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 Dependent Variable

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# Fraction Competency Packet

To use this booklet, review the glossary, study the examples, then
work through the exercises. The answers are at the end of the
booklet. When you find an unfamiliar word, check the glossary for
a definition or explanation.

Calculators are not allowed when taking the Computerized
Placement Test (CPT), nor in Fundamentals of Mathematics, Pre-
Algebra, and Elementary Algebra; therefore, do not rely on a
calculator when working the problems in this booklet.

If you have difficulty understanding any of the concepts, come to
one of the Tutoring Centers located on the Lynn, Danvers Main
and Danvers Hathorne Campuses. Hours are available at (978)
762-4000 x 5410.  The Centers are closed when school is not in
session, and Summer hours are limited.

Glossary

Boosting: Rewriting a fraction as an equivalent fraction with a higher denominator.

Denominator: Bottom number of a fraction indicating how many parts make a whole.

Difference: The result when two numbers are subtracted.

Divisor: The number after the division sign in a division problem, (i.e. 12÷7); or the bottom
number of a fraction, (i.e. ); the number "outside" the division house (i.e. ).

Equivalent Fraction:
Fractions that are found by multiplying the numerators and
denominators by the same number.

Factor: Numbers equal to or less than a given number that divides the number evenly. For
example, the factors of 12 are 1, 2, 3, 4, 6, 12.

Fraction: Any number written in the form of one whole number over another, , indicating
number of parts being considered over the number of parts that make one whole.

Fraction Bar: The line separating the numerator and denominator in a fraction, and it indicates
division.

Greatest Common Factor (GCF): The largest matching factor of two or more given numbers.
It is used to reduce fractions.

Improper Fraction: Any fraction with the numerator larger than the denominator.

Least Common Denominator (LCD):
The smallest matching multiple of two or more given
numbers. It is used to "boost" fractions. (Also called Least Common Multiple, LCM)

Mixed Number: A whole number and a fraction. (It implies addition of wholes and parts; that
is, is read "three and five sevenths".)

Multiple: (Similar to the "times table.") A multiple of a given number is equal to the given
number or greater. Multiples are found by multiplying the given number in turn by 1, 2,
3,... For example, multiples of 4 are 4, 8, 12, 16, …

Numerator: The top number of a fraction. It indicates how many parts of a certain size are
represented.

Prime Factor: Factors of a number that are only divisible by 1 and the given number. For
example, prime factors of 12 are 1 x 2 x 2 x 3. Some frequently used Prime Numbers are 2,
3, 5, 7, 11, 13.

Product: The result when two numbers are multiplied.

Proper Fraction: Any fraction when the numerator is less than the denominator.

Quotient: The solution to a division problem.

Reducing:
Dividing the numerator and the denominator by the same number to get an
equivalent fraction. Final answers of most fraction problems should be expressed
reduced to “simplest terms”; in other words, the numerator and denominator have no
more common factors.

Remainder:
The number left after a whole number division problem is complete. When
converting an improper fraction to a mixed number, the remainder is the numerator of the
fraction.

Sum: the result when two numbers are added.

Whole Number: The Numbers system including 0, 1, 2, 3,….

General Fraction Information  The fraction that represents the above picture is and is read “five sevenths”. That means
that five of the parts are shaded, and it would take seven parts of that size to make a whole. One whole can be "cut up" into equal size parts; therefore, , etc. A whole number can be written as a fraction with a denominator of 1; for example, .
Zero can be written as a fraction using zero as the numerator and any whole number as the
denominator, for example, . Any whole number may be written as a mixed number by using a zero fraction. For example, Mixed Numbers

To convert a mixed number, , to an improper fraction, :

 Whole Number Numerator Denominator Work in a clockwise direction, beginning with the denominator, (7). Multiply the denominator (7) by the whole number, (5) Add that product, (35), to the numerator (2) of the fraction. The denominator remains the same for the mixed number and the improper fraction.

Convert to Improper Fractions: Finding Equivalent Fractions with Larger Denominators
This process is sometimes called “Boosting”

Example : 56 ÷ 8 = 7 Divide the larger denominator by the smaller to find the factor used to multiply the denominator. (Note: The product of the smaller denominator and the factor is the larger denominator) Use this factor to multiply the numerator. The result is two equivalent fractions.

Note: Equal denominators are required for addition and subtraction of fractions.

Find the equivalent fractions as indicated: Equivalent Fractions with Smaller Denominators
Reducing Fractions

Example: Reduce the following fraction to lowest terms There are three common methods, DO NOT mix steps of the methods!
Method 1: The Greatest Common Factor for 90 and 105 is 15. Divide the numerator and the denominator by the GCF, 15.
Method 2: Examine the numerator and denominator for any common factors, divide both numerator and denominator by that common factor. Repeat as needed.  Both 90 and 105 are divisible by 5. Both 18 and 21 are divisible by 3.
Method 3: Express the numerator and denominator as a product of prime factors. Divide numerator and denominator by common factors, (3x5) Multiply remaining factors.

Reduce these fractions. Improper Fractions
Example: Convert to an Improper Fraction
 14 ÷ 3 = 4 Remember: Dividend ÷ Divisor = Quotient Remainder 2 Divide the numerator (14) by the denominator (3). Write the mixed number in the form: Note: Check you answer to see if you can reduce the fraction.

Convert these improper fractions to mixed numbers. Be sure to reduce when it’s
possible.

#11, 12 Hint: how many
wholes will there be?     