Long Division-With Remainders (multi-digit divisors)

Once students master the process of long division, longer problem just have more steps and require stronger multiplication skills.  The process is exactly the same.

To review the long division process (and acronyms for remembering the steps), see the lesson on Long Division with remainders (1 digit divisor).

Just like with one digit-divisors, we typically use the "house method" to do long division with multi-digit divisors.  Remember, the "house method" reverses the divisor and dividend

So, the problem: $560 \div 12=$ is written: $12\overline{)560}$ ad we follow the steps below>

1. Division: Divide the divisor into the first digit(s) of the dividend (write the answer on top of the house).
2. Multiply: Multiply the answer number times the divisor (write the answer under the portion fo the dividend you are dividing into).
3. Subtract: Subtract the product of the last step from the portion of the dividend you are working with (write the answer below).
4. Check: Make sure that answer to your subtraction problem is not greater than your divisor (if it is, then the number you wrote on the top line is too small).
5. Bring down: Bring the next digit in the dividend down and make it the ones digit of the number you just checked against the divisor.
6. Repeat: How many times does the divisor go into the new number you just created (write the answer on the top of the house, above the number you just brought down).

Let's see how the steps work with a multi-digit divisor:

Example: $981\div 11=$

Set the problem up:

\eqalign{11&\overline{)981}\qquad&&\text{Write the problem in a house}}

Do the first "division" step.  With a one digit divisor, we would ask "How many times does 11 go into the first digit in the dividend?" With a multi-digit divisor, you know before you begin that the divisor will not go into the first digit, so divide the divisor in the the first two digits, or three digits, whichever is the fewest number of digits that the divisor can go into.  Make sure that you write the first answer digit over the last digit of the first number you divide into.

\eqalign{ \quad & \quad \color{red}{8}\qquad && \color{red}{\text{11 goes into 98 eight times, write 8 over the 8}}\\11 & \overline{)981}\qquad && \quad \\&\!\! \underline{-88} \qquad && \text{8 times 11 is 88; subtract from 98}\\ &\;\; 10 \qquad && \text{98 minus 88 equals 10}}

Check the difference against the divisor.  In this case the difference (10) is smaller than the divisor (11), so you can move on.  Now, you bring down the next digit in the dividend.  Then you will divide the divisor into the new number created when you bring a new digit from the dividend and place it next to the remainder left over from the first time you multiplied and subtracted!

\eqalign{ \quad & \quad \color{red}{8}\color{green}9\qquad && \color{green}{\text{11 goes into 101 nine times, write the 9 over the 1}}\\11 & \overline{)981}\qquad && \quad \\&\!\! \underline{-99} \color{green}{\downarrow} \qquad && \\ &\;\; 10\color{green}{1} \qquad && \text{Bring down the 1 and put it next to the 10}\\&\;\underline{-99}\qquad &&\text{9 times 11 is 99}\\&\quad\;\;2\qquad&&\text{101 minus 99 is 2; you have a remainder of 2}}

You have no more digits to bring down (although if you had more digits in the dividend, you'd bring one down now).  In this case, the difference is your remainder.

The answer is now at the top of the "house."  $\color{blue}{981 \div 11 = 89 \text{ r }2}$ or $\color{blue}{981 \div 11 = 89 \dfrac{2}{11}}$

Overall, when we use the "house" to divide, we dividing a divisor into a dividend, place value by place value. By lining the answer up on top of the house, over each digit of the dividend, we help make sure that we get the place value correct (and know where to place the decimal if we have a decimal).

What to look out for?

The most common long division errors involve zeros.  Students tend to leave zeros out.  When practicing division make sure pay attention to the following trouble areas:

• When the divisor does NOT go into the dividend (or the newly created dividend after a new digit is brought down) a zero must be entered in the answer row before another digit is brought down.
• When the divisor does NOT go into the last number created when the last digit is brought down, a zero must be entered in the answer row, above the last digit of the dividend, before they write a remainder.

The other common error is that students often forget to check, after they subtract, their difference against the divisor.  Remind students to check.  Also remind students that if they want to enter two digits on the answer row (e.g., if they try to divide 39 by 3 and want to enter 13 in the answer row), they did something wrong in the prior step.  Using the house method, they will only put single digits at a time in the answer row.