Order of Magnitude Product

How many digits is the answer? Don't compute the product — just count the digits.

Walk through one

Suppose you see 823 × 612. Both numbers have 3 digits. The product has either 3+3-1 = 5 digits or 3+3 = 6 digits. Which one?

Look at just the leading digits: 8 × 6 = 48. Because that's 10 or more, we carry up to the bigger answer:

3 + 3 = 6 digits   (true: 503,676 ✓)

The rule

For a d₁-digit number times a d₂-digit number:

More examples

• 423 × 178 → leading 4×1 = 4 <10 → 3+3−1 = 5 digits (true: 75,294)
• 4327 × 891 → leading 4×8 = 32 ≥10 → 4+3 = 7 digits (true: 3,855,357)
• 2100 × 3500 → leading 2×3 = 6 <10 → 4+4−1 = 7 digits (true: 7,350,000)

Why it works

A d-digit number lives in the range [10^d−1, 10^d). Multiplying a d₁-digit by a d₂-digit lands you in [10^{d₁+d₂−2}, 10^d₁+d₂), which is exactly two possible digit-counts. The leading-digit product tells you which side of the threshold you land on.

Watch out: the lead-digit rule is an approximation

The real test isn't lead-digit × lead-digit — it's the full mantissa product. Each number is really lead.rest × 10^d−1. If those mantissas multiply to ≥10, you carry up.

142 × 750 → lead product 1×7 = 7 says 5 digits. But mantissas 1.42 × 7.5 = 10.65 → carries up to 6 digits (true: 106,500). The shortcut lied.

Rule of thumb: when the lead-digit product is 6, 7, 8, or 9 and either number has a non-trivial second digit (3+), do a quick mantissa check before committing. Lead 1×7 with second digits 4 and 5? That's really 1.4 × 7.5 ≈ 10 — carry up.

Every question has a 30-second shot clock.