Find the Error:

A student is adding [tex]1 \frac{2}{9}[/tex], [tex]3 \frac{1}{3}[/tex], and [tex]-4 \frac{5}{6}[/tex]. The first step the student performs is to find the common denominator of 9, 3, and 6. Find the student's mistake and correct it.

The least common denominator of 9, 3, and 6 is 36 because you can divide 36 by all of these numbers without getting a remainder.

Correct the mistake.

The student's mistake is finding the least common denominator incorrectly. The actual LCD is 18, not 36. By converting each fraction to have the same denominator and then performing the addition, the correct answer is 4/9.

Explanation:

The mistake made by the student is that they found the least common denominator (LCD) incorrectly. The LCD of 9, 3, and 6 is actually 18, not 36. To find the LCD, we need to find the smallest number that is divisible by all the denominators. We can do this by finding the prime factors of each denominator and multiplying the highest power of each factor.

For 9: 3 x 3 = 32

For 3: 3 x 1 = 31

For 6: 2 x 3 = 21 x 31

Thus, the LCD is 21 x 32 = 18.

After finding the LCD, the student can convert each fraction to have the same denominator and then perform the addition. In this case, the fractions become:

12/9 = 1 x (18/9) = 2/18
31/3 = 3 x (18/3) = 18/18
-45/6 = -4 x (18/6) = -12/18

Now, we can add the fractions: 2/18 + 18/18 + (-12/18) = 8/18 = 4/9

Therefore, the correct sum of 1 2/9, 3 1/3, and -4 5/6 is 4/9.