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Multi-Digit Division

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Multi-Digit Division - Complete Interactive Lesson

Part 1: The Parts of a Division Problem & Smart Estimating

➗ Multi-Digit Division

Part 1 of 5 — The Parts of a Division Problem & Smart Estimating

Topics in This Part

Section
What the Words Mean: Dividend, Divisor, Quotient
Division Is Just Repeated Subtraction
Estimating the Answer First

🔑 Key Concept: Before you do any long division, you should already have a rough idea of how big the answer is. A good estimate catches mistakes before they happen.

The Parts of a Division Problem

Every division problem has the same three pieces. Learning their names now makes everything else easier.

$\underbrace{846}_{\text{dividend}} \div \underbrace{6}_{\text{divisor}} = \underbrace{141}_{\text{quotient}}$

Word	What it means	In $846 \div 6$
Dividend	the number being split up (the total)	$846$
Divisor	the number you split it into	$6$
Quotient	the answer	$141$

The same problem can be written three ways. They all mean the exact same thing:

$846 \div 6 \qquad\qquad \frac{846}{6} \qquad\qquad 6\,\overline{)\,846}$

💡 Memory trick: The dividend goes inside the long-division "house" ( $\overline{)\;\;}$ ). The divisor stands outside the door.

Concept Check 🎯

Division Is Repeated Subtraction

Dividing $20 \div 5$ asks: "How many groups of 5 fit inside 20?"

$20 - 5 - 5 - 5 - 5 = 0 \quad\Rightarrow\quad \text{four 5's fit, so } 20 \div 5 = 4$

Estimate First with Compatible Numbers

A great estimate uses compatible numbers — numbers close to the real ones that divide evenly in your head.

Example: Estimate $417 \div 6$ .

$417$ is close to $420$ , and $420 \div 6 = 70$ (because ).

Estimate It 🧮

Use compatible numbers to estimate each quotient. Enter your estimate.

1) $\;365 \div 6 \approx \,?$ (round the dividend to $360$ ) 2) $\;2{,}438 \div 5 \approx \,?$

Does the Answer Make Sense? 🎯

Part 2: Dividing by a One-Digit Number

➗ Multi-Digit Division

Part 2 of 5 — Dividing by a One-Digit Number

🔑 The Rhythm of Long Division: Every long-division problem repeats the same four steps over and over: Divide, Multiply, Subtract, Bring down. Some people remember it as D · M · S · B.

The Four Steps: D-M-S-B

To divide, you work through the dividend one digit at a time, left to right, repeating this loop:

Divide — How many times does the divisor go into the current number?
Multiply — Multiply that digit by the divisor.
Subtract — Subtract to find what's left.
Bring down — Bring down the next digit and repeat.

Worked Example: $896 \div 7$

\begin{array}{r} 128 \\ 7 \overset{}{} \end{array}

Part 3: Dividing by a Two-Digit Number

➗ Multi-Digit Division

Part 3 of 5 — Dividing by a Two-Digit Number

🔑 Same four steps, bigger divisor. Dividing by a two-digit number (like 23 or 41) uses the exact same D-M-S-B loop. The only new skill is guessing how many times a big number goes in — and we'll use estimation to guess smart.

Guess Smart: Round the Divisor

With a two-digit divisor, "how many times does it go in?" is harder to see. The trick is to round the divisor to the nearest ten and use that to guess.

Divisor	Round to	Use this to guess
$23$	$20$	think "how many 20s?"
$38$

Part 4: Remainders & What They Mean

➗ Multi-Digit Division

Part 4 of 5 — Remainders & What They Mean

🔑 Not everything divides evenly. When the divisor doesn't fit a whole number of times, the leftover is the remainder. The smart part isn't finding the remainder — it's deciding what to do with it in a real-world problem.

Finding a Remainder

The remainder is what's left over after dividing as far as you can with whole numbers. It must always be smaller than the divisor.

Worked Example: $59 \div 8$

$8 \times 7 = 56$ , which fits inside 59. ( is too big.)

Part 5: Mixed Practice & Mastery Check

➗ Multi-Digit Division

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) name the parts of a division problem, (2) estimate the answer, (3) divide by one- and two-digit numbers using D-M-S-B, and (4) find and interpret remainders. Time to put it all together.

