Multi-Digit Whole Number Multiplication

Multiplying large numbers is an essential math skill. Whether you are multiplying a four-digit number by a two-digit number or two three-digit numbers, the standard algorithm helps you find the answer step by step.

The Standard Algorithm

The standard algorithm breaks a large multiplication problem into smaller, easier steps. Let's look at how to multiply $1{,}024 \times 56$.

Step 1: Multiply by the ones digit. Multiply the top number ($1{,}024$) by the ones digit of the bottom number ($6$). $1{,}024 \times 6 = 6{,}144$

Step 2: Multiply by the tens digit. Next, multiply the top number by the tens digit ($5$). Because the $5$ is in the tens place (meaning it is actually $50$), we must place a placeholder zero at the end of our result. $1{,}024 \times 5 = 5{,}120$ Add the placeholder zero to get $51{,}200$.

Step 3: Add the partial products. Add the results from Step 1 and Step 2 together to find the final product. $6{,}144 + 51{,}200 = 57{,}344$ So, $1{,}024 \times 56 = 57{,}344$.

Multiplying Three-Digit by Three-Digit Numbers

The process is exactly the same for larger numbers, like $345 \times 267$. You just need to keep adding placeholder zeros as you move left!

1. Multiply by the ones ($7$): $345 \times 7 = 2{,}415$
2. Multiply by the tens ($60$): Add one zero. $345 \times 60 = 20{,}700$
3. Multiply by the hundreds ($200$): Add two zeros. $345 \times 200 = 69{,}000$

Finally, add all the partial products up: $2{,}415 + 20{,}700 + 69{,}000 = 92{,}115$

Estimating Products to Check Reasonableness

Before or after you calculate an exact answer, it is highly helpful to estimate the product. This helps you check if your exact answer makes sense (is "reasonable"). To estimate, round each number to its greatest place value before multiplying.

Example: Estimate $489 \times 312$

1. Round $489$ up to the nearest hundred: $500$
2. Round $312$ down to the nearest hundred: $300$
3. Multiply the rounded numbers: $500 \times 300 = 150{,}000$

Your estimated answer is $150{,}000$. If you were to calculate the exact answer ($152{,}568$), you would see it is very close to your estimate. This means your exact math is reasonable!