To solve the expression [tex]\(\frac{5}{22} - \frac{5}{33}\)[/tex], we need to follow these steps:
1. Identify the Least Common Denominator (LCD):
The denominators of the fractions are 22 and 33. To find the LCD of these two numbers, we need to determine the smallest number that both 22 and 33 divide into evenly.
The factors of 22 are: [tex]\(2 \times 11\)[/tex]
The factors of 33 are: [tex]\(3 \times 11\)[/tex]
The LCD of 22 and 33 is the product of each distinct prime factor raised to its highest power found in the factors, which is [tex]\(2 \times 3 \times 11 = 66\)[/tex]. Therefore, the LCD is 66.
2. Convert the fractions to equivalent fractions with the LCD:
Now, we need to express [tex]\(\frac{5}{22}\)[/tex] and [tex]\(\frac{5}{33}\)[/tex] with 66 as the denominator.
- For [tex]\(\frac{5}{22}\)[/tex]:
[tex]\[
\frac{5}{22} = \frac{5 \times 3}{22 \times 3} = \frac{15}{66}
\][/tex]
- For [tex]\(\frac{5}{33}\)[/tex]:
[tex]\[
\frac{5}{33} = \frac{5 \times 2}{33 \times 2} = \frac{10}{66}
\][/tex]
3. Subtract the fractions:
Now that the fractions have a common denominator, we can subtract them.
[tex]\[
\frac{15}{66} - \frac{10}{66} = \frac{15 - 10}{66} = \frac{5}{66}
\][/tex]
4. Simplify the fraction if necessary:
The fraction [tex]\(\frac{5}{66}\)[/tex] is already in its lowest terms since 5 and 66 have no common factors other than 1.
Thus, the answer is [tex]\(\frac{5}{66}\)[/tex].
