See also the
Dr. Math FAQ:
order of operations
3D and higher
Browse Middle School Division
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Order of operations.
- How Much Money? [08/16/2001]
A man has 3 daughters, 4 sons, and a sum of money. If he divides the
money equally among his daughters $2 will be left over...
- 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?
- Introduction to Division with Negative Numbers [05/05/2004]
I don't know how to do division with negative numbers, like +18/-9
or -24/-8 or +21/-3.
- Is Integer Division Associative? [05/14/2002]
Is integer division associative?
- Least Common Multiple [10/26/1999]
What is the smallest number that is divisible by the numbers 1 through
- Least Common Multiple [11/5/1995]
A student asks about the least common multiple of 8,12,36.
- Least Common Multiple [01/24/1998]
I need to find the LCM of the numbers 16, 20, 9, and 8. I've gone up to
1,000 and still can't find it.
- Least Common Multiple [09/29/2001]
What is the least common multiple of 11, 13, and 17?
- Limits of Understanding Division by Zero [09/16/2016]
A parent disagrees with her daughter's teachers about the result of division by zero.
Invoking the floor function and comparing two similar limits, Doctor Ali closes the case
- Long Division of Small Numbers [03/26/2014]
A military enlistee struggles to make sense of dividing bigger numbers into smaller ones, as
occurs when expressing fractions as decimals. Applying the distributive property and
multiplicative identity, Doctor Ian steps through the long division algorithm.
- Making up a Division Word Problem [11/13/2001]
Write a problem that could be solved by using the division sentence
1489/28=n; then write a pair of compatible numbers and estimate the
- Multiplication and Division Equations [11/29/2001]
Solve for x: 17x = 85.
- Multiplier, Sum, Difference, Product, Quotient, Dividend [11/25/1997]
What is a multiplier? an addend? a minuend? a product? Why is the number that is being divided the "dividend"?
- Multiplying and Dividing Negatives [07/20/1998]
Can you explain multiplication and division with signed numbers? For
example what is the sign when you multiply negative numbers?
- Name of Division Symbol [7/31/1996]
What is the official name for the division symbol?
- Negative Remainders in Arithmetic and Algebra: a Difference of Degrees [05/14/2010]
An algebra teacher wonders how to explain the existence of negative polynomial
remainders to students accustomed to positive remainders. Doctor Peterson clarifies
the difference between the remainders of arithmetic and those of algebra,
underscoring polynomial degree.
- New Rule for Divisibility by 7 [11/01/1997]
Multiply the integer excluding the one's digit by 3, and add the one's
- Number Divisibility [11/01/1998]
How many natural numbers under 2,000 are divisible by 5 or 7? What's the
quickest way to figure this out?
- Order of Operations vs. Associative Property [10/10/2002]
Is the associative property an exception to the order of operations, a
corollary, or something totally unrelated?
- Order of Operations vs. Distributive Property [02/04/2005]
Does the distributive property contradict the rules of order of
operations? When we have a(b + c) order of operations says we should
add first in the parentheses, then multiply, but the distributive
property says we can multiply first to get ab + ac, then add.
- The Period of 61/97 [01/12/2001]
How can I find the period of 61/97?
- Polynomial Division Compared with Long Division [02/01/2009]
I'm having difficulty grasping the concept of polynomial division.
- Polynomial Long Division [03/17/2004]
Use long divion to divide (2x - 3) into 4x^4 - x^2 - 2x + 1. I really
need help in doing this.
- Primes and Repeating Unit Numbers [12/09/1998]
How do you prove this statement: For every prime number there exists a
repeated unit number that is a multiple of that prime.
- Quickly Dividing in Two [7/8/1996]
Are there any quick tricks for figuring out fifty percent of a number
such as 14.95 in your head?
- Rationalizing the Denominator [04/09/1997]
Must all fractions have denominators that are rational numbers?
- Remainders [07/02/1998]
Can you tell me how to do remainders?
- Roman Numerals: Division [11/05/2001]
I have found information on how to add and multiply Roman numerals, but
have not found any information on how to divide or subtract.
- Setting Up a Division Calculation Correctly [12/16/2005]
I get confused about which number goes where when I have to do something like "6 divided by 2". Which number goes in the calculator first? And which number goes on the outside and inside if I am doing it on paper?
- Small Base, Long Division [10/06/2015]
An adult struggles to divide numbers in bases other than ten. Starting with a list of
divisor multiples, Doctor Peterson steps through a similar long division before
checking his work.
- Sums Divisible by 11 [10/10/2001]
Why is the sum of a number with an even number of digits and that same
number written in reverse always divisible by 11?
- Synthetic Division [11/13/1997]
Why does synthetic division work?
- Three-Digit Numbers [01/27/1997]
If you take a three-digit number, rearrange it two other ways, add the
three numbers together, and divide by 3, why do you always get a
remainder of zero?
- To Multiply or Divide? That Is the Question. [09/02/2003]
In a story problem, how do you know whether to multiply or divide?
- Total Points from Average Points [03/12/2001]
Mandela's basketball team played six games, with an average or mean of 50
points. How many points did they score?
- Trick for Numbers Divisible by 3 or 9 [02/24/1998]
Proof of a trick for numbers that are divisible by 3 or 9.
- What Does Cancelling Mean? [01/22/2003]
My teacher says that 2k/4n can be simplified to k/2n. How can one k
cancel out 2 n's?
- Why Are Remainders Small? [07/06/2016]
A teen seeks an explanation for why polynomial division always results in remainders
smaller than divisors. With an analogy to integer division, Doctor Peterson clarifies that
it's a choice.
- Why Decimal Division Works [04/24/2001]
Can you explain why (not how) the procedure of division with decimals
- Why Does Cross Multiplication Work? [05/09/2002]
Why does the rule a/b = c/d also mean ad = bc? Why can we solve
proportions by cross multiplying?