Table of Contents
Understanding Decimals and Percentages
In arithmetic, decimals and percentages are two closely related ways of writing numbers that are not whole. They are both different ways of talking about parts of a whole, especially parts of $1$ or of $100$.
This chapter gives an overview of what decimals and percentages are and how they relate to each other. Later sections will deal with decimal notation, fraction–decimal conversion, and percent calculations in more detail.
What a Decimal Represents
A decimal is a way to write numbers using powers of $10$ smaller than $1$. The decimal point separates the whole-number part from the fractional part.
For example, in the number $47.362$:
- $47$ is the whole-number part.
- $.362$ is the decimal (fractional) part.
Each position to the right of the decimal point has a value:
- The first place to the right is the tenths place: $0.1 = \dfrac{1}{10}$
- The next is the hundredths place: $0.01 = \dfrac{1}{100}$
- The next is the thousandths place: $0.001 = \dfrac{1}{1000}$
So
$$
47.362 = 47 + \dfrac{3}{10} + \dfrac{6}{100} + \dfrac{2}{1000}.
$$
You will learn how to read, write, and work with decimal notation in the chapter “Decimal notation.” Here we only note that:
- Decimals are based on the same place-value system as whole numbers.
- Moving one place to the right divides by $10$; moving one place to the left multiplies by $10$.
What a Percentage Represents
A percentage is a way to describe a number as “out of $100$.” The symbol $\%$ means “per hundred.”
Examples:
- $25\%$ means $25$ out of $100$.
- $4\%$ means $4$ out of $100$.
- $150\%$ means $150$ out of $100$.
As fractions:
$$
25\% = \dfrac{25}{100}, \quad 4\% = \dfrac{4}{100}, \quad 150\% = \dfrac{150}{100}.
$$
Percentages are especially common in:
- Discounts and sales (e.g. “$20\%$ off”)
- Interest rates
- Test scores (e.g. “You scored $80\%$”)
- Statistics (e.g. “$60\%$ of people…”)
The chapter “Percent calculations” will show how to find a percentage of a number, how to find what percentage one number is of another, and how to work with increases and decreases.
How Decimals and Percentages Are Connected
Decimals and percentages are two different ways of writing the same idea. Percentages are based on $100$, and decimals are based on powers of $10$, so you can move between them easily.
To turn a percentage into a decimal, you divide by $100$.
To turn a decimal into a percentage, you multiply by $100$.
This is because
$$
\text{percentage} = \dfrac{\text{number}}{100},
$$
so reversing that step multiplies by $100$.
The chapter “Fraction–decimal conversion” will handle conversions involving fractions; here the focus is only on the relation between decimals and percentages.
Decimals to Percentages: The Basic Idea
Any decimal can be written as a percentage.
To turn a decimal into a percentage:
- Multiply by $100$.
- Then write the $\%$ symbol.
Conceptually, multiplying by $100$ just shifts the decimal point two places to the right.
Examples (no calculations explained in detail here, just results):
- $0.5$ as a percentage is $50\%$.
- $0.08$ as a percentage is $8\%$.
- $1.25$ as a percentage is $125\%$.
- $0.003$ as a percentage is $0.3\%$.
In each case, the size of the number does not change; only the way it is written changes.
Percentages to Decimals: The Basic Idea
Any percentage can be written as a decimal.
To turn a percentage into a decimal:
- Divide by $100$.
- Remove the $\%$ symbol.
Conceptually, dividing by $100$ shifts the decimal point two places to the left.
Examples:
- $25\%$ as a decimal is $0.25$.
- $4\%$ as a decimal is $0.04$.
- $150\%$ as a decimal is $1.50$ (or just $1.5$).
- $0.3\%$ as a decimal is $0.003$.
Again, the quantity stays the same; the representation changes.
Interpreting Percents Greater Than 100% and Less Than 1%
When working with percentages, you often see values that are not between $0\%$ and $100\%$. Decimals help you understand these.
Percentages Greater Than 100%
A percentage greater than $100\%$ means “more than the whole.”
- $100\%$ corresponds to the whole, or $1$ as a decimal.
- $150\%$ corresponds to $1.5$ (one and a half times the whole).
- $250\%$ corresponds to $2.5$ (two and a half times the whole).
Thinking in decimals:
- $150\% = 1.5$ means “$1.5$ times as large.”
- $200\% = 2.0$ means “$2$ times as large.”
Percentages Smaller Than 1%
A small percentage (less than $1\%$) corresponds to a small decimal.
Examples:
- $1\% = 0.01$.
- $0.5\% = 0.005$.
- $0.1\% = 0.001$.
These are useful when dealing with very small changes, such as tiny interest rates or small error rates.
Using Decimals and Percentages in Everyday Contexts
Decimals and percentages often describe the same situation in different forms. Being comfortable with both makes word problems and real-life situations easier to understand.
Some typical uses:
- Money and prices usually use decimals (e.g. \$3.75), while discounts and tax rates are usually given in percentages (e.g. $7.5\%$ tax).
- Test scores are sometimes given as fractions (e.g. $18$ out of $20$), sometimes as decimals (e.g. $0.9$), and often as percentages (e.g. $90\%$).
- Interest rates and growth rates are usually given in percentages, but the actual calculations often use decimals.
Later chapters will show how to calculate these precisely:
- “Decimal notation” will cover reading and writing decimal money amounts and placing the decimal point correctly.
- “Percent calculations” will step through problems like “$20\%$ off \$50” or “a $15\%$ increase.”
Rounding and Approximation (Overview)
Both decimals and percentages are often rounded to make them easier to work with or understand.
- A long decimal like $0.376482$ might be rounded to $0.38$.
- As a percentage, that same value might be rounded to $38\%$.
Even without detailed rounding rules (covered elsewhere in arithmetic), it is helpful to realize:
- Rounding changes the decimal and the percentage together.
- Being able to “see” a decimal and a percentage as the same size helps you judge whether an answer is reasonable.
For example, if a discount is “about a third,” you might think of it as:
- A decimal: about $0.33$.
- A percentage: about $33\%$.
Summary of the Relationships
Key ideas to remember from this chapter:
- Decimals and percentages are both ways to describe parts of a whole.
- A percentage tells you “how many out of $100$.”
- A decimal tells you the value using powers of $10$.
- Converting:
- From decimal to percent: multiply by $100$ and add $\%$.
- From percent to decimal: divide by $100$ (or move the decimal point two places left).
- Values greater than $100\%$ represent more than the whole; values less than $1\%$ represent very small parts, both of which can be written as decimals.
The following chapters will use these ideas to:
- Work carefully with decimal notation.
- Convert between fractions and decimals.
- Perform practical percent calculations.