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.
- Swimming Laps [04/15/2002]
John decides to swim a certain number of laps of the pool in five
days. On the first day he covers one fifth of the total. The next day
he swims one third of the remaining laps...
- Swimming Upstream and Down [09/19/1997]
A man swims 200 meters upstream in 3 minutes (180 seconds) and downstream
150 meters in .7 minutes. How fast is the man swimming and how fast is
- Symbol for Integers [5/19/1996]
Why is the symbol for integers 'z'?
- System of Equations and the Substitution Method [06/07/1998]
Can you explain the "substitution" method of solving systems of
equations, and provide an example?
- Teaching Inequalities [04/01/2009]
We're looking for the best way to help students understand the idea of
greater than or equal to, including some real-life examples of how it
might be used. Do you have any thoughts?
- Testing for Triangular Numbers [06/09/1999]
How do I figure out whether 12456, 1225, or 13456 are triangular or
- Thinking about the Mediant of Two Fractions [12/10/2009]
One of my students noticed that when given 1/a and 1/b, the fraction
2/(a+b) always seems to be in between 1/a and 1/b. Is this a new
fraction rule, or has this been discovered and proved before?
- Thirds of Seven [07/24/1997]
Seven is equal to how many thirds of seven?
- Thoughtful Trial and Error [05/21/2002]
Arrange the numbers 1 through 10 in a triangle so that all the rows of
three numbers have the same sum, and all the rows of four numbers have
the same sum.
- A Three-Legged Stool [06/26/2001]
Why is a three-legged stool steady, while a four-legged stool can be
- Three Less Than a Number [08/19/2001]
What is "three less than a number"? I believe it is x-3 but I am being
challenged that it is 3-x.
- Three Pieces of Wire [01/07/1998]
A florist cuts a wire into 3 pieces. The 1st piece is 20% of the 2nd; the
3rd piece is 110% of the 2nd. The wire is 23 cm long; how long is each
- Three Weights [12/07/1997]
A boy selling fruits has only three weights, but with them he can weigh
any whole number of pounds from 1 pound to 13 pounds inclusive. What
three weights does he have?
- Times Greater Than, Times As Much As [05/02/1999]
A number is five times greater than x. Will this number be 6x or 5x?
- Top 10 Algebra Mistakes [05/23/1998]
What do you think are the top 10 mistakes that are made in algebra?
- Total Test Questions [09/21/1997]
On her biology test, Cathy answered 5/6 of the questions correctly...
- Train and Tunnel [7/18/1996]
A train passes completely through a tunnel in 5 minutes. A second train,
twice as long, passes through the tunnel in 6 minutes...
- Train B Overtakes Train A [01/17/2002]
Train A leaves the station traveling at 30 miles per hour. Two hours
later train B leaves the same station travelling in the same direction at
40 miles per hour. How long does it take for train B to catch up to train
- A Train Breaks Down [06/30/2001]
A train breaks down an hour after starting its journey. After repairs, it
can only continue at half its original speed and arrives at its
destination two hours late...
- Train Overtaking Another Train [10/19/1996]
Two trains leave a station, one an hour after the other...
- Translating English to Algebra [06/27/2002]
Write an algebraic expression for 'Seven more than twice a number'.
- Translating Word Sentences to Math Sentences [03/10/2004]
"Six more than three times x is four times two less than x" was
translated in our textbook to 6 + 3x = 4(x - 2). Some of my students
felt it should be 6 + 3x = x - (4*2). Is there a true rule or law of
mathematics or some definite tradition as to the interpretation of
- Traveling Trains and Nonlinear Systems [07/16/1998]
A train leaves town A and travels 540 miles at a uniform speed, followed
an hour later by another train.... What were the speeds of the trains?
- Triangle and Trapezium Ratio [10/26/1998]
ABCD is a square with sides of length 4 cm. Find M on BC so that the
ratio of triangle ABM and trapezium ADCM is equal to 1/3.
- Turning an Equation Upside-Down [07/09/2001]
After this step: 2/h = (a+b)/ab, a book shows this: h/2 = (ab)/(a+b). Can
you simply invert the fractions and maintain the equivalency?
- Turning Word Problems into Equations [04/15/2002]
I don't like turning word problems into equations because it is too
confusing. Is there a technique that might help me out?
- Turning Word Problems to Equations [03/08/2001]
How do you turn English sentences into math equations?
- Two Cars on Crete [08/15/1997]
Find the speed of the car from Phaistos.
- Two Numbers with Equal Sum, Product, and Quotient? [05/07/2007]
Find two numbers such that when you find the sum, product, and
quotient of the two numbers they are all equal.
- Two Step Equations [02/24/1998]
How to solve two-step equations.
- Two Steps Forward, One Step Back [10/14/2003]
A salmon swims 30 feet upstream in 5 seconds, then rests for 2
seconds. If she moves 10 feet downstream during each rest, how long
does it take her to swim 1010 feet?
- Two Trains with a Twist [9/11/1996]
The local train heading for Westford at 60km/h and the express train
heading for Eastford at 80 km/h pass each other at Centerville while
heading in opposite directions...
- Two Ways to Solve an Equation [1/23/1996]
x + 3 = 7
- Understanding Rectangle Area and Perimeter [11/08/2002]
True or false: if the perimeter of a rectangle increases, the
rectangle's area always also increases.
- Unique Values for x, y [9/10/1996]
Find unique values for x and y so that x+y and y*x equal the same number.
- 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.