Problem:
Evaluate the expression: 3/7 - 6/11 - 8/21 + 5/22

Solution:

Step 1: Find a common denominator

To add or subtract fractions, we need to find a common denominator. In this case, the denominators are 7, 11, 21, and 22. The least common multiple (LCM) of these numbers is 2310, so we will use that as our common denominator.

Step 2: Convert each fraction to have the common denominator

To convert a fraction to have the common denominator, we multiply both the numerator and denominator by the same number. Let's convert each fraction:

3/7 = (3 * 330) / (7 * 330) = 990/2310
6/11 = (6 * 210) / (11 * 210) = 1260/2310
8/21 = (8 * 110) / (21 * 110) = 880/2310
5/22 = (5 * 105) / (22 * 105) = 525/2310

Step 3: Perform the subtraction

Now that all the fractions have the same denominator, we can subtract them:

990/2310 - 1260/2310 - 880/2310 + 525/2310

To subtract fractions with the same denominator, we subtract the numerators and keep the denominator:

(990 - 1260 - 880 + 525) / 2310

Simplifying the numerator:

(990 - 1260 - 880 + 525) = -625

Therefore, the expression simplifies to:

-625/2310

Step 4: Simplify the fraction (if possible)

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 625 and 2310 is 5, so we divide by 5:

(-625/5) / (2310/5) = -125/462

Therefore, the final answer is -125/462.

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