Ask Dr. Math

Middle School Archive

Browse Middle School Fractions & Percentages
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Finding a least common denominator (LCD).
    Dividing fractions.
    Comparing/ordering fractions; equivalent fractions.
    Reducing/simplifying fractions.
    Improper fractions and mixed numbers.
    Converting fractions to/from decimals.
    Fractions and repeating decimals.
    Finding a percentage.
    Percentage increase/decrease.

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Improper Fractions [05/20/2002]

How can 14/9 be a fraction when you can't have 14 parts of something that's been cut into 9 parts?

Inches and Feet in Decimal Form [7/16/1996]

How do I convert fractions of inches and feet into a decimal expression of the length in feet?

Increase in Chicken Consumption [8/29/1996]

In 1984 the average American ate 55.7 lbs. of chicken. This was 2.8 lbs. more than the average in 1982. What was the percent of increase in chicken consumption?

Infinite Geometric Progression [10/02/1997]

How do you go about solving .27 repeating into a fraction?

Integral Multiples [9/7/1996]

What fraction of the integral multiples of 3, between 1 and 100, are also integral multiples of 4?

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.

Invert and Multiply [11/08/1997]

I am struggling with how to explain to a class WHY we invert and multiply when dividing fractions.

Inverting and Multiplying When Dividing Fractions [9/28/1995]

I have always been taught that to divide fractions I must invert and multiply. Why does this algorithm work? Is there any other way to divide fractions?

Irwin's Theory [07/06/1998]

How can fractions be added without changing the denominators?

Is the Fraction N/N Considered Proper or Improper? [10/31/2003]

If a fraction has an equal numerator and denominator, what is it called? Is it proper, improper, or something else?

The Joy of Fractions [05/18/2002]

How come fractions are called fractions? How come they can be translated into decimals? Why do they hurt your brain when you first learn them?

LCD, LCM [08/16/1997]

What is the easiest way to find the least common denominator and least common multiple of two numbers?

Learning Fractions in the Metric System [11/27/2003]

Since the metric system doesn't use fractions, are math students in metric countries able to skip fractions and work on more useful things?

Longfellow's Bees [08/01/2002]

One-fifth of some bees fly to the rosebush, one-third fly to the apple tree, and three times the difference fly to the acorn tree. One bee is left flying around. How many bees are there altogether?

Mark-ups, Merchant Fees, and Multiplication ... or Division? [01/06/2011]

Marking up a price by p percent through division by (1 - p) yields a different result than does multiplication by (1 + p). Doctor Wallace explains methods for marking up and for offsetting credit card merchant fees, and along the way distinguishes between markup and margin.

Mathilda and Matt Share a Pie.... [01/20/2010]

Doctor Ian starts with simpler fractions, and a drawing, to model the parts, wholes, and parts-as-wholes in this problem of repeated division.

Meaning of Multiplying Fractions [12/09/1999]

What does it mean to multiply a fraction by a fraction?

Measures of Music and Fractions [01/13/1999]

How can you tell what fraction of a measure a note is? How does it relate to the time signature?

More Than 100 Percent [07/27/1999]

Is there anything larger than 100 percent, like 200 percent or 300 percent?

Mr. Thomas' Salary [9/6/1996]

Mr. Thomas spent 1/12 of his salary to buy a watch and 3/11 of the remaining to pay for a new television set... what was his salary?

Multiplying a Fraction by a Decimal [9/5/1996]

How do I simplify and multiply (40/360) by 78.5?

Multiplying a Fraction by a Variable [08/17/2006]

How do I find the product of something like (1/3) times 'a'?

Multiplying Decimal Numbers and Placing the Decimal Point [09/20/2007]

Why does counting up the total decimal places when multiplying decimal numbers work?

Multiplying Decimals and Decreasing Answers [12/02/1998]

Why does multiplying by a decimal decrease the answer? How can you have less than what you started with when you multiply?

Multiplying Decimals by Negatives [03/18/2002]

I can't find decimals on my calculator.

Multiplying Fractions [4/13/1996]

x over 2y multiplied by 4y over 2 = ?

Multiplying Fractions by Cancellation [10/09/1998]

I'm trying to multiply fractions by cancelling out numbers...

Multiplying Fractions, Dividing Common Factors: Why Does It Work? [06/10/2010]

A teacher wonders why, before multiplying fractions, it works to "factor out." Doctor Ian builds up an explanation about commutativity by walking through increasingly complicated examples.

Multiplying Mixed Numbers [04/06/2004]

Why do you need to convert mixed numbers to improper fractions before multiplying them together? What is the mathematical reason behind this operation?

Multiplying Whole Numbers by Fractions [05/31/1999]

How do you multiply whole numbers by fractions?

Name of Bar That Denotes a Repeating Decimal [08/30/2004]

What is the name of the bar that goes over a digit or group of digits that signifies that the digit or group repeats?

Name of Fraction Bar [7/5/1996]

What is the correct term for the line that separates the numerator and denominator of a fraction?

Net Result of Consecutive Percentage Increases [12/11/2003]

Suppose a quantity is increased by some percentage, and then is later increased again by another percentage. How can I calculate the overall percentage increase?

Newton's Method and Continued Fractions [10/06/1999]

Can you clarify some points on Newton's method of finding square roots without a calculator, and on the continued fraction algorithm (CFA)?

Non-terminating Decimal Representations of Fractions [11/10/1997]

Why when you take a finite, limited quantity like one-third and turn it into a decimal do you get .333... on into infinity?

Number of Places in Products of Decimals [09/19/2002]

When you multiply decimals, why is there always the same number of decimal places in the answer as in the problem?

Numbers Containing Whole Number, Decimal, and Fraction at Once [10/19/2004]

If I have a number that contains a whole number, decimal, and fraction all in one, such as 3.366 2/3, how would I convert all of that to one fraction in lowest terms?

Numbers in a Fraction [05/20/2001]

Is a fraction one number or is it two numbers?

Numerator, Denominator: Etymologies [03/25/2002]

Why are the parts of a fraction called the numerator and the denominator?

The Oneths Place [09/29/1998]

In decimal numbers, why is there no oneths place?

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