Welcome to Decimals and Fractions!

Welcome to the world of math! Up until now, you have learned a lot about "whole numbers" like \(1\), \(2\), and \(3\).
But how do we describe amounts when we cut a cake in half, or when there is just a little bit of water left in a water bottle?
To describe these "leftover amounts smaller than \(1\)," decimals and fractions come to the rescue!
At first, it might feel tricky with all the new words, but don't worry. Once you understand how they work, it becomes super fun!

1. What Are Decimals?

When measuring volume or length, you might end up with an amount that does not reach a full \(1\text{cm}\) or \(1\text{dL}\).
We use decimals to represent these leftover fractions of an amount.

Understanding \(0.1\)

When we divide \(1\) (our starting unit) into \(10\) equal parts, just \(1\) of those parts is written as \(0.1\) and read as "zero point one."
The small black dot between the numbers is called the decimal point.

Let's look at some familiar examples:
・Dividing \(1\text{cm}\) into \(10\) equal parts gives \(1\text{mm}\), which is \(0.1\text{cm}\).
・Dividing \(1\text{dL}\) into \(10\) equal parts gives \(0.1\text{dL}\).
・When you collect \(3\) of \(0.1\), it makes \(0.3\).
・When you collect \(10\) of \(0.1\), it makes a whole \(1\) again!

Place Value Names

The positions of decimal digits have special names.
Just to the left of the decimal point is the ones place (or units place), which you already know.
And the place directly to the right of the decimal point is called the tenths place.
For example, in the number \(2.3\), the ones place is \(2\) and the tenths place is \(3\) (meaning it is made up of two \(1\)s and three \(0.1\)s combined).

[Key Points]
・\(1\) divided into \(10\) equal parts (one-tenth) is \(0.1\).
・Multiplying \(0.1\) by \(10\) gives \(1\); multiplying it by \(100\) gives \(10\); multiplying it by \(1000\) gives \(100\).
・The dot between the numbers is called the decimal point.

2. Adding and Subtracting Decimals

You can add and subtract decimals just like whole numbers.
The trick is to think about "how many \(0.1\)s there are"!

Adding Decimals

Example: What is \(0.2 + 0.5\)?
・\(0.2\) is two \(0.1\)s.
・\(0.5\) is five \(0.1\)s.
Put them together, and you have \(2 + 5 = 7\) of the \(0.1\)s.
Therefore, \(0.2 + 0.5 = 0.7\).

Let's look at a problem where we regroup (carry over).
Example: \(0.6 + 0.8\)
・Together, there are \(6 + 8 = 14\) of the \(0.1\)s.
・Ten \(0.1\)s make \(1\), so the answer is \(1.4\).

Subtracting Decimals

Example: \(0.9 - 0.4\)
・Start with nine \(0.1\)s and take away four \(0.1\)s.
・\(9 - 4 = 5\) of the \(0.1\)s remain, so the answer is \(0.5\).

It works the same way when regrouping (borrowing).
Example: \(1.3 - 0.8\)
・\(1.3\) is thirteen \(0.1\)s.
・\(13 - 8 = 5\) of the \(0.1\)s remain, so the answer is \(0.5\).

[Common Mistake] Not lining up place values
When writing column calculations, always line up the decimal points vertically!
For example, when calculating \(2 + 0.3\), accidentally adding the ones digit \(2\) and the tenths digit \(3\) to get \(0.5\) is incorrect.
Because \(2\) is the same as \(2.0\), the correct answer is \(2 + 0.3 = 2.3\).

3. What Are Fractions?

Another way to express leftover amounts smaller than a whole is by using fractions.
Imagine sharing a pizza or a cake equally among friends!

How to Read and Write Fractions

When you divide one whole into \(4\) equal parts, \(1\) of those parts is written as \(\frac{1}{4}\) and read as "one-fourth" (or "one quarter").
We draw a horizontal line in the middle and write numbers above and below it.

・Denominator: The bottom number. It shows how many equal parts the whole is divided into.
・Numerator: The top number. It shows how many of those parts you have.

[Memory Trick]
Imagine a parent carrying a child on their shoulders!
The bottom is the parent (Denominator), and the top is the child (Numerator).

