Reduce Fractions

Reducing a fraction means writing it in its simplest form without changing its value. Even after we reduce a fraction, it remains equal to the original fraction, so both are called equivalent fractions. In this article, you will learn how to reduce fractions in three easy ways, including how to reduce a fraction to its lowest term.
 

Here, tools like the reduce fraction calculator can also help you practice reducing fractions and understand how to reduce a fraction quickly. These tools make it easier to find the lowest term of any fraction.

For example, 

Here, reduce 12/18 to its lowest term.
 

• Find the greatest common factor (GCF) of 12 and 18, which is 6.
   
• Divide the numerator and denominator by 6.

126 = 2, 186 = 3
 

So,

12 / 18 = 2 / 3 

Here, 

2/3 is the lowest term, and both 12/18 and ⅔ are equivalent fractions.

To reduce fractions, first find the GCF and use it to divide both the numerator and the denominator. By reducing fractions, it means to divide both numerator and denominator with its common factors and find out the simplest form. 

Example: We need to simplify \(\frac{18}{24} \)

So, first, we need to find the common factors:

18 = 1, 2, 3, 6, 9, 18

24 = 1, 2, 3, 4, 6, 8, 12      

The greatest common factor of 18 and 24 is 6. Therefore, we should divide the numerator and the denominator by 6. 
 

\(\frac{18 \div 6}{24 \div 6} = \frac{3}{4} \)
 

The answer is = \(\frac{3}{4} \).

This is how we can reduce a fraction.

Variables such as a, b, c, x, y, and z are letters that is used in mathematical expressions to represent unknown values. Fractions can also include both numbers and variables. To simplify a fraction that contains variables, follow these steps:
 

Step 1: Combine all like terms in the numerator.
For example, in the fraction (\(8a - a + 2a\)) / (12a), the like terms are all terms with a. Simplifying the numerator gives 9a, so the fraction becomes 9a / 21a.

Step 2: Factor both the numerator and the denominator, then cancel the common factors. 9a / 12a = (33a) / (34a). After canceling the common 3 and a, the simplified fraction is: 3/4.

We reduce fractions to their simplest form so that it becomes easier for us to understand and compare. There are a few methods we can use, and we will be looking at 3 of them here:

• Equivalent Fractions Method.
   
• GCF Method.
   
• Prime Factorization Method.

Equivalent fractions represent the same value, even though their numerators and denominators look different. We can use this idea to reduce a fraction to its simplest form. Here are a few steps to reduce a fraction using the equivalent fractions method:
 

Step 1: Identify a common factor of both the numerator and the denominator.

Step 2: Divide the numerator and denominator by that common factor.

Step 3: Continue dividing by common factors until the only common factor left is 1.

For example,

Reduce 45/74 using the equivalent fractions method.

Step 1: 45 and 75 are both divisible by 15.

Divide the numerator and denominator by 15.

45 ÷15 = 3, 75 ÷ 15 = 5

So we get,

45 / 75 = 3 / 5
 

Step 2: Now, check if 3 and 5 have any common factor other than 1. They do not. So, the fraction is already in its simplest form.

The GCF (Greatest Common Factor) of two or more numbers is the largest number that divides all the given numbers exactly. We can use this to quickly reduce a fraction. The steps for reducing a fraction using the GCF method are:
 

Step 1: Find the greatest common factor of the numerator and the denominator.

Step 2: Divide both the numerator and the denominator by this GCF. The result is the fraction in its simplest form.

For example,

Reduce 32 / 56 using the GCF method.
 

Step 1: Find the GCF of 32 and 56

Here, the factor of 32 is 1, 2, 4, 8, 16, 32

Factors of 56 is 1, 2, 4, 7, 8, 14, 28, 56

The greatest common factor is 8.

Step 2: Divide the numerator and denominator by the GCF

32 ÷ 8 = 4, 56 ÷ 8 = 7

So, 

32 / 56 = 4 / 7

Prime factorization means breaking a number into the product of its prime numbers. We can use this method to reduce fractions easily. The steps for reducing a fraction using the prime factorization method are as follows.
 

Step 1: First Write the numerator and the denominator as a product of their prime factors.

Step 2: Next, cross out the prime factors that appear in both the numerator and the denominator.

Step 3: Multiply the remaining factors in the numerator and denominator to get the reduced fraction.

For example,

Reduce 18/30 using prime factorization.

Step 1: Prime factorize the numerator and denominator

18 = 2 × 3 × 3

30 = 2 × 3 × 5

Step 2: Cancel out the common prime factors

Common factors: 2 and 3

Now, cancel them.
 

\(\ \frac{2 \times 3 \times 3}{2 \times 3 \times 5} \ \)
 

After canceling 2 and 3, we are left with,

 3/5.
 

