Fractions, Decimals & Percentages

1. Introduction

Imagine you cut a pizza into $8$ equal slices and eat $3$ of them. How much pizza did you eat? You could say "three eighths" ($\frac{3}{8}$), or "$0.375$ of the pizza," or "$37.5\%$ of it." These are three different costumes worn by the same number. Fractions, decimals, and percentages are simply three languages for describing parts of a whole, and being fluent in all three — and able to translate between them instantly — is one of the most useful skills in all of mathematics.

This topic shows up everywhere in real life. A store advertises $25\%$ off a jacket. A restaurant suggests an $18\%$ tip. Your phone battery shows $73\%$ charged. A weather report says there is a $40\%$ chance of rain. Your math test score might be written as $\frac{18}{24}$, which you immediately want to understand as a percent. Recipes call for $\frac{1}{2}$ cup of sugar, but your measuring cup might be marked in decimals. Interest on a savings account, tax on a purchase, statistics in the news — all of these lean on the same core idea: describing how big a part is compared to a whole.

These three forms are also the foundation for almost everything that comes later in math. Algebra uses fractions constantly. Probability is built on parts of a whole. Calculus measures rates of change that often start as "what fraction of the interval?" Solid comfort with fractions, decimals, and percents means you spend your brainpower on new ideas instead of fighting basic arithmetic.

In this article we will build mastery from the ground up. We will explain what each form really means, why the conversion rules work, how to add and multiply fractions without traps, how to solve discount and tip problems, and how to work backwards when a percent change has already happened. By the end, you should be able to read any of the three forms, convert freely among them, and choose the form that makes each problem easiest.

2. Core Concepts

2.1 What a Fraction Really Means

A fraction is written $\dfrac{a}{b}$, where the top number $a$ is the numerator and the bottom number $b$ is the denominator. The denominator tells you how many equal pieces the whole was cut into, and the numerator tells you how many of those pieces you are talking about. So $\dfrac{3}{4}$ means "the whole was split into $4$ equal parts, and we have $3$ of them."

Picture a chocolate bar divided into $4$ equal squares. Eating $3$ squares means you ate $\frac{3}{4}$ of the bar. The denominator $4$ is the total number of equal parts; the numerator $3$ counts how many you took.

A fraction is also a division problem in disguise: $\dfrac{a}{b}$ literally means $a \div b$. This single idea is the secret to converting fractions into decimals. When you write $\frac{5}{8}$, you are really asking "what is $5$ divided by $8$?"

There are a few useful types to recognize:

• A proper fraction has a numerator smaller than its denominator, so its value is less than $1$ (e.g. $\frac{2}{5}$).
• An improper fraction has a numerator greater than or equal to its denominator, so its value is at least $1$ (e.g. $\frac{7}{4}$).
• A mixed number combines a whole number and a proper fraction, like $1\frac{3}{4}$, which means $1 + \frac{3}{4} = \frac{7}{4}$.

To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator: $1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$.

2.2 Equivalent Fractions and Simplifying

Two fractions are equivalent if they represent the same amount, like $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{50}{100}$. You create equivalent fractions by multiplying (or dividing) the numerator and denominator by the same nonzero number. This works because multiplying by $\frac{2}{2}$ or $\frac{5}{5}$ is just multiplying by $1$, which never changes a number's value:

$\frac{1}{2} = \frac{1}{2} \times \frac{2}{2} = \frac{2}{4}$

Why does this work? Imagine a pizza cut into $2$ halves. Shade one half — that is $\frac{1}{2}$. Now imagine cutting each half in half again, giving $4$ quarters. The same shaded region is now $2$ out of $4$ pieces: $\frac{2}{4}$. The shaded amount did not change; only the size of the slices changed.

To simplify (or reduce) a fraction, divide the top and bottom by their greatest common divisor (GCD) — the largest number that divides both. For $\dfrac{18}{24}$, the GCD of $18$ and $24$ is $6$, so $\dfrac{18}{24} = \dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}$. A fraction is in lowest terms when the only number dividing both parts is $1$.

2.3 Multiplying and Dividing Fractions

Multiplying fractions is straightforward: multiply numerators together and denominators together:

$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

Think of it as "of": $\frac{1}{2}$ of $\frac{2}{3}$ means taking half of two-thirds, which is $\frac{1}{3}$. Check: $\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}$.

Dividing by a fraction means multiplying by its reciprocal (flip the second fraction):

$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

The phrase "how many $\frac{2}{5}$ fit into $\frac{3}{4}$?" is exactly a division problem.

2.4 What a Decimal Really Means

A decimal uses our base-$10$ place value system to write parts of a whole. Digits to the left of the decimal point stand for ones, tens, and hundreds. Digits to the right stand for tenths, hundredths, thousandths, and so on. So $0.375$ means:

$0.375 = \frac{3}{10} + \frac{7}{100} + \frac{5}{1000} = \frac{375}{1000} = \frac{3}{8}$

This is exactly why every terminating decimal can be written as a fraction with a denominator that is a power of $10$. The number of digits after the point tells you the power: one digit means tenths ($/10$), two digits means hundredths ($/100$), three digits means thousandths ($/1000$).

Decimals come in two flavors. A terminating decimal stops, like $0.25$ or $0.8$. A repeating decimal has a digit or block that repeats forever, like $\dfrac{1}{3} = 0.3333\ldots$, written with a bar as $0.\overline{3}$. Fractions whose denominators have only factors of $2$ and $5$ terminate; others repeat.

2.5 What a Percentage Really Means

A percentage is a special fraction whose denominator is always $100$. The symbol $\%$ literally means "per hundred" (from the Latin per centum). So $37\%$ means $\dfrac{37}{100}$, or "$37$ out of every $100$."

Because $100\%$ means $\frac{100}{100} = 1$, a full whole is always $100\%$. A percentage can be more than $100\%$ (e.g. $150\% = 1.5$, meaning one and a half wholes) or less than $1\%$ for tiny amounts. Saying "$200\%$ of the goal" means you doubled it — you reached twice the target.

Percentages are handy for comparison because every quantity is scaled to "out of $100$." Comparing $37\%$ to $31\%$ is immediate; comparing $\frac{37}{100}$ to $\frac{19}{50}$ requires a moment of conversion first.

2.6 Percent Increase and Percent Decrease

A percent increase adds a fraction of the original to itself. If a price rises by $20\%$, the new price is $100\% + 20\% = 120\%$ of the original, or $1.20$ times the original.

A percent decrease subtracts a fraction of the original. A $25\%$ discount means you pay $100\% - 25\% = 75\%$ of the original, or $0.75$ times the original.

The base for any percent change is always the starting value, not the ending value. This is why a $20\%$ increase followed by a $20\%$ decrease does not return to the start — each percent uses a different base.

2.7 Connecting All Three Forms

Every rational number can be written as a fraction, a decimal, and a percent (possibly with a repeating decimal). The bridges are:

• Fraction to decimal: divide numerator by denominator.
• Decimal to percent: multiply by $100$.
• Percent to fraction: write over $100$, then simplify.

Knowing that $\frac{1}{2} = 0.5 = 50\%$, $\frac{1}{4} = 0.25 = 25\%$, and $\frac{1}{5} = 0.2 = 20\%$ by heart saves enormous time on tests and in daily life.