How to Simplify Fractions

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A fraction tells us how many equal parts we have out of a whole. The top number is the numerator, and the bottom number is the denominator. For example, in \(\frac{6}{8}\), the numerator is \(6\) and the denominator is \(8\).

To simplify a fraction, we rewrite it with smaller numbers while keeping the exact same value. Think of it like reducing a recipe: \(\frac{6}{8}\) of a pizza is the same amount as \(\frac{3}{4}\) of that pizza, because both the numerator and denominator were divided by \(2\).

The important rule is this: whatever number you divide the numerator by, you must also divide the denominator by. That keeps the fraction balanced and equal to the original fraction.

Types of fractions:
Proper Fractions: The numerator is less than the denominator, such as \(\frac{3}{4}\) or \(\frac{1}{3}\).
Improper Fractions: The numerator is greater than or equal to the denominator, such as \(\frac{8}{7}\) or \(\frac{5}{2}\).
Mixed Numbers: A whole number and a fraction written together, such as \(1\frac{1}{3}\) or \(2\frac{3}{7}\).

Steps to simplify a fraction:

Find a common factor of the numerator and denominator.
For the fastest method, find the Greatest Common Factor (GCF).
Divide the numerator and denominator by that same factor.
Check your answer: if the numerator and denominator have no common factor except \(1\), the fraction is in simplest form.

Example

Start with the fraction \(\frac{9}{18}\).
The greatest common factor of \(9\) and \(18\) is \(9\).
Divide both numbers by \(9\): \(\frac{9\div 9}{18\div 9}=\frac{1}{2}\).
Since \(1\) and \(2\) have no common factor except \(1\), \(\frac{1}{2}\) is the simplest form.

WHAT IS THE GREATEST COMMON FACTOR OF 9 AND 18?

\( \frac{9}{18}=\frac{9 \div 9}{18 \div 9}=\frac{1}{2} \) , GCF(9,18) = 9

Simplifying Fractions

Think of this lesson as more than a rule to memorize. Simplifying Fractions is about number sense, equivalent forms, and careful arithmetic. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Fractions compare a part to a whole. Keep track of the numerator, denominator, and whether the pieces are the same size before adding, subtracting, or simplifying.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Simplifying Fractions

1) Simplify \({12 \over 18}\).\( \ \Rightarrow \ \)

2) Simplify \({15 \over 35}\).\( \ \Rightarrow \ \)

3) Simplify \({24 \over 30}\).\( \ \Rightarrow \ \)

4) Simplify \({45 \over 60}\).\( \ \Rightarrow \ \)

5) Simplify \({42 \over 56}\).\( \ \Rightarrow \ \)

6) Simplify \({81 \over 108}\).\( \ \Rightarrow \ \)

7) Simplify \({96 \over 144}\).\( \ \Rightarrow \ \)

8) Simplify \({125 \over 200}\).\( \ \Rightarrow \ \)

9) Simplify \({18 \over 48}\).\( \ \Rightarrow \ \)

10) Simplify \({-28 \over 70}\).\( \ \Rightarrow \ \)

11) Simplify \({150 \over 210}\).\( \ \Rightarrow \ \)

12) Simplify \({132 \over 180}\).\( \ \Rightarrow \ \)

13) A recipe uses \({7 \over 21}\) cup of oil. Write this amount in simplest form.\( \ \Rightarrow \ \)

14) In a class, \(36\) out of \(48\) students finished a project. Write the fraction in simplest form.\( \ \Rightarrow \ \)

15) A ribbon is \(72\) inches long, and \(96\) inches is the full length of the roll. Simplify \({72 \over 96}\).\( \ \Rightarrow \ \)

16) Find the missing denominator: \({18 \over n}\) simplifies to \({3 \over 5}\).\( \ \Rightarrow \ \)

17) Which fraction is equivalent to \({4 \over 9}\): \({20 \over 45}\), \({16 \over 30}\), or \({12 \over 21}\)?\( \ \Rightarrow \ \)

18) Simplify the mixed number \(2{12 \over 20}\).\( \ \Rightarrow \ \)

19) Are \({45 \over 75}\) and \({42 \over 70}\) equivalent? Simplify both to decide.\( \ \Rightarrow \ \)

20) A test has \(126\) points, and Mia earned \(84\) points. Simplify the fraction of points she earned.\( \ \Rightarrow \ \)

Answers

1) \({12 \over 18}\). The GCF(12,18) = 6. Divide both parts by \(6\): \({12 \over 18}={12\div 6 \over 18\div 6}={2 \over 3}\). Answer: \(\color{red}{{2 \over 3}}\)

