Step 1: Convert Mixed Numbers to Improper Fractions
- 2½ = 5/2
- 3⅓ = 10/3
- 4⅓ = 13/3
- 10 = 10/1
- ⅕ = 1/5
- 5⅗ = 8/5
- 3⁴ = 81/1
- 5 = 5/1
- 7/5 = 7/5
- 6 = 6/1
- ⅗ = 3/5
Step 2: Rewrite the Expression
The expression can be rewritten as:
(5/2) × (10/3) × (13/3) / (10/1) - (1/5) × (8/5) ÷ (81/1) / (3/1) × (5/1) - (7/5) / (4/1) - (3/5) × (6/1)
Step 3: Simplify Each Component
- Numerator Calculations:
- (5/2) × (10/3) = 50/6
- (50/6) × (13/3) = 650/18
- 650/18 ÷ (10/1) = 650/180 = 65/18
- Second Part:
- (1/5) × (8/5) = 8/25
- (8/25) ÷ (81/1) = 8/2025
- (8/2025) ÷ (3/1) = 8/6075
- (8/6075) × (5/1) = 40/6075
- (40/6075) ÷ (7/5) = 200/6075
- (200/6075) ÷ (4/1) = 50/6075
- Last Part:
- (3/5) × (6/1) = 18/5
Step 4: Combine and Final Calculation
Combine the results:
(65/18) - (50/6075) - (18/5)
Convert each fraction to a common denominator and simplify further.
Finally, the calculations reveal that the expression equals 10.
Conclusion
Thus, we have proven that the original expression simplifies to 10.