Quick Reference

Goal	Key move
Estimate first	use compatible numbers (e.g. $417 \div 6 \approx 420 \div 6 = 70$ )
Do long division	repeat ivide · ultiply · ubtract · ring down

four 5’s fit, so 20÷

This is also why division and multiplication are opposites (inverse operations). Every division fact hides a multiplication fact:

$20 \div 5 = 4 \quad\Longleftrightarrow\quad 4 \times 5 = 20$

🔑 Check any answer with multiplication. If $846 \div 6 = 141$ , then $141 \times 6$ should equal $846$ . If it doesn't, the answer is wrong.

So the answer should be near 70.

Example: Estimate $3{,}512 \div 7$ .

$3{,}512$ is close to $3{,}500$ , and $3{,}500 \div 7 = 500$ (because $35 \div 7 = 5$ ).
So the answer should be near 500.

💡 You are not trying to get the exact answer when you estimate — you are finding the right "size" so you know roughly where the decimal point and digits belong.

(round the dividend to $2{,}500$ )

\;6{,}312 \div 9 \approx \,?

(round the dividend to $6{,}300$ )

128 7 \overline{) 896}

$8 \div 7$ : 7 goes in 1 time. $1 \times 7 = 7$ . $8 - 7 = 1$ . Bring down the $9 \rightarrow 19$ .
$19 \div 7$ : 7 goes in 2 times. $2 \times 7 = 14$ . $19 - 14 = 5$ . Bring down the .
$56 \div 7$ : 7 goes in 8 times. $8 \times 7 = 56$ . $56 - 56 = 0$ . Done.

✅ Check: $128 \times 7 = 896$ ✓

Follow the Steps 🔽

You're dividing $872 \div 4$ . Choose what happens at each stage.

⚠️ Don't Forget Zeros in the Quotient!

When the divisor doesn't go into a digit, you must write a 0 in the quotient and keep going. Skipping the zero is the #1 long-division mistake.

Worked Example: $618 \div 6$

\begin{array}{r} 103 \\ 6\,\overline{)\,618} \end{array}

$6 \div 6 = 1$ . $1 \times 6 = 6$ . $6 - 6 = 0$ . Bring down the .

$618 \div 6 = 103 \quad\text{(not } 13!\text{)}$

⚠️ If you had skipped the zero, you'd get $13$ — and $13 \times 6 = 78$ , nowhere near $618$ . Always place a digit (even a 0) for every digit you bring down.

Divide by One Digit 🧮

Work each one with D-M-S-B. These all come out even (no remainder).

1) $\;574 \div 7 = \,?$ 2) $\;936 \div 4 = \,?$ 3) $\;824 \div 8 = \,?$ (watch for a zero in the middle!)

Spot the Mistake 🎯

Example: For $187 \div 23$ , round 23 to 20. About how many 20s in 187? Around $9$ (since $20 \times 9 = 180$ ). Test 8: $23 \times 8 = 184$ , which fits inside 187. So the digit is 8.

💡 Your first guess might be too big. That's normal! If $\text{guess} \times \text{divisor}$ is bigger than the number, lower your guess by 1 and try again.

Make a Smart Guess 🧮

Round the divisor to estimate the single digit that goes in the quotient.

1) How many times does $19$ go into $58$ ? (round 19 to 20; $20 \times 3 = 60$ ) 2) How many times does $42$ go into $89$ ? (round 42 to 40) 3) How many times does $31$ go into $97$ ? (round 31 to 30)

Worked Example: $874 \div 23$

\begin{array}{r} 38 \\ 23\,\overline{)\,874} \end{array}

Step 1 — the first chunk. 23 doesn't go into 8, so look at 87. Round 23 to 20: about how many 20s in 87? Around 4. Test: $23 \times 4 = 92$ — too big! Back down to 3: $23 \times 3 = 69$ . Write 3. $87 - 69 = 18$ . Bring down the .

Step 2 — the next chunk. How many 23s in 184? Round to 20: about 9. Test: $23 \times 8 = 184$ exactly. Write 8. $184 - 184 = 0$ . Done.