2) \({15 \over 35}\). The GCF(15,35) = 5. Divide numerator and denominator by \(5\): \({15 \over 35}={15\div 5 \over 35\div 5}={3 \over 7}\). Answer: \(\color{red}{{3 \over 7}}\)

3) \({24 \over 30}\). The GCF(24,30) = 6. \({24 \over 30}={24\div 6 \over 30\div 6}={4 \over 5}\). Since \(4\) and \(5\) share no factor except \(1\), this is simplest. Answer: \(\color{red}{{4 \over 5}}\)

4) \({45 \over 60}\). The GCF(45,60) = 15. \({45 \over 60}={45\div 15 \over 60\div 15}={3 \over 4}\). Answer: \(\color{red}{{3 \over 4}}\)

5) \({42 \over 56}\). The GCF(42,56) = 14. \({42 \over 56}={42\div 14 \over 56\div 14}={3 \over 4}\). Answer: \(\color{red}{{3 \over 4}}\)

6) \({81 \over 108}\). The GCF(81,108) = 27. \({81 \over 108}={81\div 27 \over 108\div 27}={3 \over 4}\). Answer: \(\color{red}{{3 \over 4}}\)

7) \({96 \over 144}\). The GCF(96,144) = 48. Divide both numbers by \(48\): \({96 \over 144}={96\div 48 \over 144\div 48}={2 \over 3}\). Answer: \(\color{red}{{2 \over 3}}\)

8) \({125 \over 200}\). The GCF(125,200) = 25. \({125 \over 200}={125\div 25 \over 200\div 25}={5 \over 8}\). Answer: \(\color{red}{{5 \over 8}}\)

9) \({18 \over 48}\). The GCF(18,48) = 6. \({18 \over 48}={18\div 6 \over 48\div 6}={3 \over 8}\). Answer: \(\color{red}{{3 \over 8}}\)

10) \({-28 \over 70}\). Simplify the size of the fraction using the GCF(28,70) = 14: \({-28 \over 70}={-28\div 14 \over 70\div 14}={-2 \over 5}\). The negative sign stays with the fraction. Answer: \(\color{red}{{-2 \over 5}}\)

11) \({150 \over 210}\). The GCF(150,210) = 30. \({150 \over 210}={150\div 30 \over 210\div 30}={5 \over 7}\). Answer: \(\color{red}{{5 \over 7}}\)

12) \({132 \over 180}\). The GCF(132,180) = 12. \({132 \over 180}={132\div 12 \over 180\div 12}={11 \over 15}\). Answer: \(\color{red}{{11 \over 15}}\)

13) The recipe amount is \({7 \over 21}\). The GCF(7,21) = 7. \({7 \over 21}={7\div 7 \over 21\div 7}={1 \over 3}\). Answer: \(\color{red}{{1 \over 3}\text{ cup}}\)

14) The fraction is \({36 \over 48}\). The GCF(36,48) = 12. \({36 \over 48}={36\div 12 \over 48\div 12}={3 \over 4}\). Answer: \(\color{red}{{3 \over 4}}\)

15) Simplify \({72 \over 96}\). The GCF(72,96) = 24. \({72 \over 96}={72\div 24 \over 96\div 24}={3 \over 4}\). Answer: \(\color{red}{{3 \over 4}}\)

16) \({18 \over n}\) simplifies to \({3 \over 5}\). Since \(18\div 6=3\), the denominator must also be divided by \(6\). So \(n\div 6=5\), which gives \(n=30\). Check: \({18 \over 30}={18\div 6 \over 30\div 6}={3 \over 5}\). Answer: \(\color{red}{30}\)

17) Test each choice. For \({20 \over 45}\), the GCF(20,45) = 5, so \({20 \over 45}={4 \over 9}\). \({16 \over 30}={8 \over 15}\), and \({12 \over 21}={4 \over 7}\). Answer: \(\color{red}{{20 \over 45}}\)

18) Keep the whole number \(2\) and simplify the fraction part \({12 \over 20}\). The GCF of \(12\) and \(20\) is \(4\). \({12 \over 20}={12\div 4 \over 20\div 4}={3 \over 5}\). Answer: \(\color{red}{2{3 \over 5}}\)

19) Simplify both fractions. For \({45 \over 75}\), the GCF(45,75) = 15, so \({45 \over 75}={3 \over 5}\). For \({42 \over 70}\), the GCF(42,70) = 14, so \({42 \over 70}={3 \over 5}\). Since both simplify to the same fraction, they are equivalent. Answer: \(\color{red}{\text{Yes, both equal }{3 \over 5}}\)

20) Mia earned \({84 \over 126}\) of the points. The GCF(84,126) = 42. \({84 \over 126}={84\div 42 \over 126\div 42}={2 \over 3}\). Answer: \(\color{red}{{2 \over 3}}\)

How to Simplify Fractions