I.6 Multiply 3-digit numbers by 1-digit numbers: word problems
What is multiplication?
Multiplication is a mathematical operation that combines equal groups. When we multiply, we are finding the total number of items in several groups of the same size. It is a quick way to add the same number over and over again.
- 4 + 4 + 4 = 12 is the same as 4 × 3 = 12.
- If you have 5 bags with 3 apples in each bag, you have 5 × 3 = 15 apples.
- 7 + 7 + 7 + 7 = 28 is the same as 7 × 4 = 28.
The numbers you multiply are called factors. The answer is called the product. The multiplication symbol "×" means "groups of."
Understanding place value in multiplication
Place value is the value of a digit based on its position in a number. In the number 4,326, the digit 4 is in the thousands place, so it represents 4,000. Understanding place value is essential when multiplying larger numbers because we multiply each digit separately.
- In 3,456: 3 is in the thousands place (3,000), 4 is in the hundreds place (400), 5 is in the tens place (50), and 6 is in the ones place (6).
- In 7,821: 7 is in the thousands place (7,000), 8 is in the hundreds place (800), 2 is in the tens place (20), and 1 is in the ones place (1).
When we multiply a 3-digit number by a 1-digit number, we multiply the ones, then the tens, then the hundreds. Always start with the smallest place value first.
Multiplying with regrouping (carrying)
Regrouping, also called carrying, happens when the product in one place value is 10 or greater. You write the ones digit in that place and move the tens digit to the next larger place value to add it after multiplying.
Step 1: Multiply the ones. 7 × 5 = 35. Write 5 in the ones place. Regroup 3 tens to the tens place.
Step 2: Multiply the tens. 7 × 4 = 28. Add the regrouped 3 tens. 28 + 3 = 31. Write 1 in the tens place. Regroup 3 hundreds to the hundreds place.
Step 3: Multiply the hundreds. 7 × 3 = 21. Add the regrouped 3 hundreds. 21 + 3 = 24. Write 24 in the hundreds and thousands places.
Final product: 2,415
Always remember to add the regrouped number after multiplying the current place value. It is easy to forget, so double-check your work.
Step-by-step: multiplying a 3-digit number by a 1-digit number
Follow these steps carefully every time you multiply a 3-digit number by a 1-digit number. This method works for any numbers, big or small.
Step 1: Write the numbers in a column, lining up the digits by place value. Put the larger number on top.
Step 2: Multiply the ones digit (8) by 4. 8 × 4 = 32. Write 2 in the ones place. Regroup the 3 tens to the tens column.
Step 3: Multiply the tens digit (3) by 4. 3 × 4 = 12. Add the regrouped 3 tens. 12 + 3 = 15. Write 5 in the tens place. Regroup the 1 hundred to the hundreds column.
Step 4: Multiply the hundreds digit (6) by 4. 6 × 4 = 24. Add the regrouped 1 hundred. 24 + 1 = 25. Write 25 in the hundreds and thousands places.
Final product: 2,552
Always multiply first, then add the regrouped number. This order is very important to get the correct answer.
Multiplying when the 1-digit number is in the ones place
Sometimes the 1-digit multiplier is written below the ones place of the 3-digit number. This is the standard way to set up the problem. The process is exactly the same: multiply each digit starting from the right.
9 × 2 = 18 (write 8, regroup 1)
9 × 7 = 63, plus regrouped 1 = 64 (write 4, regroup 6)
9 × 4 = 36, plus regrouped 6 = 42 (write 42)
Product: 4,248
Always keep your digits lined up neatly. Messy columns can lead to mistakes when adding regrouped numbers.
Multiplying with zeros in the 3-digit number
When a 3-digit number has a zero in one of the places, you still follow the same steps. Multiply zero by the 1-digit number, which equals zero. Then add any regrouped number from the previous step.
6 × 7 = 42 (write 2, regroup 4)
6 × 0 = 0, plus regrouped 4 = 4 (write 4, no regrouping)
6 × 3 = 18 (write 18)
Product: 1,842
5 × 0 = 0 (write 0, no regrouping)
5 × 3 = 15 (write 5, regroup 1)
5 × 4 = 20, plus regrouped 1 = 21 (write 21)
Product: 2,150
A common mistake is to skip multiplying the zero. Remember, zero times any number is zero, but you must still include it in your answer.
Estimating products to check for reasonableness
Estimation helps you check if your exact answer makes sense. Round the 3-digit number to the nearest hundred, then multiply by the 1-digit number. Your exact product should be close to your estimate.
Estimate: 672 rounds to 700. 3 × 700 = 2,100.
Exact: 3 × 672 = 2,016. 2,016 is close to 2,100, so our answer is reasonable.
Estimate: 295 rounds to 300. 8 × 300 = 2,400.
Exact: 8 × 295 = 2,360. 2,360 is close to 2,400, so our answer is reasonable.
Estimation is a quick way to catch big mistakes. If your exact answer is very different from your estimate, you should try the multiplication again.
Real-world word problems
Multiplication is used in many real-life situations. Reading word problems carefully helps you decide when to multiply.
A school has 4 classrooms. Each classroom has 235 books. How many books are there in total?
Solution: 4 × 235 = 940 books.
A delivery truck carries 386 boxes. It makes 7 deliveries. How many boxes does it deliver in total?
Solution: 7 × 386 = 2,702 boxes.
A ticket for a concert costs 149 dollars. If 5 tickets are sold, how much money is collected?
Solution: 5 × 149 = 745 dollars.
Look for keywords like "total," "in all," "each," and "every." These often tell you to multiply.
Multiplying larger 3-digit numbers
The same steps work for any 3-digit number, even those close to 1,000. Just be careful with regrouping, as the numbers can get larger.
9 × 7 = 63 (write 3, regroup 6)
9 × 5 = 45, plus regrouped 6 = 51 (write 1, regroup 5)
9 × 9 = 81, plus regrouped 5 = 86 (write 86)
Product: 8,613
6 × 8 = 48 (write 8, regroup 4)
6 × 8 = 48, plus regrouped 4 = 52 (write 2, regroup 5)
6 × 8 = 48, plus regrouped 5 = 53 (write 53)
Product: 5,328
When you regroup a number, write it small and above the next digit so you don't forget to add it.
Common mistakes to avoid
Even careful mathematicians make mistakes sometimes. Knowing the most common errors can help you avoid them.
Wrong: 4 × 257 → 4 × 7 = 28 (write 8, regroup 2), 4 × 5 = 20, but forget to add 2, so write 0, regroup 2, then 4 × 2 = 8 + 2 = 10, answer 1,008. This is incorrect.
Correct: 4 × 257 = 1,028.
If you write numbers crooked, you might add or multiply the wrong digits. Always use grid paper or line up carefully.
Always multiply the digit first, then add the regrouped number. The order matters.
Practice with small numbers first to build confidence, then try larger numbers. Checking your work with estimation also helps catch mistakes.
Practice problems with answers
Try these problems on your own. After you finish, check your answers below.
- 3 × 214 = ?
- 5 × 609 = ?
- 7 × 850 = ?
- 4 × 376 = ?
- 8 × 923 = ?
- 2 × 1,000 = ? (Remember, 1,000 is a 4-digit number, but the same rules apply)
- 3 × 214 = 642
- 5 × 609 = 3,045
- 7 × 850 = 5,950
- 4 × 376 = 1,504
- 8 × 923 = 7,384
- 2 × 1,000 = 2,000
If you got any wrong, try working through the steps again. Look for where you might have made a small mistake.
Common Core alignment: CCSS.MATH.CONTENT.4.NBT.B.5 – Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.NBT.B.5. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework.