Study Notes: Decimals (Addition & Subtraction)

Hello Maths Explorers!

Welcome to the amazing world of decimals! Have you ever seen prices like $2.50 or heard someone say they are 1.45 metres tall? Those are decimals! They are a super useful way to show numbers that are not quite whole numbers.

In this chapter, we're going to learn how to add and subtract these special numbers. It's a skill you'll use all the time, like when you're shopping or measuring things. Don't worry if it seems new, we'll go step-by-step and you'll be a decimals expert in no time!


First, A Quick Recap: What Are Decimals?

Think of a decimal as a way to write a number that is in-between whole numbers. It's like a fraction, but with a special dot!

The Mighty Decimal Point

The most important part of a decimal is the decimal point (.). It separates the whole numbers on the left from the parts of a whole on the right.

Example: In the number $$12.34$$...
- 12 is the whole number part.
- . is the decimal point.
- 34 is the decimal part (the part of a whole).

Place Value Power!

Just like whole numbers, every digit after the decimal point has its own place value. The first two are the most important for us!

- The first digit after the point is in the tenths place. (Think: 10 cents is one-tenth of a dollar)
- The second digit is in the hundredths place. (Think: 1 cent is one-hundredth of a dollar)

So, in $$5.82$$:
- 5 is in the ones place.
- 8 is in the tenths place.
- 2 is in the hundredths place.

Did you know? The word "decimal" comes from the Latin word 'decem', which means ten! This is because our decimal system is based on the number 10.

Key Takeaway: A decimal number has a whole number part and a decimal part, separated by a decimal point.


Part 1: Adding Decimals - Let's Team Up!

Adding decimals is almost exactly like adding whole numbers. There is just one simple, golden rule you MUST remember.

The Golden Rule: Line Up The Dots!

The most important trick for adding (and subtracting) decimals is to line up the decimal points vertically. Imagine the decimal points are buttons on a shirt – you have to line them up perfectly!

Step-by-Step Guide to Adding Decimals

Let's add $$12.5 + 4.3$$.

Step 1: Write the numbers one on top of the other, making sure the decimal points are lined up.

$$ \begin{array}{c} & 12.5 \\ + & \phantom{1}4.3 \\ \hline \end{array} $$

Step 2: Add the numbers just like you normally would, starting from the right.

$$ \begin{array}{c} & 12.5 \\ + & \phantom{1}4.3 \\ \hline & 16.8 \\ \end{array} $$

Step 3: Bring the decimal point straight down into your answer. And you're done!

What if the numbers have different decimal places?

Let's try $$5.6 + 3.21$$. It can look tricky, but it's easy!

Just add a placeholder zero to make the numbers the same length. Remember, adding a zero to the end of a decimal doesn't change its value! $$5.6$$ is the same as $$5.60$$.

$$ \begin{array}{c} & 5.60 & \leftarrow \text{Add a placeholder zero!} \\ + & 3.21 \\ \hline & 8.81 \\ \end{array} $$

Adding a Whole Number and a Decimal

How about $$7 + 2.54$$? Where is the decimal point in 7?

Remember: A whole number always has an invisible decimal point at the very end. So, 7 is the same as 7. or 7.00.

$$ \begin{array}{c} & 7.00 & \leftarrow \text{Make the invisible point visible and add zeros!} \\ + & 2.54 \\ \hline & 9.54 \\ \end{array} $$

Key Takeaway: To add decimals, always line up the decimal points. Use placeholder zeros if you need to!


Part 2: Subtracting Decimals - Finding the Difference

Great news! Subtraction follows the exact same Golden Rule as addition.

The Golden Rule (Again!): Line Up The Dots!

Just like before, the secret to success is lining up those decimal points before you start.

Step-by-Step Guide to Subtracting Decimals

Let's subtract $$9.8 - 3.5$$.

