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.
- How Many Stools? How Many Tables? [12/13/2002]
In their spare time, the Buffalo Bills operations staff members build
3-legged stools and 4-legged tables. Last month they used 72 legs to
build 3 more stools than tables.
- How Many Trains Meet? [09/17/2001]
Every hour, on the hour, a train leaves Tallahassee for Jacksonville,
while another train leaves Jacksonville for Tallahassee. The trip between
the two cities takes exactly two hours. How many trains going in the
opposite direction will a Tallahassee train to Jacksonville meet?
- How Many Valentines? [02/10/2003]
If everyone in your class gave a Valentine to everyone else in your
class, how many valentines would be exchanged?
- How Much Does the Youngest Boy Get? [05/25/2000]
Five brothers, each born in a different year, share a gift of $100...
- How Much Money Entering the First Store? [09/26/2002]
Timothy spent all of his money at five stores. At each store, he spent
$1 more than half of the amount he had when entering the store. How
much money did he have when he entered the first store?
- How Much Popcorn did Paul Sell? [01/28/2002]
Paul made $44.14 selling 27 items (beer and popcorn). If he made $1.22
selling popcorn and $2.62 selling beer, how many boxes of popcorn were
- How Old Are John and Julia? [05/17/2001]
Julia is as old as John will be when Julia is twice as old as John was
when Julia's age was half the sum of their present ages. John is as old
as Julia was when John was half the age he will be 10 years from now. How
old are John and Julia?
- How Wide is the Garden? [01/06/1998]
A rectangular garden has a 136-ft. fence around it. The garden is 40 feet
long. How wide is it?
- A Hundred Total of Three Varieties — for a Total Price of Exactly a Hundred [10/21/2015]
What combination of all three types of livestock can you buy for exactly 100 euros
that gives you exactly 100 animals? By bringing integer solutions and relative
primality to bear, Doctor Ali picks up where a student left off.
- Identifying Algebraic Properties [12/29/1996]
How do you identify which algebraic property has been used in each step
of a simplification process?
- Identity and Inverse Properties for Zero [01/07/2004]
If we subtract 0 from a number and get the same number, doesn't that
make 0 an identity for subtraction? Also, can't a number be its own
inverse for subtraction?
- Implied Multiplication and TI Calculators [05/02/2008]
If I type 36/6(25-11*2) into my TI-85, I get an answer of 2. If I
include a multiplication symbol to have 36/6*(25-11*2), I get an
answer of 18. I thought that when the 6 is written right up against
the parenthesis, the multiplication is implied. Why am I getting two
- The Importance of Defining Variables Clearly in Algebra [11/01/2007]
A sample of dimes and quarters totals $18.00. If there are 111 coins
in all, how many are there of each coin?
- Incorrect Application of PEMDAS and Order of Operations [09/14/2006]
My students think PEMDAS means that all addition should be done before
any subtraction is done. How can I show and convince them that they
need to do both operations at the same time working from left to right?
- Increasing the Average [02/25/2001]
Five members of a basketball team are weighed and an average weight is
recalculated after each weighing. If the average increases 2 pounds each
time, how much heavier is the last player than the first?
- Inequalities [01/24/2001]
How do you add, subtract, multiply, and divide inequalities?
- Inequalities: Add or Subtract? [10/21/2001]
Sometimes, when doing inequalities problems, I have to add or subtract
one from the answer I have calculated. I don't understand when to add,
subtract, or do nothing at all.
- Inequalities and Absolute Value - Case Method [09/01/1998]
What is the case method? How does it apply to inequalities with absolute
- Infinite Geometric Progression [10/02/1997]
How do you go about solving .27 repeating into a fraction?
- In-Out Tables for Algebraic Thinking [11/28/2015]
A younger sister sees several patterns, and even her older sister's
answer, but no path to deducing the two operations that map her given
input to output. Foreshadowing slope-intercept and point-slope
equations, Doctor Peterson builds tables to reveal the algebraic
thinking behind linear functions.
- Intercept Equation [10/07/1998]
I found a plane using the intercepts (4,0,0), (0,-5,0), and (0,0,3). Now
I want an equation for those points using Ax + By + Cz = D.
- Interpreting Slopes [09/14/1998]
Does the graph of each of these equations rise to the right, fall to the
right, is it horizontal, or is it vertical?
- Interpreting the Difference Between Two Numbers [09/22/2006]
If a question asks "What is the difference between 15 and 12" or "What is the difference between 12 and 15," would your responses be +3 and -3 respectively? Or would they both be +3?
- Interval Notation [11/01/1997]
Solve for x and write the answer in interval notation: -3 < -x < 2 / 3.
- Interval Notation for the Number Line [03/04/2009]
What does the interval [-4,10] mean on the number line? How do I graph it?
- Is y the Same on Both Sides? [9/4/1996]
3/4 + y = 0.75 + y
- Jack is Older than Jill [11/16/2001]
If you reverse the digits in Jack's age, you get Jill's age. The sum of
their ages equals 11 times the difference between them. Jack is older
than Jill. What are their ages?
- Just Plain Algebra [01/07/1998]
Do you have any advice or any programs you might recommend for learning
- K-triple [9/15/1996]
Which of these is a "k-triple", defined as (k/2, k, 3/2k)?
- Learning Addition and Multiplication with Algebra [3/16/1996]
How do you know what the (a) is in algebra? I saw this on a picture:
a(1/a) = 1/a(a) = 1...
- Learning Algebra [6/3/1996]
How do you solve for: x(x+1) + 2 (x + 1) = ?
- Length, Width of Rectangle Given Perimeter [9/4/1996]
If the perimeter of a rectangle is 300 and the length is two times the
- Letters for Variables [11/19/2001]
Why do we use letters for variables?
- Linear Equations in One Variable [10/11/2001]
Solve: 35 = 17 + 3(x-2).
- Linear Systems of Equations in Two Variables [06/18/1999]
How can I find the length of AE, EB and DC, given that parallelogram ABCD
has a perimeter of 50, trapezoid AECD has a perimeter of 39, and AE = EC?
- Linear Transformations [6/13/1996]
Determine the image of (2,-6) under (x,y) -> (x+3, y+5)
- Logarithmic Problem [2/22/1995]
Can you help me solve this: log 3^x = 6 + (1/2)log 9^x?
- Magic Triangle Puzzle [09/24/2001]
The numbers in the squares are the sums of the numbers in the circles to
which they are connected. Discover a method to help you find the numbers
to put in the circles.
- 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.