Long Multiplication Calculator
Solve long multiplication problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
a x b = sum of partial products
Long multiplication multiplies the multiplicand by each digit of the multiplier separately, shifting each partial product left by one position. The final product is the sum of all partial products. This leverages the distributive property: a x (b1 + b2 x 10 + ...) = a x b1 + a x b2 x 10 + ...
Worked Examples
Example 1: Three-Digit by Two-Digit Multiplication
Problem: Multiply 347 by 28 using long multiplication.
Solution: Step 1: 347 x 8 (ones digit)\n 7x8=56 (write 6, carry 5)\n 4x8=32+5=37 (write 7, carry 3)\n 3x8=24+3=27 (write 27)\n Partial product 1: 2,776\n\nStep 2: 347 x 2 (tens digit, shift left)\n 7x2=14 (write 4, carry 1)\n 4x2=8+1=9 (write 9)\n 3x2=6 (write 6)\n Partial product 2: 6,940\n\nSum: 2,776 + 6,940 = 9,716
Result: 347 x 28 = 9,716
Example 2: Multiplication with Area Model
Problem: Multiply 56 by 34 using the area model.
Solution: Break apart: 56 = 50 + 6, 34 = 30 + 4\nGrid products:\n 50 x 30 = 1,500\n 50 x 4 = 200\n 6 x 30 = 180\n 6 x 4 = 24\nSum: 1,500 + 200 + 180 + 24 = 1,904
Result: 56 x 34 = 1,904
Frequently Asked Questions
How do you handle carrying in long multiplication?
Carrying in long multiplication works similarly to addition but occurs during the multiplication step itself. When you multiply a digit of the multiplicand by a digit of the multiplier, you may get a product of 10 or more. Write the ones digit of that product and carry the tens digit to add to the next column product. For example, 7 times 8 equals 56: write 6 and carry 5. Then compute the next column (4 times 8 = 32, plus carry 5 = 37): write 7 and carry 3. Continue until all digits are processed, writing any final carry.
What is the area model or grid method for multiplication?
The area model (also called the grid or box method) breaks each factor into its place value components and arranges them in a grid. For 347 times 28, break 347 into 300 + 40 + 7 and 28 into 20 + 8. Create a grid with all combinations: 300 times 20 = 6000, 300 times 8 = 2400, 40 times 20 = 800, 40 times 8 = 320, 7 times 20 = 140, 7 times 8 = 56. Sum all cells: 6000 + 2400 + 800 + 320 + 140 + 56 = 9716. This method makes the distributive property visible and helps students understand why long multiplication works.
How do you multiply decimal numbers using long multiplication?
To multiply decimals, first ignore the decimal points and multiply the numbers as if they were whole integers. Then count the total number of decimal places in both original numbers combined, and place the decimal point that many places from the right in the answer. For example, 3.47 times 2.8: multiply 347 times 28 = 9716. The first number has 2 decimal places and the second has 1, totaling 3 decimal places. So the answer is 9.716. This works because multiplying by powers of 10 to remove decimals is reversed by dividing by the same powers in the final answer.
What is the lattice method of multiplication?
The lattice method (also called the Italian method or gelosia multiplication) uses a grid where each cell is divided diagonally. Each digit of one factor labels a column and each digit of the other labels a row. You multiply each pair of digits, placing the tens digit above the diagonal and the ones digit below. Then add along the diagonals from right to left, carrying as needed. For example, for 347 times 28, create a 3-by-2 grid, fill in the products, and sum the diagonals. This method automatically handles carrying and place value alignment, reducing common errors.
Why is understanding long multiplication important for algebra?
Long multiplication is directly analogous to polynomial multiplication (the FOIL method and beyond). When you multiply (3x squared + 4x + 7) by (2x + 8), you perform essentially the same steps as multiplying 347 by 28, with x representing 10. Each term of the second polynomial multiplies every term of the first, creating partial products that are then combined by adding like terms. Understanding the structure of long multiplication provides an intuitive foundation for polynomial algebra, making factoring and expansion much more accessible.
How do you verify a long multiplication result?
Several methods can verify multiplication results. The simplest is estimation: round both factors and check that the product is in the right ballpark (350 times 30 = 10,500, close to 347 times 28 = 9,716). The casting out nines method sums digits repeatedly: digit sum of 347 is 5, digit sum of 28 is 1, product of digit sums is 5, and digit sum of 9716 is 5, matching. Reverse division also works: 9716 divided by 28 should give 347. Finally, a calculator check confirms the exact result. Using two independent verification methods provides high confidence.