Long Division With Remainders Study Notes

Long division is a method used to divide large numbers into smaller groups. When a number cannot be divided exactly, we have what is called a remainder. Understanding long division helps us solve many real-world problems!

🔢 Parts of a Division Problem

Every division problem has special names for its parts:

• Dividend: The number being divided. (The total amount you have)
• Divisor: The number that divides the dividend. (How many groups you are making or how many are in each group)
• Quotient: The answer to a division problem. (How many are in each group or how many groups you made)
• Remainder: The amount left over when a number cannot be divided exactly.

For example, in \( 17 \div 5 = 3 \text{ R } 2 \):

• Dividend: \( 17 \)
• Divisor: \( 5 \)
• Quotient: \( 3 \)
• Remainder: \( 2 \)

✍️ Steps for Long Division (DMSB)

We use a special set of steps to solve long division problems. A helpful way to remember these steps is the acronym DMSB:

1. Divide
2. Multiply
3. Subtract
4. Bring Down

You repeat these steps until there are no more numbers to bring down.

💡 Example: Let's Divide \( 125 \div 3 \)

Follow these steps carefully:

Step 1: Divide the first part of the dividend.

Look at the first digit of the dividend, \( 1 \). Can \( 3 \) go into \( 1 \)? No. So, look at the first two digits, \( 12 \).

• Divide: How many times does \( 3 \) go into \( 12 \)? It goes \( 4 \) times.
• Write \( 4 \) above the \( 2 \) in \( 125 \).

Step 2: Multiply.

• Multiply: Multiply the quotient digit you just wrote (\( 4 \)) by the divisor (\( 3 \)).
• \( 4 \times 3 = 12 \)
• Write \( 12 \) directly under the \( 12 \) in the dividend.

\[ \begin{array}{r} 4 \\ 3 \overline{) 125} \\ -12 \\ \end{array} \]

Step 3: Subtract.

• Subtract: Subtract the number you just wrote (\( 12 \)) from the part of the dividend above it (\( 12 \)).
• \( 12 - 12 = 0 \)
• Write \( 0 \) below the line.

\[ \begin{array}{r} 4 \\ 3 \overline{) 125} \\ -12 \downarrow \\ 0 \end{array} \]

Step 4: Bring Down.

• Bring Down: Bring down the next digit from the dividend (\( 5 \)) next to the \( 0 \).

\[ \begin{array}{r} 4 \\ 3 \overline{) 125} \\ -12 \\ 05 \end{array} \]

Step 5: Repeat DMSB (Divide, Multiply, Subtract, Bring Down).

Now we have \( 5 \). We repeat the steps with \( 5 \):

• Divide: How many times does \( 3 \) go into \( 5 \)? It goes \( 1 \) time.
• Write \( 1 \) next to the \( 4 \) above the \( 5 \) in the dividend.

\[ \begin{array}{r} 41 \\ 3 \overline{) 125} \\ -12 \\ 05 \end{array} \]

• Multiply: Multiply the new quotient digit (\( 1 \)) by the divisor (\( 3 \)).
• \( 1 \times 3 = 3 \)
• Write \( 3 \) under the \( 5 \).

\[ \begin{array}{r} 41 \\ 3 \overline{) 125} \\ -12 \\ 05 \\ -3 \\ \end{array} \]

• Subtract: Subtract \( 3 \) from \( 5 \).
• \( 5 - 3 = 2 \)
• Write \( 2 \) below the line.

\[ \begin{array}{r} 41 \\ 3 \overline{) 125} \\ -12 \\ 05 \\ -3 \\ 2 \end{array} \]

• Bring Down: Are there any more digits to bring down from the dividend? No.

📌 What is the Remainder?

The number left at the very bottom after you've finished all the steps and have no more digits to bring down is the remainder.

In our example, the remainder is \( 2 \).

So, \( 125 \div 3 = 41 \) with a remainder of \( 2 \). We write this as \( 41 \text{ R } 2 \).

✅ Checking Your Answer

You can always check your long division answer to make sure it's correct!

The rule for checking is:

(Quotient \( \times \) Divisor) \( + \) Remainder \( = \) Dividend

Let's check our example \( 125 \div 3 = 41 \text{ R } 2 \):

• Quotient: \( 41 \)
• Divisor: \( 3 \)
• Remainder: \( 2 \)

Calculation:

\[ 41 \times 3 = 123 \] \[ 123 + 2 = 125 \]

Since \( 125 \) is our original dividend, our answer is correct!

📚 Key Terms in Long Division

Here's a quick summary of the important terms:

Term	Meaning
Dividend	The number being divided.
Divisor	The number that divides the dividend.
Quotient	The answer to the division problem.
Remainder	The amount left over after division.