Fractions for Nursing School Entrance Exam Study Guide (page 4)
The practice quiz for this study guide can be found at:
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.
Let's say that a pizza was cut into 8 equal slices and you ate 3 of them. The fraction tells you what part of the pizza you ate. The following pizza shows 3 of the 8 pieces (the ones you ate) shaded.
Three Kinds of Fractions
|Proper fraction:||The top number (numerator) is less than the bottom number (denominator):|
|The value of a proper fraction is less than 1.|
|Improper fraction:||The top number is greater than or equal to the bottom number:|
|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 denominator (2) into the numerator (13) to get the whole number portion (6) of the mixed number: 2. Write the remainder of the division (1) over the old denominator (2): 3. Check: Change the mixed number back into an improper fraction (see steps that follow).
Changing Mixed Numbers into Improper Fractions
It's easier to 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:
1. Multiply the whole number (2) by the denominator (4): 2 × 4 = 8 2. Add the result (8) to the numerator (3): 8 + 3 = 11 3. Put the total (11) over the denominator (4):
4. Check: Reverse the process by changing the improper fraction into a mixed number. If you get the number you started with, your answer is right.
Reducing a fraction means writing it in lowest terms, that is, with the smallest numbers possible. For instance, 50¢ is of a dollar, or of 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 the numerator and the denominator.
- Divide that number into the numerator, and replace the numerator with the quotient (the answer you got when you divided).
- Repeat the same division step for the denominator.
- Repeat steps 1–3 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 numerator and denominator both end in zeros, cross out the same number of zeros in both numbers to begin the reducing process. For example reduces to when you cross out two zeros 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.
Raising Fractions to Higher Terms
Sometimes 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:
1. Divide the old denominator (3) into the new one (24): 2. Multiply the answer (8) by the old numerator (2): 2 ×8 = 16 3. Put the answer (16) over the new denominator (24): 4. Check: Reduce the new fraction to see if you get back the original one:
If the fractions have the same denominators, just add the numerators together and write the total over the denominator.
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 denominators, say :
1. Add the fractions: 2. Change the improper fraction into a mixed number: 3. Add the whole numbers: 2 + 1 = 3 4. 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 denominator, you will have to raise some or all of the fractions to higher terms so that they do; this number is then called the common denominator. All of the original denominators 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 into which all the denominators evenly divide:
- First, see if all the denominators divide evenly into the biggest one.
- Inspect multiples of the largest denominator until you find a number into which all the other ones evenly divide.
- When all else fails, multiply all the denominators together.
|1. Find the LCD. Multiply the denominators:||3 × 5 = 15|
|2. Raise each fraction to 15ths:|
|3. Add as usual:|
If the fractions have the same denominators, just subtract the numerators and write the difference over the denominator.
If the fractions you want to subtract don't have the same denominator, you will have to raise some or all of the fractions to higher terms so that they all have the same denominator, or LCD. If you forgot how to find the LCD, just reread the section on adding fractions with different denominators.
1. Raise each fraction to 12ths because 12 is the LCD, the smallest number that 6 and 4 both divide into evenly: 2. Subtract as usual:
Subtracting mixed numbers with the same denominator is similar to adding mixed numbers.
1. Subtract the fractions: 2. Subtract the whole numbers: 4 – 1 = 3 3. Add the results of steps 1 and 2:
Sometimes, there is an extra "borrowing" step when you subtract mixed numbers with the same denominators, say :
1. You can't subtract the fractions the way they are because is bigger than . So you borrow 1 from the 7, making it 6, and change that 1 to because 5 is the denominator: 2. Add the numbers from step 1: 3. Now you have a different version of the original problem: 4. Subtract the fractional parts of the two mixed numbers: 5. Subtract the whole number parts of the two mixed numbers: 6 – 2 = 4 6. Add the results of the last two steps together:
Multiplying fractions is actually easier than adding them. All you do is multiply the numerators and then multiply the denominators. You do not need to find a common denominator.
Sometimes, you can cancel before multiplying. Canceling is a shortcut that makes the multiplication go faster because you're multiplying with smaller numbers. It's very similar to reducing: If there is a number that divides evenly into the numerator and denominator, do that division before multiplying. If you forget to cancel, you will still get the right answer, but you will have to reduce it.
|1. Cancel the 6 and the 9 by dividing 3 into both of them: 6 ÷ 3 = 2 and 9 ÷ 3 = 3. Cross out the 6 and the 9 and replace with the reduced numbers:|
|2. Cancel the 5 and the 20 by dividing 5 into both of them: 5 ÷ 5 = 1 and 20 ÷ 5 = 4. Cross out the 5 and the 20 and replace with the reduced numbers:|
|3. Multiply across the new numerators and the new denominators:|
To multiply a fraction by a whole number, first rewrite the whole number as a fraction with a denominator of 1.
Example: (Optional: Convert to a mixed number:
To multiply with mixed numbers, it's easier to change them to improper fractions before multiplying.
1. Convert to an improper fraction: 2. Convert to an improper fraction: 3. Cancel and multiply the fractions: 4. Optional: Convert the improper fraction to a mixed number:
To divide one fraction by a second fraction, invert the second fraction (that is, flip the top and bottom numbers; this is called the reciprocal) and then multiply. That's all there is to it!
|1. Invert the second fraction :|
|2. Change the division sign (÷) to a multiplication sign (×):|
|3. Multiply the first fraction by the new second fraction:|
To divide a fraction by a whole number, first change the whole number to a fraction by putting it over 1. Then follow the division steps above.
When the division problem has a mixed number, convert it to an improper fraction and then divide as usual.
|1. Convert to an improper fraction:|
|2. Rewrite the division problem:|
|3. Change ÷ to ×, cancel, and multiply:|
The practice quiz for this study guide can be found at:
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