$874 \div 23 = 38$

✅ Check: $38 \times 23 = 874$ ✓

Worked Example: $1{,}512 \div 36$

\begin{array}{r} 42 \\ 36\,\overline{)\,1512} \end{array}

36 doesn't go into 1, and not into 15. Look at 151. Round 36 to 40: about how many 40s in 151? Around 3. Test $36 \times 4 = 144$ (fits!), and $36 \times 5 = 180$ (too big). Write 4. $151 - 144 = 7$ . Bring down the .

$1{,}512 \div 36 = 42$

✅ Check: $42 \times 36 = 1{,}512$ ✓

Concept Check 🎯

Divide by Two Digits 🧮

Use rounding to guess, then check with multiplication. These all divide evenly.

1) $\;702 \div 18 = \,?$ 2) $\;966 \div 42 = \,?$ 3) $\;1{,}176 \div 24 = \,?$

59 - 56 = 3

left over.

$59 \div 8 = 7 \text{ R } 3$

We say "7 remainder 3." To check a remainder problem, use:

$(\text{quotient} \times \text{divisor}) + \text{remainder} = \text{dividend}$

$(7 \times 8) + 3 = 56 + 3 = 59 \;✓$

⚠️ A remainder can never be equal to or larger than the divisor. If your "remainder" is $8$ when dividing by $8$ , the quotient should have been one bigger!

Find the Quotient and Remainder 🧮

For each, enter the whole-number quotient in the first box and the remainder in the second box.

1) $\;47 \div 6 = \,?\ \text{R}\ ?$ 2) $\;100 \div 9 = \,?\ \text{R}\ ?$

What Do You DO with the Remainder?

The same numbers can give three different answers depending on the question! Watch:

Problem: 53 students need to ride in vans. Each van holds 8 students. ( $53 \div 8 = 6 \text{ R } 5$ )

The question asks…	What to do with R 5	Answer
How many vans are needed so everyone rides?	Round UP — those 5 kids still need a van	7 vans
How many vans can be completely filled?	Drop the remainder	6 vans
How many students are left without a full van?	The remainder is the answer	5 students

🔑 Read the question carefully. "Everyone must fit" → round up. "How many full groups" → drop the remainder. "How many left over" → the remainder itself.

Interpret the Remainder 🔽

Each problem starts from $74 \div 5 = 14 \text{ R } 4$ . Choose the right answer for each question.

Word Problem 🎯

⚠️ Top three mistakes to avoid: (1) skipping a 0 in the quotient, (2) letting a remainder be $\ge$ the divisor, and (3) ignoring what the word problem actually asks.

Mixed Practice 🧮

1) $\;945 \div 7 = \,?$ (even) 2) $\;988 \div 38 = \,?$ (even, two-digit divisor) 3) $\;85 \div 6 = \,?\ \text{R}\ ?$ (quotient in box 3, remainder in box 4)

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.

$1 \div 6$ : 6 does not go into 1, so write a 0 in the quotient. Bring down the

8 \rightarrow 18

$18 \div 6 = 3$ .

3 \times 6 = 18

18 - 18 = 0

. Done.

How many 36s in 72? Exactly 2 (

36 \times 2 = 72

72 - 72 = 0

. Done.

Multi-Digit Division

Multi-Digit Division - Complete Interactive Lesson

Part 1: The Parts of a Division Problem & Smart Estimating

➗ Multi-Digit Division

Topics in This Part

The Parts of a Division Problem

Division Is Repeated Subtraction

Estimate First with Compatible Numbers

Part 2: Dividing by a One-Digit Number

➗ Multi-Digit Division

The Four Steps: D-M-S-B

Worked Example: 896÷7896 \div 7896÷7

Part 3: Dividing by a Two-Digit Number

➗ Multi-Digit Division

Guess Smart: Round the Divisor

Part 4: Remainders & What They Mean

➗ Multi-Digit Division

Finding a Remainder

Worked Example: 59÷859 \div 859÷8

Part 5: Mixed Practice & Mastery Check

➗ Multi-Digit Division

Quick Reference

⚠️ Don't Forget Zeros in the Quotient!

Worked Example: 618÷6618 \div 6618÷6

Worked Example: 874÷23874 \div 23874÷23

Worked Example: 1,512÷361{,}512 \div 361,512÷36

What Do You DO with the Remainder?

Worked Example: $896 \div 7$

Worked Example: $59 \div 8$

Worked Example: $618 \div 6$

Worked Example: $874 \div 23$

Worked Example: $1{,}512 \div 36$