- The Sailor's Reward Problem and Prime Numbers [12/11/2004]
Does the answer to the classic Sailor's Reward problem have to be a
- Same Day Off? [12/08/2002]
One person has every ninth day off and started on Sunday, and
another person has every sixth day off and started on Monday. When
will they both be off on the same day?