For the nonzero numbers a, b, and c, define
J(a,b,c) = a/b + b/c + c/a.
Find J(2,12, 9).
To find J(a,b,c), we substitute the given values of a, b, and c in the equation:
J(a,b,c) = a/b + b/c + c/a
Using a = 2, b = 12, and c = 9:
J(2,12,9) = 2/12 + 12/9 + 9/2
We can simplify the fractions before adding them:
2/12 can be simplified to 1/6,
12/9 can be simplified to 4/3, and
9/2 cannot be further simplified.
Now we substitute the simplified fractions back into the equation:
J(2,12,9) = 1/6 + 4/3 + 9/2
To add these fractions, we need to find a common denominator. The least common multiple of 6, 3, and 2 is 6. So we convert each fraction:
1/6 = 1/6,
4/3 = 8/6, and
9/2 = 27/6.
Now the fractions have a common denominator, and we can add them:
J(2,12,9) = 1/6 + 8/6 + 27/6
Adding the numerators:
J(2,12,9) = (1 + 8 + 27)/6
Simplifying the numerator:
J(2,12,9) = 36/6
Dividing the numerator by the denominator:
J(2,12,9) = 6
Therefore, J(2,12,9) equals 6.
To calculate J(a, b, c) = a/b + b/c + c/a, we need to substitute the given values of a, b, and c into the expression.
Given:
a = 2
b = 12
c = 9
Substituting these values into the expression:
J(2, 12, 9) = 2/12 + 12/9 + 9/2
To add fractions, we need to find a common denominator. The least common multiple (LCM) of 12, 9, and 2 is 36.
Converting the fractions:
2/12 = (2 * 3) / (12 * 3) = 6/36
12/9 = (12 * 4) / (9 * 4) = 48/36
9/2 = (9 * 18) / (2 * 18) = 162/36
Now, we can calculate:
J(2, 12, 9) = 6/36 + 48/36 + 162/36
Adding the fractions:
J(2, 12, 9) = (6 + 48 + 162) / 36
Calculating the numerator:
6 + 48 + 162 = 216
Substituting the numerator into the expression:
J(2, 12, 9) = 216 / 36
Now, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 12.
Dividing the numerator and denominator by 12:
216 ÷ 12 = 18
36 ÷ 12 = 3
Simplified result:
J(2, 12, 9) = 18 / 3
Finally, we can simplify the fraction further by dividing both the numerator and denominator by 3:
18 ÷ 3 = 6
3 ÷ 3 = 1
Final answer:
J(2, 12, 9) = 6 / 1
J(2, 12, 9) = 6