Grade 4 fractions lesson
Simplifying Fractions: Every Type Explained Step by Step
Simplify every common type of fraction by dividing the numerator and denominator by the same common factor until no further reduction is possible.
What does simplifying a fraction mean?
To simplify a fraction means to write an equivalent fraction using the smallest possible whole-number numerator and denominator. This is also called writing the fraction in simplest form or lowest terms.
For example, \(\frac{6}{8}\) and \(\frac{3}{4}\) name the same amount. Dividing both \(6\) and \(8\) by \(2\) gives \(\frac{6\div2}{8\div2}=\frac{3}{4}\).
Review the Types of Fractions lesson if you need help recognizing proper, improper, mixed, unit, equivalent, decimal, and zero fractions.
This lesson is only about simplifying. It does not teach addition, subtraction, multiplication, or division of one fraction by another.
The one rule that works every time
Divide the numerator and denominator by the same nonzero whole number. This keeps the value unchanged because both parts are being reduced by the same factor.
Keep simplifying until the greatest common factor of the numerator and denominator is \(1\). Then the fraction is in simplest form.
Use this four-step check
- Find a common factor: choose a number that divides both the numerator and denominator exactly.
- Divide both: never divide only the top number or only the bottom number.
- Write the equivalent fraction: its value must match the original fraction.
- Check again: stop only when the new numerator and denominator have no common factor greater than \(1\).
Method 1: divide by the greatest common factor
The greatest common factor, or GCF, is the largest whole number that divides both numbers exactly. Using it usually simplifies a fraction in one step.
Solved example: Simplify \(\frac{24}{36}\).
The factors of \(24\) are \(1,2,3,4,6,8,12,24\). The factors of \(36\) include \(1,2,3,4,6,9,12,18,36\). Their GCF is \(12\).
\[\frac{24}{36}=\frac{24\div12}{36\div12}=\frac{2}{3}\]
The GCF of \(2\) and \(3\) is \(1\), so \(\frac{2}{3}\) is fully simplified.
Method 2: divide by smaller common factors
You do not have to spot the GCF immediately. You may divide by smaller common factors more than once, as long as you divide the numerator and denominator by the same number each time.
Solved example: Simplify \(\frac{48}{60}\).
Both numbers are even, so divide by \(2\): \(\frac{48}{60}=\frac{24}{30}\). Divide by \(2\) again: \(\frac{24}{30}=\frac{12}{15}\). Both are divisible by \(3\): \(\frac{12}{15}=\frac{4}{5}\).
Now \(4\) and \(5\) have no common factor greater than \(1\), so \(\frac{4}{5}\) is the simplest form.
Method 3: use prime factors
Prime factors can reveal every factor shared by the numerator and denominator. This is useful when the numbers are larger.
Solved example: Simplify \(\frac{84}{126}\).
Write the prime factors: \(84=2\times2\times3\times7\) and \(126=2\times3\times3\times7\).
The shared prime factors are \(2\), \(3\), and \(7\). Together they make the GCF \(42\).
\[\frac{84}{126}=\frac{84\div42}{126\div42}=\frac{2}{3}\]
Simplify a proper fraction
A proper fraction has a numerator smaller than its denominator. Simplify it by dividing both numbers by their GCF.
Solved example: Simplify \(\frac{18}{24}\). The GCF of \(18\) and \(24\) is \(6\).
\[\frac{18}{24}=\frac{18\div6}{24\div6}=\frac{3}{4}\]
The result remains a proper fraction because \(3\lt4\).
Simplify an improper fraction
An improper fraction has a numerator greater than or equal to its denominator. The simplifying rule does not change.
Solved example: Simplify \(\frac{42}{30}\). The GCF is \(6\).
\[\frac{42}{30}=\frac{42\div6}{30\div6}=\frac{7}{5}\]
The simplified answer \(\frac{7}{5}\) is still improper. Changing it into a mixed number is a different skill, explained in the Improper Fractions lesson, and is not required for simplification.
Simplify a mixed number
A mixed number contains a whole-number part and a proper fraction part. Keep the whole number unchanged and simplify only the fraction part.
Solved example: Simplify \(5\frac{14}{21}\). The GCF of \(14\) and \(21\) is \(7\).
\[5\frac{14}{21}=5\frac{14\div7}{21\div7}=5\frac{2}{3}\]
The whole-number part stays \(5\). No fraction operation is needed.
Simplify a unit fraction or make a unit fraction
A unit fraction has numerator \(1\), such as \(\frac{1}{9}\). A valid unit fraction is already in simplest form because \(1\) has no factor greater than \(1\).
Another fraction may simplify to a unit fraction. For example, the GCF of \(5\) and \(35\) is \(5\):
\[\frac{5}{35}=\frac{5\div5}{35\div5}=\frac{1}{7}\]
Simplify a decimal fraction
A decimal fraction has a denominator such as \(10\), \(100\), or \(1000\). It uses the same common-factor rule.