Unit Fractions and Multiples of Parts

A fraction that represents just "one part" when a whole is divided equally is called a unit fraction (for example: \(\frac{1}{3}\), \(\frac{1}{5}\), etc.).
Every fraction's size can be understood by "how many unit fractions are collected."
・Collecting three \(\frac{1}{5}\)s gives \(\frac{3}{5}\) (three-fifths).
・Collecting four \(\frac{1}{4}\)s gives \(\frac{4}{4}\). Because this is exactly the whole thing, it is equal to \(1\)!
In other words, whenever the denominator and the numerator are the same number, such as \(\frac{4}{4} = 1\) or \(\frac{5}{5} = 1\), it equals \(1\).

[Key Points]
・When using fractions, it is essential that all parts are divided into equal sizes!
・The bottom number is the denominator, and the top number is the numerator.
・When the numerator and denominator are the same, the value is \(1\).

4. Adding and Subtracting Fractions

Adding and subtracting fractions with the same denominator is super simple.
That's because you only need to calculate the numerators (the top numbers)!

Adding Fractions

Example: What is \(\frac{2}{5} + \frac{1}{5}\)?
・\(\frac{2}{5}\) is two \(\frac{1}{5}\)s.
・\(\frac{1}{5}\) is one \(\frac{1}{5}\).
Together, you have \(2 + 1 = 3\) of the \(\frac{1}{5}\)s.
So the answer is \(\frac{3}{5}\).

Subtracting Fractions

Example: \(\frac{4}{7} - \frac{1}{7}\)
・You have four \(\frac{1}{7}\)s and take away one of them.
・That leaves \(4 - 1 = 3\) of them, so the answer is \(\frac{3}{7}\).

Example: \(1 - \frac{2}{3}\)
When subtracting from \(1\), transform the \(1\) into a fraction!
Since the denominator of the fraction being subtracted is \(3\), change \(1\) into \(\frac{3}{3}\).
Then you can calculate: \(\frac{3}{3} - \frac{2}{3} = \frac{1}{3}\).

[Common Mistake] Adding the denominators together!
Many students accidentally calculate \(\frac{2}{5} + \frac{1}{5} = \frac{3}{10}\)!
The denominator tells us "what kind of parts" (how many pieces it was split into), so you must never add or subtract it.
Keep the denominator the same and only calculate the numerators.

5. Comparing Sizes on a Number Line

Using a number line makes the relationship between decimals and fractions clear at a glance!
Imagine a number line from \(0\) to \(1\) divided into \(10\) equal parts.

・Each tick mark is \(0.1\) as a decimal.
・Each tick mark is \(\frac{1}{10}\) (one-tenth) as a fraction.
This means \(0.1\) and \(\frac{1}{10}\) represent the exact same value!

・\(0.3\) and \(\frac{3}{10}\) are also the exact same size.
・\(0.7\) and \(\frac{7}{10}\) are the exact same size, too.

Equals Signs and Inequality Signs

We use the following symbols to compare the values of numbers:
・Equals sign (\(=\)): Used when values are equal.
　(Example: \(0.1 = \frac{1}{10}\))
・Inequality signs (\(>\), \(<\)): Used when values are different. The open mouth points to the larger number!
　(Example: \(0.4 < 0.7\), \(\frac{5}{8} > \frac{3}{8}\))

[Fun Fact]
Decimals are great when reading precise markings for length or volume, while fractions are handy for cleanly dividing pizzas or intervals of time into equal parts.
Both are super helpful tools in our everyday lives!

Summary (Review Checklist)

・Decimals: One of \(10\) equal parts of \(1\) is \(0.1\). The position immediately to the right of the decimal point is the tenths place.
・Fractions: A way to show equal parts of \(1\). The bottom is the denominator, and the top is the numerator.
・Addition and Subtraction: For decimals, line up the decimal points. For fractions with the same denominator, keep the denominator unchanged and only calculate the numerators.
・Connection: \(0.1\) and \(\frac{1}{10}\) are the exact same amount!

Once you can use decimals and fractions, the world of numbers opens up so much more. Keep practicing problems, and build your confidence!