INTRODUCTION

1. What is a fraction?

A fraction consists of a numerator (part) on top of a denominator (total) separated by a horizontal line.

For example, the fraction of the circle which is shaded is:

\(\frac{2 (parts shaded)}{4 (total parts)}\)




In the square, the fraction shaded is \(\frac{3}{8}\) and the fraction unshaded is \(\frac{5}{8}\)




2. Equivalent Fractions – Multiplying

The three circles on the right each have equal parts shaded, yet are represented by different but equal fractions. These fractions, because they are equal, are called equivalent fractions. Any fraction can be changed into an equivalent fraction by multiplying both the numerator and denominator by the same number.





3. Equivalent Fractions – Dividing (Reducing)

Equivalent fractions can also be created if both the numerator and denominator can be divided by the same number (a factor) evenly. This process is called “reducing a fraction” by dividing a common factor (a number which divides into both the numerator and denominator evenly).

\(\frac{4}{8} \div \frac{4}{4} = \frac{1}{2}\)

\(\frac{27}{81} \div \frac{9}{9} = \frac{3}{9}\)


4. Simplifying a Fraction (Reducing to its Lowest Terms)

It is usual to reduce a fraction until it can’t be reduced any further. A simplified fraction has no common factors which will divide into both numerator and denominator. Notice that, since 27 and 81 have a common factor of 9, we find that \(\frac{3}{9}\) is an equivalent fraction. But this fraction has a factor of 3 common to both numerator and denominator. So, we must reduce this fraction again. It is difficult to see, but if we had known that 27 was a factor (divides into both parts of the fraction evenly), we could have arrived at the answer in one step

\(\frac{8}{24} \div \frac{8}{8} = \frac{1}{3}\)



TYPES OF FRACTIONS

1. Common Fractions

A common fraction is one in which the numerator is less than the denominator (or a fraction which is less than the number 1). A common fraction can also be called a proper fraction.

\(\frac{88}{93},\frac{8}{15}\),..............are al common fractions


2. Fractions that are Whole Numbers

Some fractions, when reduced, are really whole numbers (1, 2, 3, 4… etc). Whole numbers occur if the denominator divides into the numerator evenly

\(\frac{8}{4}\) is the same as \(\frac{8}{4} \div \frac{4}{4}=\frac{2}{1}\) or \(2\)


3. Mixed Numbers

A mixed number is a combination of a whole number and a common fraction.

e.g. \(2 \frac{3}{5}\) (two and three-fifths)



4. Improper Fractions

An improper fraction is one in which the numerator is larger than the denominator. From the circles on the right, we see that \(1 \frac{3}{4}\) (mixed number) is the same as \(\frac{7}{4}\) (improper fraction). An improper fraction, like \(\frac{7}{4 }\) can be changed to a mixed number by dividing the denominator into the numerator and expressing the remainder (3) as the numerator.

e.g. \(\frac{16}{5}=3 \frac{1}{5}\)

A mixed number can be changed to an improper fraction by changing the whole number to a fraction with the same denominator as the common fraction.

\(2 \frac{3}{5} = \frac{10}{5} and \frac{3}{5}=\frac{13}{5}\)

A simple way to do this is to multiply the whole number by the denominator, and then add the numerator.

e.g. 4 \(\frac{5}{9} = \frac{4 \times 9 + 5 }{9} = \frac{36+5}{9 }= \frac{41}{9}\)


5. Simplifying fractions

All types of fractions must always be simplified (reduced to lowest terms).

e.g. \(\frac{6}{9}=\frac{2}{3}\)

Note that many fractions can not be reduced since they have no common factors.


COMPARING FRACTIONS

Students can use several strategies to compare fractions. These strategies are fairly elementary. We will outline below the most common ways to confront fractions.


1. Using Visual Models

Area models can be used to compare fractions, drawing pictures: they could be circles, stripes, boxes, and so on.



In this case , regardless of the value of the two fractions, \(\frac{3}{4} >\frac{5}{8}\) because the colored part is more significant in the first strip than the second one.


2. Using the number line

(a) - More and less than a benchmark such as one-half or one whole.

Comparing fractions to a benchmark such as one-half or one whole is another strategy for comparing two fractions. For example, when comparing 3/8 and 5/6, we know that 3/8 is less than 1/2 and that 5/6 is more than 1/2. Thus, i is the greater fraction. Geometrically, we can illustrate this strategy by placing both fractions on the number line as shown below.




While any two fractions that represent distinct numbers can be compared to a benchmark, this is a strategy that works particularly well when one fraction is more than a common benchmark such as 1/2 or 1 and the other fraction is less than this benchmark. Given two fractions greater than 1/2 but less than 1, finding an appropriate benchmark can be problematic. When comparing 5/6 and 13/16, it is difficult to determine a benchmark fraction.


(b) - Distance from a benchmark such as one-ha/for one-whole.

The second strategy, comparing two fractions to a benchmark, is limited when both fractions are either greater than or less than the chosen benchmark. A strategy in these cases is to compare the distances of the fractions from the benchmark. For example, 5/6 and 7/8 are both less than the benchmark 1 and so we can compare these two fractions by considering their distance from 1.

5/6 is 1/6 less than 1 and 7/8 is 1/8 less than 1. We can then compare 1/6 and 1/8 using our strategy for comparing two fractions that have the same number of parts, but parts of different sizes. Since 1/6 is greater than 1/8, then 1/6 is a greater distance from 1 than 1/8. Therefore, 7/8 is greater than 5/6 (or 5/6 is less than 7/8). Geometrically, we can illustrate this strategy using the number line as shown below.







3. Common Numerator or Denominator

Common Numerator or Denominator
Common Numerator	Common Denominator	Improper fraction > Proper fraction
\(\frac{3}{5} > \frac{2}{5}\) because \(4<5\)	\(\frac{3}{5} > \frac{2}{5}\) because \(3 > 2\)	\(\frac{3}{2 }> \frac{24}{25}\)
\(\frac{1}{3} < \frac{1}{2}\) because \(3>2\)	\(\frac{9}{14} < \frac{11}{14}\) because \(9 > 11\)	\(\frac{9}{10 }> \frac{10}{9}\)

ADDING/SUBTRACTING/MULTIPLYING/DIVIDING

Adding and Subtracting Fractions:

1. Common (like) denominators are necessary, so change all unlike fractions to equivalent fractions with like denominators. To make equivalent fractions, multiply the numerator and denominator by the same number.
2. Keep mixed numbers; DO NOT change mixed numbers into improper fractions.
3. Add (or subtract) the numerators, put the numerator answer over the common denominator. If any improper fractions arise in the answer, change the improper portion to a mixed number. (In Math 105, answers are often left in improper form)

Multiplying and Dividing Fractions:

1. Common denominators are NOT needed.
2. Always change mixed numbers to improper fractions.
3. CANCEL (reduce) between any numerator and any denominator if you can, but cancel only when a multiplication sign is present: Never cancel when you have a division sign.
4. TO MULTIPLY: Multiply numerator times numerator, denominator times denominator. Reduce answer to a mixed number in lowest terms. (In Math 105, answers are often left in improper form)
5. TO DIVIDE: Change the divide sign to a multiplication sign, then invert the second fraction and multiply as in Step 4.