What is the doubling-and-halving trick, and when does it help?

Double one factor and halve the other — the product is unchanged — to turn an awkward factor into an easy one.

Since (2a)(b⁄2) = ab:

• 16 × 35 → 8 × 70 = 560.
• 15 × 12 → 30 × 6 = 180.

Check: 16 × 35 = 560 ✓, 15 × 12 = 180 ✓. Best when it produces a round factor (×5→×10, ×25→×100). You can repeat it: 56 × 25 → 28 × 50 → 14 × 100 = 1400 — but only while the halved factor stays a whole number.