Round each number to something easy, compute, and track whether you rounded up or down so you know which way the true answer leans.

• 312 + 488 → 300 + 500 = ~800 (actual 800).
• 47 × 19 → 47 × 20 = 940, but you rounded 19 up by 1 (≈ one 47 too many), so the true value is a bit under 940 (actual 893).

Rounding both numbers the same direction inflates or deflates the estimate predictably; rounding in opposite directions tends to cancel and gives a tighter estimate.

Because a number and its digit sum leave the same remainder when divided by 9 — the digital root is just the number mod 9.

Since 10 ≡ 1 (mod 9), every digit contributes only itself to the remainder, so any number ≡ its digit sum (mod 9). Arithmetic is preserved under "take the remainder mod 9": if a·b = c, then (a mod 9)·(b mod 9) ≡ c (mod 9). Casting out nines is exactly checking that congruence. (Digital roots run 1–9; a multiple of 9 gives 9, standing in for remainder 0.)