Decimal Operations

Decimals Are Everywhere 🔢

A decimal is a way to show parts of a whole — just like a fraction. The little dot is the decimal point, and it separates the whole number part (on the left) from the part smaller than one (on the right).

You already use decimals every single day:

• Money: a snack costs $2.75
• Sports: a sprinter runs the 100m in $10.4$ seconds
• Cooking: a recipe needs $1.5$ cups of sugar

In this topic you'll master all four operations with decimals: add, subtract, multiply, and divide. The good news? You already know how to do these with whole numbers — decimals just add a couple of careful rules.

Place Value Is the Secret 🗝️

Each spot to the right of the decimal point is ten times smaller than the one before it. Let's break down the number $12.345$:

So $12.345$ means: 1 ten, 2 ones, 3 tenths, 4 hundredths, and 5 thousandths.

Why this matters: when we add and subtract, we must line up matching place values. That's why we line up the decimal points — it keeps tenths over tenths and hundredths over hundredths.

⭐ The Big Ideas

Here is the heart of every decimal operation:

• Adding & Subtracting: Line up the decimal points vertically. Add zeros as placeholders so every number has the same number of decimal places. Then add or subtract normally and bring the decimal point straight down.
• Multiplying: Ignore the decimal points and multiply like whole numbers. Then count the total decimal places in both factors and put that many decimal places in your answer.
• Dividing: Set it up so you are dividing by a whole number, then divide normally, keeping the decimal point lined up.

Adding $12.5 + 3.84$? Line them up and fill in a zero:

$\begin{aligned} 12.50 \\ +\ 3.84 \\ \hline 16.34 \end{aligned}$

Notice the $0$ we added to $12.5$ to make it $12.50$ — that placeholder keeps everything lined up.

Quick Concept Check ✅

Let's make sure the most important rule stuck.

Worked Example: Adding & Subtracting ➕➖

Add: $5.2 + 13.456 + 0.89$

Step 1 — Line up the decimal points and fill in zeros so every number has 3 decimal places:

$\begin{aligned} 5.200 \\ 13.456 \\ +\ 0.890 \\ \hline 19.546 \end{aligned}$

Answer: $19.546$ ✅

Subtract: $10 - 3.76$

A whole number like $10$ has an invisible decimal: it's $10.00$. Now line up and subtract, borrowing just like with whole numbers:

$\begin{aligned} 10.00 \\ -\ 3.76 \\ \hline 6.24 \end{aligned}$

Answer: $6.24$ ✅

Worked Example: Multiplying ✖️

Multiply: $2.5 \times 1.3$

Step 1 — Ignore the decimals and multiply as whole numbers:

$25 \times 13 = 325$

Step 2 — Count the decimal places in the problem:

• $2.5$ has 1 decimal place
• $1.3$ has 1 decimal place
• Total: $1 + 1 = 2$ decimal places

Step 3 — Place the decimal so the answer has 2 decimal places (count 2 from the right):

$325 \rightarrow 3.25$

Answer: $3.25$ ✅

Tip: With small decimals like $0.4 \times 0.3$, multiply $4 \times 3 = 12$, then place 2 decimal places to get $0.12$.

Your Turn ✍️

Solve each one. Type just the number (no extra symbols).

• Box 1: $8.5 + 2.34 = ?$
• Box 2: $9 - 4.7 = ?$
• Box 3: $1.2 \times 0.5 = ?$

Guided Practice: Pick the Answer 🎯

Work each one carefully. Remember the rules for each operation!

Finish Each Statement 🧩

Choose the option that correctly completes each sentence.

Decimals in the Real World 🌎

Decimals show up constantly when we deal with money, measurements, and sharing.

Shopping trip: Maya buys a notebook for $3.49 and a pen for $1.25.

• Total cost: $3.49 + $1.25 = $4.74
• Change from $10: $10.00 - $4.74 = $5.26

Splitting evenly: If 4 friends share a $12.80 pizza equally, each pays $12.80 ÷ 4 = $3.20.

The math is exactly the same as before — just remember to line up for $+$ and $-$, count places for $\times$, and keep the decimal lined up for $\div$. Money answers should always have two decimal places (cents).

Solve the Story Problems 🛒

Type just the number. For money answers, you can write it like $5.26$ (no dollar sign needed).

• Box 1: A book costs $8.99 and a bookmark costs $2.50. What is the total cost?
• Box 2: You pay with a $20 bill for an item that costs $13.45. How much change do you get?
• Box 3: Four friends split a $9.60 bill equally. How much does each person pay?

One More Real-World Problem 🏃

🏆 Putting It All Together

You've now worked with all four decimal operations. Here's a quick summary to lock it in:

Remember the difference: for $+$ and $-$ you line up the points, but for $\times$ you ignore the points and count places at the end. Mixing these two up is the most common mistake — so pause and ask yourself which operation you're doing!

Final Challenge: Mixed Operations 🎓

These mix everything together. Take your time and apply the right rule for each!