Dividing Decimals

➗ Dividing Decimals

Part 1 of 5 — What Division With Decimals Means

Topics in This Part

🔑 Key Concept: Dividing decimals works exactly like dividing whole numbers. The only new job is keeping the decimal point in the right place. Master that, and you've mastered the whole topic.

What Does Dividing Mean?

Division answers the question "how many equal groups?" or "how big is each share?"

When you see $8.4 \div 4$, you can read it two ways:

• Sharing: Split $8.4$ into $4$ equal parts. How big is each part?
• Grouping: How many groups of $4$ fit into $8.4$?

The parts of a division problem have names:

💡 Tip: $8.4 \div 4$ and $\dfrac{8.4}{4}$ mean the exact same thing — a fraction bar is a division sign.

Concept Check 🎯

Place Value: Why the Decimal Point Matters

Each spot in a decimal number has a value ten times smaller than the spot to its left:

The number $3.25$ means $3$ ones $+$ $2$ tenths $+$ $5$ hundredths.

⚠️ Watch out: $0.6$, $0.06$, and $6.0$ all use the digit $6$, but they are very different numbers. The decimal point tells you which place that $6$ lives in. Getting the point in the wrong spot can make an answer $10$ or $100$ times too big!

Name That Place 🔽

For each number, choose the place value of the digit $7$.

Estimate First — Then You Can Check Yourself

Before dividing, make a quick estimate by rounding to friendly whole numbers. Your real answer should land near the estimate.

Example: Estimate $18.6 \div 3$

Round $18.6$ to $18$. Since $18 \div 3 = 6$, the answer should be about $6$.

(The exact answer is $6.2$ — nice and close, so you know you're on the right track.)

Example: Estimate $41.7 \div 6$

Round $41.7$ to $42$ (because $42 \div 6 = 7$ is easy). The answer should be about $7$.

💡 Estimating is your safety net. If you ever calculate $18.6 \div 3 = 62$, your estimate of "about $6$" instantly warns you the decimal point slipped.

Estimation Check 🎯

You're Ready

You now know what division asks, the names of each part, why place value matters, and how to estimate a sanity-check answer.

In Part 2, we keep things simple: dividing a decimal by a whole number, where the decimal point barely has to move at all.

➗ Dividing Decimals

Part 2 of 5 — Dividing a Decimal by a Whole Number

🔑 The Idea: When the divisor is a whole number, you divide exactly like normal long division — then bring the decimal point straight up into the answer.

The One Golden Rule

When you divide a decimal by a whole number:

🔑 Bring the decimal point straight up. Place the point in the answer (quotient) directly above the point in the dividend — before you start dividing the digits.

Worked Example: $7.5 \div 5$

$\begin{array}{r} 1.5 \\ 5\,\overline{\smash{)}\,7.5} \end{array}$

1. The decimal point in $7.5$ sits between the $7$ and the $5$. Put a point straight up in the answer.
2. Divide as usual: $7 \div 5 = 1$ remainder $2$.
3. Bring down the $5$ to make $25$. Then $25 \div 5 = 5$.
4. Answer: $\mathbf{1.5}$.

✅ Check: $1.5 \times 5 = 7.5$ ✓

Place the Point 🔽

The digits of each answer are already correct — you just choose where the decimal point belongs.

Worked Example: $9.6 \div 8$

Point goes straight up. Then divide:

$9.6 \div 8 = 1.2$

• $9 \div 8 = 1$ remainder $1$.
• Bring down the $6$ to make $16$. Then $16 \div 8 = 2$.
• Answer: $\mathbf{1.2}$.

Worked Example: $14.4 \div 4$

$14.4 \div 4 = 3.6$

• $14 \div 4 = 3$ remainder $2$.
• Bring down the $4$ to make $24$. Then $24 \div 4 = 6$.
• Answer: $\mathbf{3.6}$.

💡 Notice the digits never changed how you divide — the decimal point just rode straight up from the dividend into the answer.

Concept Check 🎯

Sometimes You Add a Zero

What if the division doesn't come out evenly with the digits you have? You can add zeros after the decimal point — they don't change the value of the dividend, but they give you more digits to bring down.

Worked Example: $3 \div 4$

$3$ doesn't have a decimal part, but $3 = 3.00$. Write the point and add zeros:

$3 \div 4 = 0.75$

• $3 \div 4 = 0$ remainder $3$ (write $0$, point, then keep going).
• Bring down a $0$: $30 \div 4 = 7$ remainder $2$.
• Bring down another $0$: $20 \div 4 = 5$.
• Answer: $\mathbf{0.75}$.

💡 Adding zeros to the right of the last decimal digit never changes a number: $3 = 3.0 = 3.00 = 3.000$. This trick lets you keep dividing until the remainder is $0$.

Divide It 🧮

Divide each decimal by the whole number. Bring the point straight up!

1) $4.8 \div 4 = \,?$ 2) $13.5 \div 5 = \,?$ 3) $6 \div 8 = \,?$ (add zeros — answer is a decimal less than 1)

➗ Dividing Decimals

Part 3 of 5 — Dividing by Powers of 10 & Moving the Point

🔑 The Shortcut: Multiplying or dividing by $10$, $100$, or $1000$ doesn't require long division at all — you just slide the decimal point. This is the secret that makes Part 4 possible.

Dividing by 10, 100, 1000

Each time you divide by $10$, every digit gets ten times smaller, so the decimal point slides one place to the LEFT.

Count the zeros in the divisor — that's how many places the point hops left.

💡 If you run out of digits, fill the empty spots with zeros: $7 \div 100 = 0.07$ (two hops left, fill with a leading zero).

