TABE Level D Math: Mixed Numbers, Fractions, and Whole Numbers

Fractions
A. proper fraction is part of a whole.

For example, the fraction is 5 out of 8. In baseball, this could mean 5 hits out of 8 times at bat. Or it could mean 5 slices of a pizza that is cut into 8 slices. The top number is called the numerator, and the bottom number is called the denominator. The numerator of a proper fraction is smaller than the denominator.

An improper fraction has a numerator that’s larger than its denominator. Improper fractions are bigger than 1.

Usually an improper fraction is changed to a mixed number.

A mixed number such as is a number that falls between two whole numbers.

It has a whole number part (2 in this example) and a fractional part ( in this example).

For the TABE D, you need to know how to add, subtract, multiply, divide, and reduce fractions. You also need to be able to work with mixed numbers. 

Reducing Fractions:
You can reduce a fraction by finding a number that divides evenly into both the numerator and denominator.

For example, you reduce the fraction to by dividing the numerator and denominator by 2.

You can divide by 2 again to reduce further to and again by 2 to reduce to

This is as far as you can go.

The original has now been reduced to lowest terms

If you had divided numerator and denominator by 8 to begin with, you would have reduced to lowest terms in one step instead of three. Although you can use this method to reduce both proper and improper fractions, only proper fractions need to be reduced on the TABE D. A. reduced fraction is in the same proportion as the original fraction. If you get 16 problems right on a 24-problem test, you would expect to get 8 right on a 12-problem test, 4 right on a 6-problem test, or 2 right on a 3-problem test. 

Practice Reducing Fractions:
Reduce each fraction to lowest terms.
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Adding and Subtracting Fractions with the Same Denominators
You can only add or subtract fractions that have the same denominators. The denominator of your answer is the same as the common denominator of the two fractions you add. Then you simply add the numerators to get the numerator of your answer. The same idea applies to subtracting fractions, except that you subtract the numerators instead of adding them.

Examples:

You can write a problem that adds or subtracts fractions in horizontal form (as done here) or in vertical form, such as  
If you can reduce a fraction after adding, be sure to do so. For example and reduces to

Practice Adding and Subtracting Fractions with the Same Denominators
Add or subtract as indicated, and reduce the answer to lowest terms if possible.
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Adding and Subtracting Fractions with Different Denominators
Fractions with different denominators can’t be added or subtracted directly. You first have to find a common denominator. This is a denominator that the two different denominators both divide into evenly.

For example, two fractions that have denominators of 2 and 3 have 6 as a common denominator because 2 and 3 both divide evenly into 6. Once you have a common denominator, you have to get new numerators.
For example, suppose you have to subtract
The denominators 3 and 2 both go evenly into 6, so 6 is a common denominator. Write
The first fraction is 2/3. Divide the denominator 3 into 6 and multiply by the answer (2) by the numerator 2.

This answer is 4, and it goes in the numerator of the first fraction:
Do the same for the other fraction. Divide the 2 into 6 and multiply by the 1. The answer is 3. Put this in the numerator of the second fraction:
Now you can subtract the fractions:

Practice Adding and Subtracting Fractions with Different Denominators
Add or subtract as indicated, and reduce if possible.
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Changing Improper Fractions to Mixed Numbers
So far, we’ve talked about adding and subtracting proper fractions with the same and different denominators. In all the sample problems, answers were also proper fractions.

Two proper fractions might add to an improper fraction, such as , which add to .

Usually, you change the improper fraction to a mixed number.
To change to a mixed number, divide 5 by 4, and get 1 with a remainder of 1.

The remainder can be written as the numerator of a fraction. The denominator of this fraction is the divisor.

This gives you the mixed number .

Before moving on, you should practice changing improper fractions to mixed numbers.

Practice Changing Improper Fractions to Mixed Numbers
Change each improper fraction to a mixed number.
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Adding and Subtracting Mixed Numbers, Fractions, and Whole Numbers
Before looking at these problems, you need to know how to change mixed numbers to improper fractions.

Multiply the denominator by the whole number and add the numerator. Then put this answer over the denominator.

Example: . Multiply 3 by 2 and add 1, to get 7, so
 

To add or subtract mixed numbers, follow these steps:
1. Change each mixed number to an improper fraction.
2. Add or subtract the improper fractions just as you would proper fractions.
3. Change the answer back to a mixed number.

Example A.

1. Change to improper fractions.
2. Get a common denominator and add.
3. Change to a mixed number.

Example B.

1. Change to improper fractions.
2. Get a common denominator and subtract.
3. Change to a mixed number.

To add a fraction and a whole number, just put the fraction next to the whole number to make a mixed number.

Example C.

To add a whole number and a mixed number, just add the whole number parts and leave the fractional part with the answer.

Example D.

To subtract a fraction or a mixed number from a whole number, follow these steps:
1. Write the whole number as an improper fraction with a denominator of 1.
2. Write the mixed number as an improper fraction.
3. Subtract as you would with two fractions.

Example E.

1. Change 7 to
2. Change to
3. Do the subtraction

Practice Adding and Subtracting Mixed Numbers, Fractions, and Whole Numbers
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Multiplying Fractions
Multiplying fractions is easier than adding or subtracting them. When you multiply two proper fractions, the answer is a proper fraction. Multiply the two numerators to get the numerator of the answer. Do the same thing with the denominators.

Example F.

and

The answer is
Sometimes it is possible to reduce the answer.

Example G.

and The answer is but this can be reduced to by dividing the numerator and denominator by 6.

Multiplying a Whole Number and a Fraction
To multiply a whole number and a fraction, multiply the numerator by the whole number, and leave the denominator alone.

Example H.

You might get an improper fraction as an answer. In a multiple-choice question, the correct answer choice might be the improper fraction or the equivalent mixed number.

Example I.

Dividing Fractions
To divide fractions, you turn the second fraction (the divisor) upside down and multiply.
Example J. ( is changed to ).

This is the same as the mixed number Either one could be the correct answer choice.

Practice Multiplying and Dividing Fractions
Multiply or divide as indicated. Reduce answers and change improper fractions to mixed numbers.
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Mixed Practice on Fractions
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Answers: Practice Reducing Fractions
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Answers: Practice Adding and Subtracting Fractions with the Same Denominators
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Answers: Practice Adding and Subtracting Fractions with Different Denominators
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Answers: Practice Changing Improper Fractions to Mixed Numbers
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Answers: Practice Adding and Subtracting Mixed Numbers, Fractions, and Whole Numbers
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Answers: Practice Multiplying and Dividing Fractions
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Answers: Mixed Practice on Fractions
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