To calculate \( 7 \frac{1}{2} - 1 \frac{5}{6} \), we first convert both mixed numbers into improper fractions.
-
Convert \( 7 \frac{1}{2} \): \[ 7 \frac{1}{2} = \frac{(7 \times 2) + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2} \]
-
Convert \( 1 \frac{5}{6} \): \[ 1 \frac{5}{6} = \frac{(1 \times 6) + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6} \]
Now we can subtract the two improper fractions: \[ \frac{15}{2} - \frac{11}{6} \]
To subtract these fractions, we need a common denominator. The least common multiple of 2 and 6 is 6.
- Convert \( \frac{15}{2} \) to a fraction with a denominator of 6: \[ \frac{15}{2} = \frac{15 \times 3}{2 \times 3} = \frac{45}{6} \]
Now we can perform the subtraction: \[ \frac{45}{6} - \frac{11}{6} = \frac{45 - 11}{6} = \frac{34}{6} \]
- Simplify \( \frac{34}{6} \): \[ \frac{34}{6} = \frac{17}{3} \]
Next, we convert \( \frac{17}{3} \) back into a mixed number.
- Divide 17 by 3:
- 3 goes into 17 five times (5 x 3 = 15)
- Remainder is 2.
So, \( \frac{17}{3} = 5 \frac{2}{3} \).
Thus, \( 7 \frac{1}{2} - 1 \frac{5}{6} = 5 \frac{2}{3} \).
The answer in the required format is: \[ 5 \frac{2}{3} \]