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.
- The Unknown Angle that Didn't Matter [10/01/2014]
A geometry word problem leads to the confounding result 120° equals
0. Doctor Ian takes a closer look at how to interpret this puzzle's statement —
and empty solution sets generally.
- Use Factoring to Find Two Consecutive Integers... [7/31/1996]
Find two consecutive integers whose product is 56.
- Using Euclid's Algorithm with Three Numbers [11/05/2003]
How do I find the GCD of three integers using Euclid's Algorithm? I
am not sure where you plug the third integer into the algorithm.
- Using Guess and Check to Start an Algebra Problem [02/15/2008]
Laura is three times as old as Maria was when Laura was as old as
Maria is now. In two years Laura will be twice as old as Maria was two
years ago. Find their present ages. Dr. Ian shows that by first making
a guess and checking it you can then easily write the algebraic
equations needed to solve the problem.
- Using Interval Notation to Express Answers [12/03/1996]
Solve for x in the inequality x/(x+3) >= 1/(x-1) and express the answer
in interval notation.
- Using Variables to Solve Word Problems [05/10/1998]
Susan puts only dimes and quarters in her coin bank ...
- Variable Expressions with Parentheses [9/9/1996]
How do you solve 2(x-1)-3(2-3x)-(x+1) using the distributive property?
- A Variable for Each Digit [11/28/2001]
A husband's age is the inverse of and greater than his wife's age... Some
examples are 82,28 or 54,45. Is there a way to solve this without simple
trial and error?
- Velocity and Time [02/11/1999]
A car averages 30 km/hr the first half of a trip, and 50 km/hr the second
half of the trip. What is the car's average velocity for the trip?
- A Velocity-Time Problem [02/16/1999]
If you drive 40 mph instead of 30 mph, you save 30 seconds per mile, but
if you make the same 10 mph increase from 70 mph to 80 mph, you only save
5.6 seconds per mile. Why?
- Venn Diagram - Fast Food Restaurants [05/09/2002]
From a list of how many restaurants serve what kinds of food, deduce
the total number of restaurants.
- Verifying the Solution to an Inequality ... by Substituting? by Checking Boundaries? [04/28/2010]
A student is confused about how to verify work on a linear inequality in one variable,
which also features fractions and sign changes. Doctor Peterson responds with an
explanation of how to check a solution to an inequality: by testing its boundaries.
- Water Main Section Lengths [01/09/2003]
A water main is being laid using pipe that comes in either 18-foot or
20-foot sections. The water main will require 14 fewer sections of 20-
foot pipe than if 18-foot sections are used. Find the total length of
the water main.
- Weighted Average of the Velocities [01/30/2003]
Finding the average velocity for each of the three phases of the
- Weighted Averages [11/30/2001]
An advertisement for an orange drink claims that the drink contains 10%
orange juice. How much pure orange juice would have to be added to 5
quarts of the drink to obtain a mixture containing 40% orange juice?
- What are Like Terms? [11/20/2001]
Examples of like terms.
- What Are Variables and How Do They Work? [06/29/2004]
I don't understand the meaning of even basic algebra, like what 2y or
x means. What do variables mean?
- What Are Variables For? [02/09/2001]
Why is it important to be able to figure out the values of variables?
- What Does It Mean to Distribute [08/25/2007]
I understand the basics of the distributive property, but I'm confused
about what the 'over' means, like in the distributive property of
multiplication over addition or over subtraction.
- What Do Input and Output Mean in Math? [04/09/2007]
What is the relationship between the input and output on a table? I'm
not quite sure what an input or output is.
- What Do Parentheses Mean? [08/15/2002]
Can you help me find ways to figure out what parentheses mean?
- What Is a Coefficient? [01/29/2004]
What is a coefficient?
- What is 'Algebra'? [8/6/1996]
What is the meaning of the word 'algebra'?
- What is Algebra? [08/24/1997]
Could you explain algebra from the ground up?
- What is Algebra? [07/20/2001]
Algebra is like arithmetic, but some of the numbers have names instead of
- What is a Partial Fraction? [01/08/1999]
What do partial fractions have to do with polynomials?
- What is the Value of K? [01/24/2003]
For positive integer walues on N, define N! to be the sum of integers
1 to N inclusive.
- What is x? [01/07/2002]
Can you tell me how to work out 5x + 3 = 38 ?
- What's the Value of x? [07/25/1997]
2x = x+18+27 and 3x = 2x+7+9.
- What Time Did Ajay Start? [08/06/2003]
Three persons, Ajay, Bhanu, and Chandu, plan to travel from P and reach
Q at 4.00 PM...
- What Time Does She Get to Work? [04/30/2002]
Lillian drives to work at an average speed of 45 miles per hour. If
she travels 30 miles to work and leaves at 8:00 A.M., what time does
she get to work?
- When Do You Flip an Inequality Sign? [12/19/2005]
What does it mean to change the sense of an inequality? When do you
have to do that?
- When FOIL Fails [03/22/2001]
Find the product - (2c-3)3rd power. I can do FOIL on a question to the
second power, but I cannot figure out how to do a question to the third
- When Hands of a Clock Meet [8/29/1996]
To the nearest second, what is the first time after 12:00 that the hour
hand and the minute hand of a clock are together?
- When Was King George Born? [08/23/2002]
King George died 180 years after King Alfred was born. Their combined
age when they died was 73 years. King Alfred died in 315 A.D. In what
year was King George born?
- Which is more 'simple'? [09/24/2001]
Which is considered more simple (when simplifying): 2x+2y (two terms and
three operations) or 2(x+y) (one term and two operations, but the higher
operation of parentheses)?
- Which Property? [10/11/2005]
A look at the general ideas of commutative, associative, and
distributive properties with the goal of understanding them and being
able to remember which one is being used in a math problem.
- Who Picked the Most? [09/12/1998]
Arrange the names of the people in the order of the number of peaches
that each picked, starting with the person who picked the most.
- Who Will Win the Race? [11/26/2002]
Who will win the race and by how many meters?
- Why Algebraic Expressions with Parentheses? [03/18/2003]
How would you write these algebraic expressions without parentheses?
-(2x-3y-6) and -(5x-13y-1).