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