Math Skills: Fractions

Lesson 1: Lesson 1: Fractions Pretest and Terminology

Fraction Terminology

First, let's go over Ratio and Proportion.

A ratio is a division of any two numbers, AS LONG AS THE BOTTOM NUMBER ISN'T ZERO! You cannot divide by 0; this is considered "undefined" (since any number times 0 = 0). A ratio is a part to a whole.

2/4 is a ratio. 2 : 4 is also a ratio written a bit differently. But any ratio is a part to a whole. Even whole numbers are ratios! Isn't 4/4 (or 1 whole) a ratio?

Believe it or not, ANY number that yields a whole number, fraction and non-repeating/terminating decimal, any definable number, is a ratio. Most of the numbers we are familiar with are ratios and called "rational" numbers. Irrational numbers like the square root of a negative number or non-terminating decimals like the Greek letter pi (used in circumferences of circles) are NOT ratios!

All fractions that behave this way are ratios.

Put two of them together separated by an equals sign, you have a proportion. The thing is, is the proportion a true proportion? If the two ratios equal each other, it is.

2/4 = 1/2 is a true proportion.

2/4 = 1/3 is NOT a true proportion, and thus we won't concern ourselves with it.

You have seen me use the expression "improper fraction" on the pretest. Let's first say what a "proper fraction" is. A proper fraction is one where the top number is smaller than the bottom number. 4/5 is a proper fraction.

14/5 is "improper", because the top number is larger than the bottom number. BTW, 4/4 is also improper.

This concept related to Mixed Numbers: mixed numbers have a whole number and a fraction with it. Thus 2 4/5 is a mixed number. But when we start multiplying and dividing fractions, it is more convenient to change mixed numbers into improper fractions first to make it easier to work (it is VERY HARD to multiply mixed numbers the usual way!!!).

How do you turn mixed numbers into improper fractions? Multiply the whole number by the bottom number, then add the top number. Thus: 2 4/5 means 2 x 5 = 10, add 4, get 14/5. Remember the number you got goes over the bottom number.

And what is the "top number"? It is called the numerator. The "bottom number" is called the denominator.

Remember this if you ever go on "Weakest Link"!

Reducing Fractions:

To reduce a fraction--and ALL fractions you get as answers in this course MUST be reduced!--find a number that is a factor (factors are numbers you multiply; the old fashioned terms for these are "multiplicand" and "muliplier") of both numerator and denominator, and find the "greatest" factor--this is your greatest common factor.

Example: 9/12.

List the factors of 9: 1, 3, 9. List the factors of 12: 1, 2, 3, 4, 6, 12. The common factors are 1 and 3. The greatest of these is 3, so you divide 9 by 3 and get 3; you divide 12 by 3 and get 4. Thus 9/12 reduces to 3/4.

If the greatest common factor is 1, then the fraction CANNOT be reduced!

This leads to Prime and Composite numbers. All even numbers greater than 2 are composite. Many other odd numbers are also composite because they have various other factors than 1 and itself. 9 is odd but is composite because, besides 1 and 9, it has a factor of 3.

2, 3, 5, 7, 11, 13, 17, 19, for example, are PRIME because they only have factors of one and itself.

One is neither prime nor composite.

When adding or subtracting fractions you MUST have a common (like) denominator. To get a common denominator, find a number both (or all three, whatever) denominators go into evenly, and when you find the LOWEST number they go into, you have your common denominator.

Example: 3/4 + 2/3. The answer is NOT 5/7!!!

Find the Lowest common denominator of 4 and 3. That is, find the LEAST COMMON MULTIPLE of 4 and 3, a LEAST number both go into. 3 and 4 both go into 12, but they also go into 24, 36, etc. The lowest is the one you want, 12, which is the common denominator.

Thus 3/4 becomes 9/12 (4 times 3 is 12; 3 times 3 is 9) and 2/3 becomes 8/12 (3 x 4 = 12; 2 x 4 = 8). And 9/12 + 8/12 = 17/12.

Finally, a reciprocal.

To divide fractions, you turn the divisor (second fraction) into the reciprocal, then multiply. You INVERT or "flip the fraction over". A reciprocal is simply a "flipped over" faction.

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