Find $\frac{3}{7}+\left(\frac{-6}{11}\right)+\left(\frac{-8}{21}\right)+\left(\frac{5}{22}\right)$ .

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Q. Find

\frac{3}{7}+\left(\frac{-6}{11}\right)+\left(\frac{-8}{21}\right)+\left(\frac{5}{22}\right)

Identify LCD: Identify the least common denominator (LCD) for the fractions. $\newline$ The denominators are $7$ , $11$ , $21$ , and $22$ . The LCD for these numbers is the smallest number that each of the denominators can divide into without leaving a remainder. We can find the LCD by finding the least common multiple (LCM) of the denominators. $\newline$ LCM of $7$ and $11$ is $77$ , LCM of $77$ and $21$ (which is $3\times7$ ) is $11$ $0$ , and LCM of $11$ $0$ and $22$ is $11$ $3$ . $\newline$ So, the LCD is $11$ $3$ .
Convert to Equivalent Fraction: Convert each fraction to an equivalent fraction with the LCD as the denominator. $\newline$ For $(\frac{3}{7})$ , we multiply the numerator and denominator by $66$ to get $(\frac{3\times66}{7\times66}) = (\frac{198}{462})$ . $\newline$ For $(\frac{-6}{11})$ , we multiply the numerator and denominator by $42$ to get $(\frac{-6\times42}{11\times42}) = (\frac{-252}{462})$ . $\newline$ For $(\frac{-8}{21})$ , we multiply the numerator and denominator by $22$ to get $(\frac{-8\times22}{21\times22}) = (\frac{-176}{462})$ . $\newline$ For $(\frac{5}{22})$ , we multiply the numerator and denominator by $66$ $0$ to get $66$ $1$ .
Add Equivalent Fractions: Add the equivalent fractions together. $\newline$ Now we add the numerators of the equivalent fractions and keep the common denominator: $\newline$ $(\frac{198}{462}) + (\frac{-252}{462}) + (\frac{-176}{462}) + (\frac{105}{462})$ .
Perform Numerator Addition: Perform the addition of the numerators. $198 - 252 - 176 + 105 = -125$ . So, the sum of the fractions is $(-125/462)$ .
Simplify Fraction: Simplify the fraction if possible. $\newline$ We look for the greatest common divisor (GCD) of $125$ and $462$ to simplify the fraction. The GCD of $125$ and $462$ is $1$ , so the fraction is already in its simplest form.