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Word Problems

Browse Elementary Multiplication
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
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    Learning to multiply.
    Least common multiple, Greatest common factor.
    Number sentences.
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Algebraic Explanation for Wrist Multiplication [09/10/2004]
The trick behind the technique of wrist-multiplying is explained by use of a standard algebraic multiplication formula.

Algebraic Proof That a*0 = 0 [11/02/2003]
If 'a' is any element of all real numbers, then a*0 = 0. I would like to understand the specific theorem or proof behind this idea.

Alternative to Memorizing Times Tables [01/13/2004]
Memorizing times tables is the standard approach, but there are some helpful ways of thinking about them that will help you understand the math behind them and learn them quickly.

Arranging Blocks in Groups [06/13/1998]
Elise wants to make a rectangular arrangement out of 24 squares. One side must be 8 squares. What will be the dimensions of the rectangle?

Big Number Multiplication [05/04/1997]
What does 4000 x 500 x 32 x 768 x 4 equal?

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?

Canceling and Reducing When Multiplying Several Fractions [01/25/2008]
I need help when multiplying three or more fractions. How can I cancel numbers out when they are not next to each other?

Canceling Instead of Multiplying and Dividing [09/25/2003]
Find the value of the product 3/5 x 5/7 x 7/9 x 9/11 x 11/13 x 13/15.

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...

Can You Multiply Three Numbers Simultaneously? [03/15/2004]
If I have the problem 24 + 342 + 506 I can line the numbers up and add them at the same time. Can I do this with the problem 24 x 342 x 506? If I can, how do you do it? If I can't, why not?

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?

Common Multiples of 4 and 7 [11/19/1997]
What are the numbers that are common multiples of 4 and 7?

Common PEMDAS Errors [12/28/2001]
What does PEMDAS mean?

The Commutative Property Around Us [11/24/1998]
Which operations are commutative... to put on your coat and to pick up your boots...

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.

Counting Decimal Places [05/22/2001]
When multiplying decimals, why does placing the decimal in the answer by counting the number of decimal places in the problem and counting from the right the same number of places work?

Defining Multiplication [08/16/2002]
What is the distinction between 3*4 and 4*3?

Defining Multiplication, Redux [10/09/2014]
Parents, students, and teachers alike continue to wrestle with distinctions between multipliers and multiplicands.

Differences of Squares in the Diagonals of Times Tables [02/17/2013]
An adult wonders about a quirk found within the times tables. Doctor Peterson goes on to clarify the relationship between odd numbers and multiples of four that have remainders of 0, 1, or 3.

Distributive Property over Addition vs. Multiplication [12/11/2002]
Why can't you use the distributive property when the integers in the parenthesis are being multiplied?

Divisibility Algorithm [11/26/2001]
Is there a trick for figuring out multiples of seven?

Estimating when Multiplying [08/19/1999]
How do you estimate when multiplying by a 1-digit number?

Estimation in 3rd Grade [12/16/1996]
How do you learn to estimate? What is its importance?

Evaluating Parentheses, Brackets, Braces and Grouping Symbols [01/20/2007]
How do I evaluate or simplify an expression with lots of grouping symbols, like (18/2){[(9 x 9 - 1)/ 2] - [5 x 20 - (7 x 9 - 2)]}?

Even More on Order of Operations [10/26/2017]
How can 8/4(3-1) have two answers? Doctor Peterson unpacks the ambiguities that have made the order of operations a persistent lightning rod for controversy — and misunderstanding.

Evens and Odds in the Multiplication Table [09/11/2005]
In the multiplication table, are there the same number of even numbers as odd numbers, or is there more of one kind than the other?

Explaining Finger Multiplication [02/13/2002]
Open both palms toward you. To calculate 7 x 9, put 7-5=2 fingers down on the left hand and 9-5=4 fingers down on the right hand...

Explaining Order of Operations [09/06/2001]
How would you explain the order of operations to a 5th or 6th grader who has not yet studied the subject?

Factors and Rectangles [06/25/1998]
I'm trying to figure out what a rectangle and factors have in common.

Finding Least Common Multiple by Using Prime Factors [11/05/2003]
Can you explain how the prime factorization of a pair of numbers can help you find the least common multiple of those numbers?

Finding Least Common Multiple (LCM) by Factoring [10/19/2001]
Find the least common multiple of 6, 8, and 9: figure prime factors and find the product of the greatest power of each prime factor.

Finding What Cancels [03/26/2003]
(1 1/2)(1 1/3)(1 1/4)(1 1/5)... Find how many factors it will take to reach the product of 1000.

Finger Multiplication [04/18/1997]
Do you know a method for teaching blind children "finger math"?

Finger Multiplication for the 9s, Explained [04/22/2002]
I know the nine times table 'finger trick' but am puzzled as to why it works.

Further Musings on "Multiplicand" and "Multiplier" [06/09/2011]
A reader shares thoughts on clearer usage of "multiplicand" and "multiplier." Doctor Peterson responds with some guidelines.

Gift Wrap Combinations [04/20/1999]
Mary has 3 different colors of ribbon and 2 different colors of wrapping paper. How many many different ways can she wrap a gift?

Girls, Cats, and Kittens [11/05/2007]
There are seven girls on a bus. Each girl has one sack and each sack has seven cats inside. Each cat has a litter of seven kittens. How many legs are on the bus?

Greatest Common Factors, Least Common Multiples [8/29/1996]
Please explain greatest common factors, how to find them, and how they are different from lowest common multiples.

Groups in Multiplication [12/04/2002]
When multiplying factors vertically, which factor represents the number of groups? Is it the top factor or the bottom factor next to the multiplication symbol?

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