Concept Check 🎯

Multiplying by 10, 100, 1000 (the opposite)

Multiplying makes numbers bigger, so the point slides to the RIGHT.

⚠️ Direction matters! Dividing → point moves left (number shrinks). Multiplying → point moves right (number grows). Mixing these up is the #1 power-of-10 mistake.

Slide the Point 🔽

For each one, pick the correct result.

Why This Will Matter Next

Here's the big plan for the hardest case (Part 4): when the divisor is a decimal, we'll multiply both numbers by a power of $10$ to turn the divisor into a whole number. Then we're back to the easy case from Part 2.

For example, to handle $\div 0.4$, we'll multiply by $10$ to get $\div 4$. That's why sliding the point fluently is so important right now.

🔑 Fairness rule (preview): Whatever you multiply the divisor by, you must multiply the dividend by the same amount. It's like a fraction — scaling top and bottom equally keeps the value the same.

Move the Point 🧮

Slide the decimal point the right number of places.

1) $73.5 \div 10 = \,?$ 2) $2.6 \times 1000 = \,?$ 3) $4.5 \div 100 = \,?$

➗ Dividing Decimals

Part 4 of 5 — Dividing by a Decimal

🔑 The Master Move: Never divide by a decimal directly. First turn the divisor into a whole number by sliding its point right — then slide the dividend's point the same number of places. Now it's a Part 2 problem.

The 3-Step Method

To compute (dividend) $\div$ (decimal divisor):

1. Count how many places the point must move right to make the divisor whole.
2. Move the point that many places in both the divisor and the dividend.
3. Divide as usual (whole-number divisor), bringing the point straight up.

Worked Example: $8.4 \div 0.4$

• Step 1: $0.4$ needs to move $1$ place right to become $4$.
• Step 2: Move both points $1$ place right: $\;8.4 \to 84\;$ and $\;0.4 \to 4$.
• Step 3: $84 \div 4 = 21$.

$8.4 \div 0.4 = 84 \div 4 = \mathbf{21}$

✅ Check: $21 \times 0.4 = 8.4$ ✓

Worked Example: $9.36 \div 0.04$

• The divisor $0.04$ needs $2$ places right to become $4$.
• Move both points $2$ places: $\;9.36 \to 936\;$ and $\;0.04 \to 4$.
• Divide: $936 \div 4 = 234$.

$9.36 \div 0.04 = 936 \div 4 = \mathbf{234}$

Worked Example: $1.5 \div 0.5$

• $0.5$ needs $1$ place right to become $5$.
• Move both: $\;1.5 \to 15\;$ and $\;0.5 \to 5$.
• Divide: $15 \div 5 = 3$.

$1.5 \div 0.5 = 15 \div 5 = \mathbf{3}$

💡 Surprise: $1.5 \div 0.5 = 3$, which is bigger than the number you started with! Dividing by a number less than $1$ makes the answer grow — because lots of tiny pieces fit inside.

Set Up the Problem 🔽

You're solving $7.2 \div 0.6$. Choose what happens at each step.

Two Things to Watch

When you move the points, keep these in mind:

1. Move BOTH the same amount. If the divisor moves $2$ places, the dividend moves $2$ places. Moving only one of them changes the answer.
2. The dividend may need extra zeros. To compute $8.4 \div 0.04$, the divisor $0.04 \to 4$ needs $2$ places, so $8.4$ must move $2$ places too: $8.4 \to 840$ (you append a zero).

💡 Big-picture sense check: dividing by a number less than $1$ always makes the answer bigger than the dividend, while dividing by a number greater than $1$ makes it smaller.

Concept Check 🎯

Your Turn — Recap the Steps

Before the drill, run the method in your head one more time:

1. Count the decimal places in the divisor.
2. Slide both points right that many places.
3. Divide the new whole-number problem and bring the point straight up.

🔑 If the divisor is already whole (like $\div 4$), there's nothing to slide — you're back in Part 2.

Divide by a Decimal 🧮

Make the divisor whole, move both points, then divide.

1) $6.3 \div 0.7 = \,?$ 2) $4.8 \div 0.6 = \,?$ 3) $8.4 \div 0.04 = \,?$

➗ Dividing Decimals

Part 5 of 5 — Word Problems, Mixed Practice & Mastery Check

You can now divide a decimal by a whole number, slide the point for powers of $10$, and divide by a decimal. Let's use these skills on real situations and then prove your mastery.

Quick Reference

⚠️ Two habits to keep: (1) Always estimate first so you catch a misplaced point. (2) When dividing by a decimal, move both points the same number of places — never just one.

Word Problems

Example: Sharing money

Four friends split a $9.60 lunch bill equally. How much does each pay?

$9.60 \div 4 = 2.40$

Each friend pays $2.40. (Check: $2.40 \times 4 = 9.60$ ✓)

Example: How many pieces?

A ribbon is $7.5$ meters long. How many $0.5$-meter pieces can you cut?

$7.5 \div 0.5 = 75 \div 5 = 15$

You get $15$ pieces. (Here dividing by $0.5$, a number under $1$, gives an answer bigger than $7.5$ — that makes sense, since each piece is short.)

💡 Decide which number is the divisor: "split among $4$ people" → divide by $4$. "How many $0.5$-m pieces" → divide by $0.5$.

Word Problem Check 🎯

Mixed Practice 🧮

One of each type — pick the right method for each!

1) $15.6 \div 4 = \,?$ (decimal ÷ whole number) 2) $28.3 \div 100 = \,?$ (power of 10) 3) $5.6 \div 0.8 = \,?$ (÷ a decimal)

Exit Quiz ✅

Answer all three to finish the lesson.