Fractions

Fractions are ratios of whole numbers which allow us to express numbers which are between the whole numbers. For example,

Fractions represent "part of a whole". Imagine that we have a freight car with eight equal sized compartments. If three of these compartments are full of grain, we would indicate that we have three eighths of a freight car’s worth of grain. This is illustrated below.

Consider a car with eight compartments of which two are full. The fraction of a full car is two eighths. If we look at the same car split into four equal compartments, this same amount of grain fills one fourth of the car. We arrive at the following result.

We say that such equal fractions while they "look different" are equivalent. To generate equivalent fractions, we can multiply or divide both numerator (the top number) and denominator (the bottom number) by a common number. So we have the following fractions equivalent to two thirds.

Similarly, This same result could be stated in terms of "canceling" the common factor of 6 between the numerator and denominator: If a fraction has no common factors between its numerator and denominator, the fraction is in lowest terms.

There are three types of fractions:

1. Proper fractions have the numerator less than (symbolized by < ) the denominator. All proper fractions are less than 1.

2. Improper fractions have the numerator greater than (symbolized by > ) the denominator. All improper fractions are greater than 1. Improper fractions can be expressed as a mixed number, which is a whole number plus a proper fraction. For example,
   

   

Unit fractions have the numerator equal to the denominator. All unit fractions are equal to one. For example,

Note: when we write , we are using a shorthand notation. There really is a + sign between the 4 and the one sixth that is understood, but unstated. Working backwards we can convert a mixed number into an improper fraction. For example,

Fractions can be entered on both the Casio fx-300W and the TI-30Xa using the key, which is found in the second row, second column of the Casio fx-300W, and the seventh row, first column of the TI-30Xa. For example, to enter the fraction , use the following keystrokes:

14 24

7 û 12 will then appear in the display as the fraction reduced to lowest terms. For a mixed number such as , enter the following keystrokes:

11 5 16

The Casio fx-300W displays 11û5û16, while the TI-30Xa displays 11_5û16. To change this answer to the improper fraction , enter on the Casio fx-300W, or on the TI-30Xa. If a fraction has already been entered and appears on the display of the Casio fx-300W, entering converts it to a decimal and if is pressed a second time the decimal is converted back to a fraction. Fraction to decimal conversions are performed on the TI-30Xa by entering . The key is the back space key, which deletes characters in the display. It is found in the eighth row, first column of the TI-30Xa.

As an application, solve for the following missing numerator: .
As a first step reduce to lowest terms as . So, . The first denominator 12 is three times the second denominator 4, so the missing numerator must be three times the second numerator 3. The answer is that the missing numerator is 9.

To compare two fractions and determine which is larger, we can use the following procedure:

1. If the fractions involve mixed numbers with proper fractions, the number with the larger whole number is the larger number. For example,
   

   

2. If the fractions are both proper fractions or mixed numbers with equal whole numbers, then convert the fractions into decimals. The number with the larger decimal is the larger number. For example,
   

   

To add or subtract fractions we need a common denominator. Consider adding to . Since one fourth is equivalent to two eighths, we have the following solution:

If mixed numbers are involved, we first deal with the whole numbers, then the fractions. For example,

To multiply fractions we form the product of the numerators over the product of the denominators. For example,

If the product involves mixed numbers, we first convert them to improper fractions. For example,

Consider the division problem 8 ¸ 2 = 4. This is the same as

.

This means that division by the number b is equivalent to multiplication by the fraction . The fraction is called the reciprocal of . To form the reciprocal of a number we exchange the numerator with the denominator. In summary, division by a non-zero number equals multiplication by the reciprocal of that number.

In a division problem 0 is never allowed as the denominator or divisor. The reason for this is as follows. Suppose 20 ¸ 0 = made sense. Then there would be some number a which is the answer to this division problem. Restating this as a multiplication problem would give a × 0 = 20. But any number times zero gives zero! So no sensible answer to 20 ¸ 0 = exists.

Another way of explaining this goes to the very meaning of division.

Consider now the division . From the diagram below it is clear that one fourth contains 2 one eighths. So the answer must be 2.

The following shows that this result is consistent with the multiplication by the reciprocal definition of division.

If the division involves mixed numbers, we first convert them into improper fractions. For example,

In expressions which combine operations the standard order of operations apply as shown in the following:

These calculations are all easily performed on either the Casio fx-300W or the TI-30Xa. The keystokes for the previous calculation are as follows:

2 3 2 1 2 1 1 4 3

More involved calculations with grouping symbols are also possible. For example,

This is keystroked as follows:

3 3 4 5 3 16 3 7 8 2 1 2

Exercises:

Write as an improper fraction.

1. = __________

2. = __________

Write as a mixed number reduced to lowest terms.

3. = __________

4. = __________

Reduce to lowest terms.

5. = __________

6. = __________

Supply the missing numerators.

7. = __________

8. = __________

Indicate which number is larger.

9.            Larger = __________

10. 3      2      Larger = __________

Perform the indicated operations and express the answer as a fraction in lowest terms.

11. = __________

12. = __________

13. = __________

14. = __________

15. = __________

16. = __________

17. = __________

18. = __________

19. = __________

20. = __________

21. = __________

22. = __________

Solve and state all results as fractions reduced to lowest terms.

23. How many pieces of inch thick plywood are in a stack 35 inches high?
    Number = __________

24. A lumberyard sells lumber only in even foot lengths. What is the shortest single board of lumber from which a carpenter could cut three feet long and two feet long pieces?
    Length = __________

25. A cubic foot contains about gallons. How many cubic feet are there in 120 gallons?
    Number = __________

26. A nail inches long goes through a board long supporting a joist. How far into the joist does the nail extend?
    Distance = __________

27. A part is measured as inches on a scale drawing. If the scale is one foot to , how long is the actual part?
    Length = __________

Answers:

1. 
2. 
3. 
4. 
5. 
6. 
7. ; missing numerator is 9
8. ; missing numerator is 30
9. >
10. >
11. 
12. 9
13. 1
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 112 pieces
24. 16 feet
25. 16 cubic feet
26. 
27. 

	© 2002 by Al Lehnen; HTML-ized by Kevin Mirus (kmirus@madison.tec.wi.us or kjmirus@execpc.com).
	This document was last modified Wednesday, August 21, 2002, 4:10 PM.