Fraction Review for Armed Services Vocational Aptitude Battery (ASVAB) Study Guide (page 2)
Problems involving fractions may be straightforward calculation questions, or they may be word problems. Typically, they ask you to add, subtract, multiply, divide, or compare fractions.
Working with Fractions
A fraction is a part of something.
Example: Let's say that a pizza was cut into eight equal slices and you ate three of them. The fraction tells you what part of the pizza you ate. The pizza below shows this: 3 of the 8 pieces (the ones you ate) are shaded.
Three Kinds of Fractions
|Proper fraction:||The top number is less than the bottom number:|
|The value of a proper fraction is less than 1.|
|Improper fraction:||The top number is greater than or equal to the bottom number:/td>|
|The value of an improper fraction is 1 or more.|
|Mixed number:||A fraction written to the right of a whole number:|
|The value of a mixed number is more than 1; it is the sum of the whole number plus the fraction.|
Changing Improper Fractions into Mixed or Whole Numbers
It's easier to add and subtract fractions that are mixed numbers rather than improper fractions. To change an improper fraction, say , into a mixed number, follow these steps:
- 1. Divide the bottom number (2) into the top number (13) to get the whole number portion (6) of the mixed number:
- Write the remainder of the division (1) over the old bottom number (2):
- Check: Change the mixed number back into an improper fraction (see steps below).
Changing Mixed Numbers into Improper Fractions
It's easier multiply and divide fractions when you're working with improper fractions rather than mixed numbers. To change a mixed number, say , into an improper fraction, follow these steps:
- Multiply the whole number (2) by the bottom number (4): 2 × 4 = 8
- Add the result (8) to the top number (3): 8 + 3 = 11
- Put the total (11) over the bottom number (4):
- Check: Reverse the process by changing the improper fraction into a mixed number. If you get back the number you started with, your answer is right.
Reducing a fraction means writing it in lowest terms, that is, with smaller numbers. For instance, 50¢ is of a dollar, or of a dollar. In fact, if you have 50¢ in your pocket, you say that you have half a dollar. Reducing a fraction does not change its value.
Follow these steps to reduce a fraction:
- Find a whole number that divides evenly into both numbers that make up the fraction.
- Divide that number into the top of the fraction, and replace the top of the fraction with the quotient (the answer you got when you divided).
- Do the same thing to the bottom number.
- Repeat the first 3 steps until you can't find a number that divides evenly into both numbers of the fraction.
For example, let's reduce . We could do it in two steps ; then . Or we could do it in a single step .
Shortcut: When the top and bottom numbers both end in zeroes, cross out the same number of zeroes in both numbers to begin the reducing process. For example reduces to when you cross out two zeroes in both numbers.
Whenever you do arithmetic with fractions, reduce your answer. On a multiple-choice test, don't panic if your answer isn't listed. Try to reduce it and then compare it to the choices.
Reduce these fractions to lowest terms.
Raising Fractions to Higher Terms
Before you can add and subtract fractions, you have to know how to raise a fraction to higher terms. This is actually the opposite of reducing a fraction.
Follow these steps to raise to 24ths:
- Divide the old bottom number (3) into the new one (24):
- Multiply the answer (8) by the old top number (2): 2 × 8 = 16
- Put the answer (16) over the new bottom number (24):
- Check: Reduce the new fraction to see if you get back the original one:
Raise these fractions to higher terms.
If the fractions have the same bottom numbers, just add the top numbers together and write the total over the bottom number.
Examples: Reduce the sum: .
. Change the sum to a mixed number: ; then reduce: .
There are a few extra steps to add mixed numbers with the same bottom numbers, say :
- Add the fractions:
- Change the improper fraction into a mixed number:
- Add the whole numbers: 2 + 1 = 3
- Add the results of steps 2 and 3:
Finding the Least Common Denominator
If the fractions you want to add don't have the same bottom number, you will have to raise some or all of the fractions to higher terms so that they all have the same bottom number, called the common denominator. All of the original bottom numbers divide evenly into the common denominator. If it is the smallest number that they all divide evenly into, it is called the least common denominator (LCD).
Here are a few tips for finding the LCD, the smallest number that all the bottom numbers evenly divide into:
- See if all the bottom numbers divide evenly into the largest bottom number.
- Check out the multiplication table of the largest bottom number until you find a number that all the other bottom numbers evenly divide into.
- When all else fails, multiply all the bottom numbers together.
- Find the LCD. Multiply the bottom numbers: 3 ×5 = 15
- Raise each fraction to 15ths:
- Add as usual:
Try these addition problems:
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