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.
- 2a/5 or 2a Divided by 5? [12/13/2001]
Why, when writing a division problem, do you write 2a over 5 instead of
2a divided by 5?
- (9/2)/(3/4): A Picture [03/27/2003]
We were trying to find a division fraction question and everything we
came up with turned into a multiplication fraction problem. Why is
- Basic Real Number Properties [07/31/1997]
I need information about basic real number properties: associative,
commutative, closure, identity, inverse, distributive.
- Decimal in the Denominator Visualized [12/13/2013]
A young adult struggles to visually model and evaluate a fraction that has a decimal in
its denominator. Doctors Jerry and Peterson suggest equivalent fractions, inverses,
and scale models.
- Division [11/25/1997]
How do you divide 1,000,000,000,000 by 999,999,999,999,999,999,999?
- Division by Zero [11/2/1994]
Why can't you divide a number by 0?
- Division Direction [03/30/2015]
An adult wonders why the standard algorithm for division does not
follow the right-to-left order of the algorithms for the other basic
operations. With copious examples, Doctor Peterson reveals that
efficiency explains why manual division proceeds faster when performed
left to right.
- Explaining Fractions [11/06/2001]
I am a practicum student in a 8th grade math class. The teacher and I
don't know how to explain to the students why you can do this manuever.
c=d/g; c/d=g; d/c=g.
- Least Common Multiple Puzzle [03/24/2003]
What is the smallest number divisible by 1,2,3,4,5,6,8,9,10 that is
- Least Common Multiple Word Problem [12/10/1996]
Each of three businesses receives different sized cartons of glasses. If
all three receive the same number of glasses and none of them receive
more than a thousand glasses, how many glasses and cartons are in each
- Long Division [08/23/1998]
My biggest problem is long division. For example, what is 73/4?
- Manipulating Positive and Negative Numbers [8/9/1996]
Could you explain the rules for adding, subtracting, multiplying, and
dividing positive and negative numbers?
- Multiples of 9 Number Puzzle [03/23/2003]
If you take a two-digit number, add the digits, and subtract the sum
from the original number, why is the answer always a multiple of 9?
- Order of Operations: Math vs. English; Calculators [11/13/1996]
What is 144 divided by 3(8+4)? 4 or 576?
- Probability of Sky Falling is 0/0? [3/3/1995]
When asked the probability that the "sky will fall" one student responded
"0/0". What should we have told this 8th grade student about dividing by
- Reciprocals [12/19/1996]
What is the use of reciprocals?
- Restrictions on Variables in Division [8/4/1996]
What are the restrictions on the variables in (y^3+8)/(y^2+3y+2)?
- Wording Division Problems [08/10/1998]
How would you clarify the wording of division problems like "divide into"
or "divide by"?
- Working with Integers [01/12/1999]
Could you explain how to divide, multiply, add, and subtract integers?
- 1 Divided by Very Small Number [8/6/1996]
1/0.00000000000000000000000000000000001 = ?
- Adding and Subtracting Roman Numerals [10/26/1999]
How can I add and subtract Roman Numerals without converting them to
regular numbers? What did the Romans do to solve this kind of problem?
- Are Subtraction and Division Commutative and Associative Operations? [05/15/2008]
I don't agree that subtraction and division are not commutative
operations. If we define a - b as a + (-b), then it commutes correctly
into (-b) + a. So why does everyone say subtraction is not commutative?
- Brackets [01/29/2001]
Please explain how brackets  work.
- Calculating Percentage Markup Versus Profit [01/23/2004]
In order to figure a profit of 10% I would normally take my cost and
then multiply by 1.1 to find the selling price. My employer tells me
that in order to find a 10% profit I should divide my cost by 0.9, but
I don't understand how this is 10%. Can you please explain?
- Cancelling Fractions [06/03/2003]
I am reading a book on algebra. The author states that cancelling is
division too and then shows the following without any explanation of
how it's done...
- Cars in Rows, with Remainders [01/22/2013]
A student struggles to figure out what positive integer smaller than 1000 groups into
fives with remainder one, or into four with remainder one, or nine evenly. Doctor Ian
introduces divisibility, methods — and several kinds of shortcuts.
- Casting Out Nines to Check Arithmetic [08/26/1999]
Can you explain how to use the 'casting out 9's' technique in a way that
a 6th grader can understand?
- Comments on Dividing Fractions [03/15/2007]
A question on how to divide fractions leads into comments on how
division and subtraction can be thought of as multiplication and
addition of an "inverted" number.
- Commutative Property and Division [02/21/2002]
I am trying to prove that the commutative property following certain
rules will work in all circumstances for division...
- Compatible Number Estimating [01/31/2002]
My homework is on compatible numbers to estimate quotients - for example,
1,625 divided by 38.
- Compatible Number Method for Estimating [09/10/2001]
We are supposed to estimate using compatible numbers. How does one
estimate something like 574/8?
- Confusion over Interpretation of PEMDAS [03/24/2005]
Students take PEMDAS letter by letter, tending to do all
multiplication before any division, rather than working from left to
right. Here's a suggestion on how to help avoid that confusion.
- Converting Bits to Megabytes [02/12/2001]
The human brain will store 100 trillion bits in its lifetime. How many
million megabytes can your brain store?
- Counting Days of the Week [02/02/1999]
If today is Thursday, what day would it be 120 days from now?
- Counting Techniques [12/29/1997]
Find the number of positive divisors of 17640, inclusive of 1 and 17640
itself. Find the number of odd integers between 2000 and 7000 that do not
have repeated digits.
- Cutting An Equal Number of Slices of Cake [07/13/1998]
One cake is 35 slices and another is 42 slices. If the cakes need to have
the same number of slices, how many slices should they have?
- Deciding How to Express the Answer to a Division Problem [07/22/2006]
When dividing something like 7 by 2, is it best to express the answer
as 3 1/2, 3.5, or 3 remainder 1? How do I decide which form to use?
- Definition of Quotient [11/13/2008]
If I divide 15 by 2, I always thought that the quotient was 7 and the
remainder was 1. But I've seen that the quotient can also be 15/2. How
can quotient have different meanings?
- Definition of Vinculum [10/21/2004]
Is the bar placed over the set of repeated digits in a repeating
decimal called a vinculum? I've heard that but I can't confirm it.
- Determining If a Large Number is Divisible by 11 [10/22/2003]
I just learned a trick to decide whether a large number is divisible by
11 or not. Why does the trick work?