So, 18/30 = 3/5

We already know how to place whole numbers on a number line. In the same way, we can also show fractions and identify fractions using a number line. Follow the steps below.
 

Step 1: First, draw 6 number lines, each with two whole numbers marked at the ends.

Step 2: Divide each number line into equal sections. For example, the first line is divided into 2 equal parts, showing the fraction 1/2. The second line is divided into 3 equal parts, showing 1/3 and 2/3. Continue this pattern for all the number lines, marking the fractions on each one.

Step 3: Compare the distance of each fraction from zero. Fractions that reach the same point on the number line are equivalent. For example, 1/2, 2/4, 3/6, and 4/8 all land at the same distance from 0, so they are equivalent fractions. Similarly, 1/3 and 2/6 represent the same position, so they are equivalent as well.

Step 4: By observing the positions of fractions on the number line, we can easily identify which fractions are equivalent.

Variables are usually represented by letters like a, b, c, x, y, z, etc. They represent changeable or unknown values.

Step 1: Factor both the numerator and the denominator, including variables.

Step 2: Avoid all common factors from the numerator and the denominator.

Step 3: The resultant is in simple form.

An example will help us understand better: 

Find the simplest form of the fraction \(\frac{6x^2y}{3xy^2} \)

Step 1: Numerator = 6x²y

The prime factorization of 6 is \(2 × 3\). 

Therefore, \(6 = 2 × 3\)

Similarly, the prime factorization of x² is x × x 

So,\( x^2 = x × x \)

y can be written as it is.

In other words, \(6x^2y = 2 × 3 × x × x × y.\)

Let’s do the same in the denominator.

Denominator = 3xy²

Since 3 is a prime number, it cannot be factorized further. So write 3 as it is. 

\(x = x\)

\(y^2 = y × y \)

So \(3xy^2 = 3 × x × y × y\)

Step 2: Let’s rewrite the fraction and then cancel out the common factors.

\(6x^2y = 2 × 3 × x × x × y\) and \(3xy^2 = 3 × x × y × y \)

Therefore, the fraction is \(\frac{2 \times 3 \times x \times x \times y}{3 \times x \times y \times y} \)

Now, canceling out common factors (x, y, 3)

x → \(\frac{2 \times x \times y}{y \times y} \)

y → \(\frac{2x}{y} \), we get: 

\(\frac{2x}{y} \).

A fraction with variables raised to powers can be simplified by applying the rules of exponents. The rule is applied to both the numerator and denominator until the fraction is reduced to its simplest form. 

The key rule is:\(\frac{a^m}{a^n} = a^{m-n} \) where a is not equal to 0. 

Example: \(\frac{x^5}{x^2} = x^{5-2} \)

The final result = x³

Sometimes, reducing fractions and the different methods involved can seem difficult. That’s why we have some tips and tricks to help reduce the difficulties.

• Find the greatest common factor in both the numerator and the denominator. 

• Use prime factorization to break down a large number.
   
• Don’t forget to reduce numbers and variables.
   
• Use the exponent rule carefully while applying.
   
• Always check that the fraction is already in its simplest form.
   
• Parents can encourage children to use blocks, coins, or paper pieces to group and reduce fractions.
   
• Teachers can teach divisibility rules early to help children simplify fractions faster.
   
• Children can practice using snacks or toys to visualize fractions.
   

Reduce fractions have applications in real life too. From cooking, traveling, and shopping. Fractions are a little hard to understand, but they make smart decisions.
 

1. In cooking: While cooking, you need to know how to reduce fractions to simplify the measurements of the ingredients. Reducing fractions are also used to scale the recipes up or down depending on the requirement. For example, if a recipe for 4 people says \(\frac{2}{3} \) cup of sugar, then while cooking for 2 people, we need to divide \(\frac{2}{3} \) by 2. So,\(\frac{2}{3} \div 2 = \frac{2}{6} \). Now, reducing the fraction further, we get \(\frac{1}{3} \). Now we know that we must add \(\frac{1}{3} \) cup of sugar for 2 people.
 

2. While sharing food: If you cut a pizza into 12 slices, and you eat 6 pieces. So you eat \(\frac{6}{12} \) of the pizza, which means \(\frac{6}{12} \) is equal to \(\frac{1}{2} \). In short, you eat half a pizza.
 

3. In traveling: A group is going for a 60 km trip. You drive 30 km among the 60 km. So it can be considered as \(\frac{30}{60} \)0. It is \(\frac{1}{2} \) of the distance. So you covered half the distance. 
 

4. In shopping: If an item costs 50₹. If you buy an item for 25₹ instead of 50₹, the fraction representing the price is \(\frac{25}{50} = \frac{1}{2} \).

5. Time management: If you take 20 minutes out of 60 minutes, then the fraction is \(\frac{20}{60} \). Reducing it, we get \(\frac{1}{3} \), meaning you used one-third of the time.