2. Algebra
GRE_Testing

# 1.2. Fractions

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fraction is a number of the form $\frac{c}{d}$, where and are integers and d ≠ 0. The integer is called the numerator of the fraction, and is called the denominator. For example, $\frac{-7}{5}$ is a fraction in which -7 is the numerator and 5 is the denominator. Such numbers are also called rational numbers. Note that every integer is a rational number, because is equal to the fraction $\frac{n}{1}$.

If both the numerator and the denominator d, where d ≠ 0, are multiplied by the same nonzero integer, the resulting fraction will be equivalent to $\frac{c}{d}$.

Example 1.2.1: Multiplying the numerator and denominator of the fraction $\frac{-7}{5}$ by 4 gives

Multiplying the numerator and denominator of the fraction $\frac{-7}{5}$ by -1 gives

For all integers and d, the fractions $\frac{-c}{d}$, $\frac{c}{(-d)}$  and $-(\frac{c}{d})$ are equivalent.

Example 1.2.2: $\frac{-7}{5}=\frac{7}{-5}=-\frac{7}{5}$

If both the numerator and denominator of a fraction have a common factor, then the numerator and denominator can be factored and the fraction can be reduced to an equivalent fraction.

Example 1.2.3: $\frac{40}{72}=\frac{(8)(5)}{(8)(9)}=\frac{5}{9}$

To add two fractions with the same denominator, you add the numerators and keep the same denominator.

Example 1.2.4:  $-\frac{8}{11}+\frac{5}{11}=\frac{-8+5}{11}=\frac{-3}{11}=-\frac{3}{11}$

To add two fractions with different denominators, first find a common denominator, which is a common multiple of the two denominators. Then convert both fractions to equivalent fractions with the same denominator. Finally, add the numerators and keep the common denominator.

Example 1.2.5: To add the two fractions $\frac{1}{3}$ and $-\frac{2}{5}$ first note that 15 is a common denominator of the fractions.

Then convert the fractions to equivalent fractions with denominator 15 as follows.

$\frac{1}{3}=\frac{1(5)}{3(5)}=\frac{5}{15}$ and $-\frac{2}{5}=-\frac{2(3)}{5(3)}=-\frac{6}{15}$

Therefore, the two fractions can be added as follows.

The same method applies to subtraction of fractions.

#### Multiplying and Dividing Fractions

To multiply two fractions, multiply the two numerators and multiply the two denominators. Here are two examples.

Example 1.2.6: $(\frac{10}{7})(\frac{-1}{3})=\frac{(10)(-1)}{(7)(3)}=\frac{-10}{21}=-\frac{10}{21}$

Example 1.2.7: $(\frac{8}{3})(\frac{7}{3})=\frac{56}{9}$

To divide one fraction by another, first invert the second fraction (that is, find its reciprocal), then multiply the first fraction by the inverted fraction. Here are two examples.

Example 1.2.8:  $\frac{17}{8}\div \frac{3}{5}=(\frac{17}{8})(\frac{5}{3})=\frac{85}{24}$

Example 1.2.9:

#### Mixed Numbers

An expression such as  is called a mixed number. It consists of an integer part and a fraction part, where the fraction part has a value between 0 and 1; the mixed number means

To convert a mixed number  to a fraction, convert the integer part to an equivalent fraction with the same denominator as the fraction, and then add it to the fraction part.

Example 1.2.10: To convert the mixed numberto a fraction, first convert the integer 4 to a fraction with denominator 8, as follows.

Then add $\frac{3}{8}$ to it to get

#### Fractional Expressions

Numbers of the form $\frac{c}{d}$, where either c or d is not an integer and d ≠ 0, are called fractional expressions. Fractional expressions can be manipulated just like fractions. Here are two examples.

Example 1.2.11: Add the numbers $\frac{\pi }{}2$ and  $\frac{\pi }{3}$

Solution: Note that 6 is a common denominator of both numbers.

The number $\frac{\pi }{}2$  is equivalent to the number $\frac{3\pi }{6}$ and the number $\frac{\pi }{3}$ is equivalent to the number $\frac{2\pi }{6}$

Therefore

Example 1.2.12:   Simplify the number

Solution: Note that the numerator of the number is $\frac{1}{\sqrt{2}}$ and the denominator of the number is $\frac{3}{\sqrt{5}}$. Note also that the reciprocal of the denominator is $\frac{\sqrt{5}}{3}$

Therefore,

which can be simplified to

Thus, the number simplifies to the number