How do you calculate 305 * 70 quickly using mental math or shortcuts?

1. Treat zeros as place‑holders and multiply the compact parts

A common, fast mental rule is to ignore trailing zeros, multiply the remaining digits, then re‑attach the zeros. For 305×70 you can see 70 as 7×10, so do 305×7 = 2,135, then multiply that by 10 to get 21,350. Teaching resources and mental‑math primers recommend breaking numbers into place value parts exactly this way to make multi‑digit multiplication manageable in your head ^{[1] [2]}.

2. Use distributive splitting when a number isn't round

If you prefer distributive decomposition: 305×70 = (300+5)×70 = 300×70 + 5×70 = 21,000 + 350 = 21,350. Texts on mental math and multiplication tricks repeatedly suggest separating numbers into easy chunks (e.g., hundreds + tens + units) and multiplying each chunk before summing the results ^{[2] [3]}.

3. Why the “half and double” or FOIL ideas aren’t needed here — but are useful elsewhere

Some mental‑math guides teach half‑and‑double or FOIL-style breakdowns for specific patterns (e.g., multiply 26×25 by halving one factor and doubling the other, or expanding (a+b)(c+d)) but for 305×70 the place‑value shortcut is simpler and faster: convert 70 to 7×10 or split 305 into 300+5. FOIL/distributive expansions are helpful for other two‑ and three‑digit multiplications, as illustrated in several educational resources ^{[4] [5] [3]}.

4. Step‑by‑step mental path you can speak or hold in your head

A compact verbal script: “305 times 7 is 2,135; add a zero for times 70 → 21,350.” Or mentally: 300×70=21,000; 5×70=350; 21,000+350=21,350. Instructional sites teach exactly these small‑step decompositions to reduce working memory load and avoid errors ^{[2] [1]}.

5. Common beginner errors and how the shortcut prevents them

Beginners often misplace carries or lose track of zeros in multi‑digit multiplication. Breaking the task into “easy products” (300×70, 5×70) or removing zeros first (305×7 then ×10) reduces those risks because each intermediate product is a round number or a familiar small multiplication, a method recommended across mental‑math resources ^{[2] [3]}.

6. Alternatives and practice strategies to get faster

If you want speed beyond a single trick, practice multiplying by 5s, 7s and 10s until they are automatic, and practise splitting numbers into round chunks; many mental‑math curricula and trick lists emphasize repeated practice of these basic patterns to make shortcuts instinctive ^{[2] [6]}. Competitors in mental‑calculation circles use more elaborate cross‑multiplication for very large numbers, but for everyday three‑digit by two‑digit tasks the place‑value split is the recommended, practical approach ^{[7] [3]}.

Limitations and sourcing note: available sources cover the place‑value and distributive strategies and give similar worked examples (split into hundreds/tens/units or remove zeros first) but do not provide this specific numeric narration verbatim; the numerical result and the techniques above are consistent with the cited mental‑math instructional material ^{[1] [2] [3]}.