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.
- Nested Negative Exponents [8/2/1996]
How do I deal with nested negative exponents?
- Net Result of Consecutive Percentage Increases [12/11/2003]
Suppose a quantity is increased by some percentage, and then is later
increased again by another percentage. How can I calculate the
overall percentage increase?
- Nonlinear Factors [10/12/1998]
I have been told that factoring the sum of two squared numbers is not
possible, but factoring x^4 + 4 is possible. Please explain.
- Normalization [08/01/2001]
How do I figure out: 90 + 70 + 88 + 94 + x / 5 = 85 ?
- Number Puzzle: 26, 70 [09/12/2001]
The sum of two numbers is 26; 4 times the first number plus 2 times the
second number equals 70. What are the two numbers?
- Number * Sum of Remaining Four Numbers [04/03/2003]
Find 5 numbers such that when each number is multiplied by the sum of
the remaining 4 numbers, the following values will result: 152, 245,
297, 320, 360.
- An Odd Inequality Reversal [08/03/2017]
After correctly factoring a quadratic into two linear expressions that always multiply to a negative value, a young adult gets turned around. Doctor Ian makes sense of what the algebra means, then examines cases.
- One Variable Equations [04/26/1997]
How do you solve 14 = 7d - 2 + d?
- Operator Precedence [08/13/2003]
Since the following statement is true: (1+1)**(5-2) is 8, why is the
following statement true and not false: 2**1+1 is 3 and not 4, and
3*1**3 is 3 and not 27...' ?
- Orange and Grape Sodas [08/24/2002]
Douglas bought a total of 50 cans of orange and grape sodas. He bought
12 more cans of grape than orange. How many of each did he buy?
- Ordering Exponents and Variables [04/08/2000]
Is there a rule for putting terms in descending order if the variables
have the same exponent? What about negative exponents and descending
- Order in Linear Expressions [11/20/2001]
Can you give me a convincing argument as to why, in an equation such as
y=13-7x, where y is a function of x, it should be written as y=-7x+13?
- Order of Addition, Subtraction [9/5/1996]
If an equation has both addition and subtraction in it, which do I solve
- Order of Algebraic Operations [5/25/1995]
How do I solve: (9*4-7*5+6*7-10)/( 6/6+7*3-4*3-9) and 2+2*3-(2/2)*(4-3)+1
- Order of Operation in Contexts [08/02/2014]
Two adults disagree about how to evaluate an ambiguous arithmetic expression.
Starting with an analogy to natural language, Doctor Ian shows how context sometimes
suggests an appropriate order of operations, then stresses the value of grouping
- Order of Operations [05/19/1999]
Given a, b, x, and y, find ax/by.
- Order of Operations and Square Roots [10/09/2009]
At what step in the order of operations (PEMDAS) does one work out a
square root? Is it an exponent?
- Order of Operations Disorder, and Reverse Order [04/28/2016]
A parent struggles to reconcile how her son could apply the order of
operations but still end up with answers different from her correct
ones, when she does not even know why she performed her algebraic steps
in the order she did. With imagery and worked examples, Doctors Greenie
and Peterson illuminate how the PEMDAS convention has two complementary
- Order of Operations Dispute [09/26/2000]
The problem reads: N divided by (division sign)ml where n=12, m=6, and
l=3. I believe the correct answer should be .6666, as 12 divided by 18
- Order of Operations for Solving Equations [10/12/1995]
How would you multiply a question like this: -2(x-y)^2?
- Order of Operations in an Equation [9/13/1995]
Let c = 4. Find: 16 divided by 2c. Is the answer 2, since 16 divided by 8
is 2, or is it 32 because order of operations tells us to multiply or
divide from left to right?
- Order of Operations: Negative Sign, Parenthesis [8/25/1996]
How do I explain why (-2)^2 = 4 and -2^2 = -4?
- Order of Operations: Polynomials and Exponents [9/10/1996]
I'm stuck on 3*5-4^2, (3*5-4)^2, -6^2*2, (-6*2)^2 .
- Order of Operations without Parentheses [8/3/1996]
How do you solve a problem like 21+82-59+29*86*66/50*7/87?
- Order of Operations with Parentheses [08/04/2001]
I know the answers to the problems but not how to get the answers.
- Order of Operations with Percentages [04/05/2001]
Why does the order of operations exclude percentage, square roots, etc.?
- Origin of the word Quadratic [04/07/1998]
Why is a second degree polynomial called quadratic?
- Parallel and Perpendicular Lines [01/14/1999]
How do you tell without graphing whether the graphs of these equations
are parallel, perpendicular, or neither?
- Parallel Lines [12/31/1998]
What are some ways of proving lines parallel - geometrically and
- Parallel Lines [9/3/1996]
The lines y=kx+4 and 3y=(k+3)x-5 are parallel. Find k.
- Parentheses from the Inside Out [03/03/2003]
I don't understand order of operations and evaluating expressions.
- Pascal's Triangle Pattern [04/22/1999]
What pattern does multiplying each entry by 1, 2, 3, 4, 5... in order,
and adding the products yield?
- Passing Cyclists [08/31/2003]
Two cyclists ride on the same road in opposite directions, passing
each other twice...
- Percentage Increase in Cost of Eggs [8/4/1996]
When eggs went from 17 to 39 cents, by what percent did the price
- Plus or Minus Sign [03/08/2002]
What does this equation mean: y = +- k ? The - sign is directly under the
- Polynomial Factoring Rules [04/02/1997]
How do I apply the polynomial factoring rules to t^21+1 and 25y^2-144 =
- Polynomials with x [9/10/1996]
How do I solve problems like 3x-x(3-5x)=(5x-2)(x+4)?
- Population and Percentage [03/07/1999]
Given population data, find the number of women in two different years.
- Positive Domains [5/19/1996]
What are the values for x in x > -2 when the domain is positive?
- Pre-calculus Extra Credit Projects [6/5/1996]
A high school pre-calculus student asks for suggestions for an extra