Evaluate: 121⁄2% of 560 km + 662⁄3% of 240 km

Concept — A percentage written as a mixed fraction is first turned into a simple fraction, and "a% of N" means (a ÷ 100) × N. Two percentage terms that act on different base quantities cannot be combined by adding their rates: each term is evaluated on its own base, and the two resulting quantities are then added.

The two percentage-to-fraction equivalents needed here:

Application — Evaluate each term on its own base, then add the two quantities.

1. Convert the first rate: 121⁄2% = 25⁄2% = (25⁄2) ÷ 100 = 1⁄8.

2. First term: 1⁄8 × 560 km = 70 km.

3. Convert the second rate: 662⁄3% = 200⁄3% = (200⁄3) ÷ 100 = 2⁄3.

4. Second term: 2⁄3 × 240 km = 160 km.

5. Add the two quantities: 70 km + 160 km = 230 km.

Cross-check — In decimal form the same two terms give 0.125 × 560 km = 70 km and (2⁄3) × 240 km = 160 km, a total of 230 km again. Adding the two rates first is not valid here, because they act on different base quantities: 121⁄2% + 662⁄3% = 791⁄6%, and 791⁄6% of 560 km is 4431⁄3 km while 791⁄6% of 240 km is 190 km, neither of which is the value of the expression.

Result — The expression evaluates to 230 km.