Algebra I All-in-One For Dummies. Mary Jane Sterling

      

The denominator of a fraction, or bottom number, tells you the total number of items. The numerator, or top number, tells you how many of that total (the bottom number) are being considered.

      In all the cases using fractions, the denominator tells you how many equal portions or pieces there are. Without the equal rule, you could get different pieces in various sizes. For example, in a recipe calling for cup of flour, if you didn’t know that the one part was one of two equal parts, then there could be two unequal parts — one big and one little. Should the big or the little part go into the cookies?

      Along with terminology like numerator and denominator, fractions fall into one of three types — proper, improper, or mixed — which I cover in the following sections.

Converting Improper Fractions and Mixed Numbers

      An improper fraction is one where the numerator (the number on the top of the fraction) has a value greater than or equal to the denominator (the number on the bottom of the fraction) — so the fraction is top heavy. Improper fractions can be written as mixed numbers or whole numbers, and vice versa. A mixed number contains both a whole number and a fraction.

      To convert a mixed number to an improper fraction: Multiply the integer (A) times the denominator (D) and add the numerator (N). Put that result over the denominator (D).

      To convert an improper fraction to a mixed number: Divide the numerator (N) by the denominator (D). Put the quotient (Q) in front as the integer, the remainder (R) as the numerator, and the denominator (D) in its usual place.

      You sometimes want an improper fraction, and other times a mixed number; it just depends on what you’re doing at the time. You can easily change from one form to the other.

      

Q. After the party, Maria puts all the leftover pieces of pizza together. There are 15 pieces, each of a pizza. How much pizza does Maria have left?

      A. Maria has a whole pizza plus seven pieces more: . (So she’ll have to save two pizza boxes to put the leftovers in.)

      Q. A recipe calls for cup of sugar, but you want to double the recipe (you have a hungry family). Doubling the sugar requires cups. If you’re using a 1-cup measuring cup, your cup will runneth over. How much more than a cup of sugar is this?

      A. , so you’ll need a full cup plus cup more.

      1 Change the improper fraction to a mixed number.

      2 Change the mixed number to an improper fraction.

      3 Change the mixed number to an improper fraction.

      4 Change the mixed number to an improper fraction.

      5 Change the improper fraction to a mixed number.

      6 Change the improper fraction to a mixed number.

Finding Fraction Equivalences

      In algebra, all sorts of computations and manipulations use fractions. In many problems, you have to change the fractions so that they have the same denominator or so that their form is compatible with what you need to solve the problem. Two fractions are equivalent if they have the same value, such as and . To create an equivalent fraction from a given fraction, you multiply or divide both the numerator and denominator by the same number. This technique is basically the same one you use to reduce a fraction.

      Rewriting fractions

      When you multiply or divide the numerator and denominator of a fraction by the same number, you don’t change the value of the fraction. In fact, you’re basically multiplying or dividing by 1 because any time the numerator and denominator of a fraction are the same number, it equals 1.

      For example, how much of a 32-ounce package are you using if your recipe calls for 12 ounces? You’re using of the package. If you divide both the 12 and the 32 by 4, you’re basically dividing by , which equals 1. The same goes for multiplying the numerator and denominator by the same number. So, you’re using of the package.

      Here’s another example: Create two new versions of the fraction — one with smaller numbers and one with larger numbers.

      To come up with smaller numbers, you need to find something that divides both the 16 and the 20 evenly. There are two possibilities: 2 and 4.

      The fraction is said to be in “lowest possible terms” because there are no common factors available. No whole numbers can divide the two numbers evenly except for 1.

      There is no end to the possibilities when creating multiples of a particular fraction:

       and and , and so on. You choose the numbers that work best for you in the situation.

      However, not all fractions with large numbers can be changed to smaller numbers.