Multi-Digit Division

➗ Multi-Digit Division

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

Topics in This Part

🔑 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.

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

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?"

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

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.

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 $42 \div 6 = 7$).
• 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.

Estimate It 🧮

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

1) $\;365 \div 6 \approx \,?$ (round the dividend to $360$) 2) $\;2{,}438 \div 5 \approx \,?$ (round the dividend to $2{,}500$) 3) $\;6{,}312 \div 9 \approx \,?$ (round the dividend to $6{,}300$)

Does the Answer Make Sense? 🎯

➗ 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:

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

Worked Example: $896 \div 7$

• $8 \div 7$: 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 $6 \rightarrow 56$.
• $56 \div 7$: 7 goes in 8 times. $8 \times 7 = 56$. $56 - 56 = 0$. Done.

$896 \div 7 = 128$

✅ 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: $618 \div 6$

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

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

⚠️ 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 🎯

➗ 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.

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: $874 \div 23$

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 $4 \rightarrow 184$.

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: $1{,}512 \div 36$

• 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 $2 \rightarrow 72$.
• How many 36s in 72? Exactly 2 ($36 \times 2 = 72$). $72 - 72 = 0$. Done.

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

✅ 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) $\;1{,}176 \div 24 = \,?$

➗ 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. ($8 \times 8 = 64$ is too big.)
• $59 - 56 = 3$ 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. ($53 \div 8 = 6 \text{ R } 5$)

🔑 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 🎯

➗ 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

⚠️ 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.