Solved example: Simplify \(\frac{45}{100}\). The GCF of \(45\) and \(100\) is \(5\).
\[\frac{45}{100}=\frac{45\div5}{100\div5}=\frac{9}{20}\]
The answer does not need to keep a denominator of \(10\), \(100\), or \(1000\). The goal is the smallest equivalent numerator and denominator.
Fractions equal to zero, one, or a whole number
A fraction with numerator \(0\) and a nonzero denominator equals \(0\). For example, \(\frac{0}{18}=0\).
When the numerator and denominator are equal, the fraction equals \(1\): \(\frac{16}{16}=1\).
Some improper fractions simplify to another whole number. For example, \(\frac{24}{8}=3\). Writing the whole number is simpler than keeping a denominator of \(1\).
A denominator can never be \(0\). An expression such as \(\frac{5}{0}\) is undefined and cannot be simplified.
Simplify a negative fraction
Negative fractions are usually studied after the main Grade 4 fraction work, but their number parts simplify in exactly the same way. Simplify the positive numerator and denominator, then keep one negative sign.
Solved extension example: Simplify \(-\frac{28}{42}\). The GCF of \(28\) and \(42\) is \(14\).
\[-\frac{28}{42}=-\frac{28\div14}{42\div14}=-\frac{2}{3}\]
The forms \(-\frac{2}{3}\) and \(\frac{-2}{3}\) mean the same thing. A negative sign on both numerator and denominator would make a positive fraction.
What about like, unlike, and equivalent fractions?
Like and unlike describe how the denominators of two or more fractions compare. Equivalent fractions have the same value. These labels do not create a new simplification method.
Simplify each fraction separately. For example, \(\frac{4}{8}\) and \(\frac{6}{12}\) both simplify to \(\frac{1}{2}\), showing that they are equivalent.
This example compares simplified forms only. It does not add, subtract, multiply, or divide the fractions.
Quick reference: every common fraction type
| Fraction type | Original form | Common factor used | Simplest form |
|---|---|---|---|
| Proper | \(\frac{18}{24}\) | \(6\) | \(\frac{3}{4}\) |
| Improper | \(\frac{42}{30}\) | \(6\) | \(\frac{7}{5}\) |
| Mixed number | \(5\frac{14}{21}\) | \(7\) for the fraction part | \(5\frac{2}{3}\) |
| Unit result | \(\frac{5}{35}\) | \(5\) | \(\frac{1}{7}\) |
| Decimal fraction | \(\frac{45}{100}\) | \(5\) | \(\frac{9}{20}\) |
| Zero fraction | \(\frac{0}{18}\) | Write its value | \(0\) |
| Equals one | \(\frac{16}{16}\) | \(16\) | \(1\) |
| Equals a whole number | \(\frac{24}{8}\) | Write its value | \(3\) |
| Negative fraction | \(-\frac{28}{42}\) | \(14\) | \(-\frac{2}{3}\) |
| Already simplest | \(\frac{7}{11}\) | GCF is \(1\) | \(\frac{7}{11}\) |
Fast divisibility clues
Before listing every factor, look for easy shared divisors. If both numbers are even, try \(2\). If both digit sums are divisible by \(3\), try \(3\). If both numbers end in \(0\) or \(5\), try \(5\).
The Grade 4 Number Sense lessons for divisibility by 2, divisibility by 3, and divisibility by 5 explain these tests in detail.
Common simplifying mistakes
- Dividing only one number: \(\frac{6}{8}\) does not become \(\frac{3}{8}\). Divide both numerator and denominator by \(2\).
- Subtracting the same number: changing \(\frac{6}{8}\) to \(\frac{4}{6}\) changes the value. Simplifying uses division by a common factor.
- Cancelling matching digits: digits cannot simply disappear. Use factors of the complete numerator and denominator.
- Stopping too early: \(\frac{12}{18}=\frac{6}{9}\), but \(\frac{6}{9}\) still simplifies to \(\frac{2}{3}\).
- Changing the whole-number part: in \(5\frac{14}{21}\), simplify only \(\frac{14}{21}\).
- Forgetting the sign: a negative fraction stays negative when only its number parts are simplified.
How to know the answer is fully simplified
List or test the factors of the final numerator and denominator. If their only common factor is \(1\), the fraction is in simplest form.
Use the Greatest Common Factor Calculator to confirm the GCF after finding it yourself. The Fraction Calculator can also check a simplified form, but the common-factor reasoning should come first.
The main idea is simple: divide the numerator and denominator by the same common factor, keep going until the GCF is \(1\), and use exactly the same rule for every kind of fraction.
Once simplest form is secure, continue with Adding Fractions, Subtracting Fractions, or Multiplying Fractions. The Fraction Operations overview compares their rules first.