What is the step-by-step multiplication of 305 by 70 by hand?
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Executive summary
A clear, reliable hand method solves 305 × 70 by reducing it to simpler steps: multiply 305 by 7, then append a zero because multiplying by 70 equals multiplying by 7 then by 10; the final product is 21,350 (this follows the standard long‑multiplication and place‑value rules) [1][2][3].
1. Stack the numbers and recognize place value before starting
Long multiplication begins by writing the multiplicand above the multiplier , aligned at the right so digits occupy the correct place values, a standard starting point in the traditional algorithm [1][4].
2. Multiply by the ones digit of the bottom number (which is 0) and note the trivial partial product
Following long‑multiplication rules, multiply the rightmost digit of the bottom number by each digit of the top number , writing that partial product on the first row of work; each individual product is zero so the first partial row is 000, and any carried values are zero [5][6].
3. Move to the tens digit and shift one place left before multiplying (the tens digit is 7)
The next digit in the multiplier is the 7 in the tens place, so per the algorithm the partial product formed by multiplying 7 by 305 must be written shifted one place to the left (equivalently, add one trailing zero to the row before final summation) to reflect that 7 represents 70, not 7 [5][2].
4. Multiply 305 by 7 digit‑by‑digit, carrying where needed
Work right to left: 7 × 5 = 35, write 5 and carry 3; 7 × 0 = 0 plus carried 3 gives 3, write 3 and carry 0; 7 × 3 = 21, write 21 at the leftmost positions — that yields the intermediate partial product 2,135 (before shifting) which is the direct result of multiplying 305 by 7 following the long‑multiplication digit‑by‑digit method [1][4].
5. Apply the place‑value shift for the tens digit and write the shifted partial product
Because that 7 was in the tens place , the 2,135 partial product must be written one place to the left (or equivalently append a zero) yielding 21,350 as the shifted partial product row; this shifting step is the mechanical reflection of multiplying by 10 and is emphasized in long‑multiplication guides [2][3].
6. Add the partial products to obtain the final product
Add the first partial row (000 from multiplying by the ones digit) to the shifted second partial row ; summing these rows gives the final product 21,350, which is the expected outcome when following the standard algorithm and place‑value rules [1][5].
7. Alternative view: use distributive property for a quicker mental check
The same result follows from distributive reasoning: 305 × 70 = 305 × (7 × 10) = (305 × 7) × 10 = 2,135 × 10 = 21,350, a commutative and distributive justification that calculators and teaching resources use to explain why appending a zero after multiplying by 7 is valid [3][7].
8. Common pitfalls and how the procedural sources warn against them
Instructional sources caution about misalignment of digits when writing shifted partial products, failure to carry correctly during the column multiplications, and forgetting the place‑value shift when a multiplier digit occupies tens or hundreds places — mistakes that produce incorrect sums even if each small multiplication was done correctly [5][6][4].
9. Final authoritative statement of the answer
Using the standard long‑multiplication algorithm and place‑value shifting produces the final product 305 × 70 = 21,350; this result matches the stepwise procedures taught in long‑multiplication tutorials and calculators that display intermediate steps [1][8].