To solve the expression [tex]\(\frac{6}{21} - \frac{5}{7}\)[/tex], follow these steps:
1. Identify a Common Denominator:
First, find a common denominator for the fractions. The denominators are 21 and 7. The least common multiple (LCM) of 21 and 7 is 21. So, our common denominator will be 21.
2. Rewrite the Fractions with the Common Denominator:
Convert each fraction to an equivalent fraction with the denominator 21.
[tex]\(\frac{6}{21}\)[/tex] already has the denominator 21, so it remains:
[tex]\[\frac{6}{21}\][/tex]
For the second fraction, [tex]\(\frac{5}{7}\)[/tex]:
- We need to convert it to a fraction with the denominator 21. Multiply both the numerator and denominator by 3 (since [tex]\(21 \div 7 = 3\)[/tex]):
[tex]\[
\frac{5 \times 3}{7 \times 3} = \frac{15}{21}
\][/tex]
3. Subtract the Fractions:
Now, subtract the second fraction from the first, ensuring both fractions have the same denominator:
[tex]\[
\frac{6}{21} - \frac{15}{21}
\][/tex]
Subtract the numerators by keeping the common denominator:
[tex]\[
\frac{6 - 15}{21} = \frac{-9}{21}
\][/tex]
4. Simplify the Resulting Fraction:
Simplify [tex]\(\frac{-9}{21}\)[/tex] by finding the greatest common divisor (GCD) of 9 and 21, which is 3. Divide both the numerator and the denominator by 3:
[tex]\[
\frac{-9 \div 3}{21 \div 3} = \frac{-3}{7}
\][/tex]
So after performing the operations step-by-step, the result is:
[tex]\[
\frac{6}{21} - \frac{5}{7} = -\frac{3}{7}
\][/tex]
To express this as a decimal:
[tex]\[
-\frac{3}{7} \approx -0.42857142857142855
\][/tex]
Thus, the simplified fraction is [tex]\(-\frac{3}{7}\)[/tex], which is approximately [tex]\(-0.42857142857142855\)[/tex] when written as a decimal.