Step 1: Line up the decimal points.

$$ \begin{array}{c} & 9.8 \\ - & 3.5 \\ \hline \end{array} $$

Step 2: Subtract normally from right to left, and bring the decimal point straight down.

$$ \begin{array}{c} & 9.8 \\ - & 3.5 \\ \hline & 6.3 \\ \end{array} $$

Watch Out! Subtraction and Placeholder Zeros

Placeholder zeros are even more important in subtraction! You can't just ignore an empty space.

Let's solve $$15.7 - 4.28$$.

$$ \begin{array}{c} & 15.70 & \leftarrow \text{You MUST add a placeholder zero here!} \\ - & \phantom{1}4.28 \\ \hline & 11.42 \\ \end{array} $$ You have to borrow from the 7 to do '0 minus 8'. Without the zero, you might make a mistake!

Common Mistake Alert!

The trickiest problem is subtracting a decimal from a whole number. Let's try $$20 - 5.67$$.

WRONG WAY:
Don't just stick the decimal on the end!

RIGHT WAY:
Remember the invisible decimal point after the 20 and add placeholder zeros!

$$ \begin{array}{c} & 20.00 & \leftarrow \text{VERY IMPORTANT!} \\ - & \phantom{1}5.67 \\ \hline & 14.33 \\ \end{array} $$

Key Takeaway: To subtract decimals, line up the decimal points. Always fill empty spots with placeholder zeros.


Part 3: Mixing It Up! Addition and Subtraction Together

Sometimes you'll see a problem with both addition and subtraction, like $$9.5 + 3.1 - 2.0$$. It's easy!

The rule is simple: Work from left to right.

Step 1: Solve the first part of the problem ($$9.5 + 3.1$$).

$$ \begin{array}{c} & 9.5 \\ + & 3.1 \\ \hline & 12.6 \\ \end{array} $$

Step 2: Now use that answer for the next part of the problem ($$12.6 - 2.0$$).

$$ \begin{array}{c} & 12.6 \\ - & \phantom{1}2.0 \\ \hline & 10.6 \\ \end{array} $$

So, $$9.5 + 3.1 - 2.0 = 10.6$$. Easy peasy!


Part 4: Decimals in the Real World - Solving Problems

Let's use our new skills to solve some real-life problems!

Problem 1: You go to the shop and buy a chocolate bar for $1.50 and a drink for $2.25. How much do you spend altogether?

Thinking: The word "altogether" tells me I need to add.
$$ \begin{array}{c} & 1.50 \\ + & 2.25 \\ \hline & 3.75 \\ \end{array} $$ Answer: You spend $3.75 altogether.

Problem 2: You pay with a $5.00 note. How much change do you get?

Thinking: The word "change" tells me I need to subtract.
$$ \begin{array}{c} & 5.00 \\ - & 3.75 \\ \hline & 1.25 \\ \end{array} $$ Answer: You get $1.25 in change.


Part 5: Super Skill - Estimating Your Answers

Estimating means making a smart guess. Before you solve a problem, it's a great idea to estimate the answer. This helps you check if your final answer makes sense!

A simple way to estimate is to round the decimals to the nearest whole number.

Example: Let's solve $$8.9 + 12.2$$.
- Estimate first: $$8.9$$ is very close to $$9$$. $$12.2$$ is very close to $$12$$. My answer should be around $$9 + 12 = 21$$.
- Now, solve for real:

$$ \begin{array}{c} & \phantom{1}8.9 \\ + & 12.2 \\ \hline & 21.1 \\ \end{array} $$

Our answer is $$21.1$$, which is very close to our estimate of $$21$$. It looks correct!


Chapter Summary - You've Got This!

Wow, you've learned so much! Here are the most important things to remember:

- Decimals are numbers with parts of a whole, separated by a decimal point.
- THE GOLDEN RULE: Always, always, always line up the decimal points when you add or subtract!
- Use placeholder zeros to make numbers the same length, especially when subtracting.
- A whole number has an invisible decimal point at the end (e.g., 12 is 12.00).
- For mixed problems, just work from left to right.
- Estimating by rounding is a great way to check your work.

Practice is the key to becoming a maths master. Keep trying, and you'll find that working with decimals is fun and useful! Well done!