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Whole Numbers and Fractions

Now you know about regular fractions with numerators on top and denominators on the bottom. Fractions can be positive or negative. Until now, we have just looked at fractions that are less than one. It didn't matter whether it was one half (1/2) or nine hundred ninety-nine one-thousandths (999/1000), the values were still less than one.

When you have a fraction that is greater than one, it can come in two formats. Mixed numbers have a whole number followed by the fraction (2 1/2). You would say "two and one half." The other format is an improper fraction where the numerator is greater than the denominator (5/2). Mathematicians would say that is five halves. You will find both types of fractions in your problems. Both of those examples represent the same value (2 1/2 = 5/2). Here's why...

• 2 1/2 is the same amount as two whole objects and half of a third.
• You could break those two whole objects into halves as well. If each object had two halves, then two objects would give you a total of four pieces (2x2=4).
• That means you have four halves from the whole objects and one-half of a third.
• You would have a total of five halves. You can write five halves as 5/2.

Different Ways of Writing Mixed Numbers

As we just told you, a mixed number is a whole number with the fraction written to its right side.
• 2 1/2, 5 8/13, 6 4/7

You will usually see the fraction written in its simplest form.
• 6 4/7 correct
• 6 8/14 incorrect (needs to be reduced)

You might have problems that ask you to simplify mixed numbers. Don't worry. You usually don't worry about the whole number. When you simplify, just worry about the fraction.

Example:
Simplify 8 4/8
• Just worry about simplifying the fraction 4/8.
• We'll divide by the common factor 4 and get 1/2.
• The simplified mixed number is 8 1/2.

Another format for a fraction that is greater than one is the improper fraction. They are called "improper", because fractions are always supposed to be written with a numerator (number on top) that is less than the denominator (number on the bottom). 2/3 is a proper fraction. 3/2 is an improper fraction because 3 > 2. They are useful in many problems when you add fractions and subtract fractions. You will also use improper fractions when you work with reciprocals.

Is 1/3 an improper fraction? No. 1 < 3.

Is the reciprocal of 8/11 an improper fraction? Yes. The reciprocal is 11/8. 11/8 is an improper fraction.

So, we can make an improper fraction by finding the reciprocal. You may also need to make an improper fraction from a mixed number. We just did it above, but let's do it one more time...

Example:
Write 2 1/2 as an improper fraction.
• The equivalent of two is 2/1 or 4/2. If you are unsure, check using division.
• Rewrite the mixed number as an addition problem. 2 1/2 = 2 + 1/2
• Substitute the new value of two. 2 + 1/2 = 4/2 + 1/2
• Using a little fraction addition... 4/2 + 1/2 = 5/2
• 5/2 is an improper fraction that is equivalent to 2 1/2.

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