Lesson 80: Division with Zeros in Three-Digit Answers

New Concept

Sometimes the answer to Step 1 is zero, and we will have a zero in the answer.

What’s next

Next, you’ll apply the four-step division process to problems where a zero placeholder is a key part of the correct answer.

The pencil-and-paper method for dividing numbers has four steps: Step 1: Divide. Step 2: Multiply. Step 3: Subtract. Step 4: Bring down. Think of these steps as a repeatable dance routine. For every digit you bring down from the dividend, you must perform this entire four-step sequence until you have no numbers left to bring down.

Example 1: Solve $5 \overline{)525}$. First, divide 5 by 5 to get 1. Then bring down the 2. Since 5 doesn't go into 2, put a 0 in the quotient. Bring down the 5 to make 25. Divide 25 by 5 to get 5. The answer is $105$.
Example 2: Solve $3 \overline{)912}$. Divide 9 by 3 to get 3. Bring down the 1. Since 3 cannot go into 1, place a 0 in the quotient. Bring down the 2 to make 12. Divide 12 by 3 to get 4. The final answer is $304$.

Mastering division is all about remembering this simple four-step loop: Divide, Multiply, Subtract, Bring Down. It's a cycle you'll repeat for each digit in your number, making even big division problems manageable and straightforward. Just keep the rhythm going!

Every time we bring a number down, we return to Step 1. Sometimes the answer to Step 1 is zero, and we will have a zero in the answer. This zero acts as a crucial placeholder, ensuring all other digits in our final answer end up in the correct place value. Without it, the entire answer would be wrong.

Example 1: In $4 \overline{)816}$, after dividing 8 by 4 to get 2, you bring down 1. Since 4 cannot go into 1, you must write a 0 in the answer. Then bring down the 6, making it 16. Divide 16 by 4 to get 4. Your answer is $204$.
Example 2: For $6 \overline{)1254}$, divide 12 by 6 to get 2. Bring down 5. Since 6 is larger than 5, place a 0 in the quotient. Now, bring down the 4 to make 54. Divide 54 by 6 to get 9. The correct answer is $209$.

What happens when you bring a digit down and it's too small to be divided? Don't just skip it! You must place a zero in the quotient. This zero holds the spot and says, 'Nope, can't divide here, let's bring down the next friend!'

When there are no more digits to bring down, the division is complete. If the number left over after the final subtraction step is not zero, this leftover amount is called the remainder. It signifies the part of the dividend that could not be divided evenly by the divisor, so we write it separately with our answer.

Example 1: Solve $4 \overline{)1283}$. Divide 12 by 4 to get 3. Bring down 8. Divide 8 by 4 to get 2. Bring down 3. Since 4 can't go into 3, place a 0 in the quotient. The leftover 3 is your remainder. The answer is $320 \text{ R } 3$.
Example 2: In $5 \overline{)2534}$, divide 25 by 5 to get 5. Bring down 3. Since 5 won't go into 3, place a 0. Bring down 4 to make 34. Divide 34 by 5 to get 6 with 4 left over. The answer is $506 \text{ R } 4$.

Think of a remainder as the 'leftovers' from a division party. After sharing everything out as evenly as possible, the remainder is what's left on the plate because you couldn't make another full group. Just write it next to your answer with a capital 'R'!

When dividing large numbers that end in multiple zeros, like 6000, you can simplify the problem. Focus on dividing the non-zero digits first, such as dividing 6 by 4. Then, use the standard division steps for the remaining zeros. This trick makes intimidatingly large numbers much easier to handle by breaking them down into simpler parts.

Example 1: To solve $4 \overline{)2000}$ mentally, first divide 20 by 4, which is 5. Then, simply attach the two leftover zeros from 2000 to your answer to get 500. So, $2000 \div 4 = 500$.
Example 2: For $3 \overline{)1200}$, focus on $12 \div 3 = 4$. Now, take the two zeros from 1200 and add them to the end of your result. The final answer is $400$.

Don't let giant numbers with lots of zeros scare you! The secret is to ignore the zeros at first. Just divide the front part of the number (like $20 \div 5 = 4$), then bring down the remaining zeros to finish the problem. It's a great mental math shortcut!