Reduce Fractions

For reducing fractions, first find GCF to divide both numerator and 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 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. 

18÷6/24÷6 = 3/4 

The answer is 18/24 = 3/4.

This is how we can reduce a fraction.

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.

This method involves finding the common factors of the numerator and the denominator. The second part of this method is to divide both the numerator and the denominator by a common factor. 

Example: 36/60 (36 and 60 are divisible by 6)

We can now divide the numerator and the denominator by 6
36 ÷ 6/60 ÷ 6 = 6/10

Repeat the process until there are no other common factors left, except 1. 

6/10 (6 and 10 can be divisible by 2).

6 ÷ 2/10÷ 2 = 3/5 This result had no other common factor than 1.

GCF or Greatest Common factor is the method to find out the common factor of numerator and denominator and divide them by the common factor. To find the GCF of the numerator 18 and denominator 24 in this example 18/24.

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

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24

The GCF is 6.

Dividing both the numerator and the denominator by 6.

18 ÷ 6 = 3

24 ÷ 6 = 4

Reduced fraction is 34.

Prime factorization is a method of breaking a number into a product of prime factors. 

Step 1: We have to find the prime factorization of the numerator and the denominator.

Step 2: Cancel the common prime factors of the numerator and the denominator.

Step 3: Multiply the remaining factors to get the shortest and simplest form of the fraction.

Example: Find the reduced factor of 6/24.

62/4 = 2 × 3/2 × 3 × 2 × 2

Cutting all common factors and taking the remaining factors as we mentioned in the above steps. Canceling the common factors, we will be left with 1/2 × 2. Therefore, we can conclude that 6/24 = 1/4.

Variables are usually represented by English alphabets 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 (6x ²y) /(3xy2)

Step 1: Numerator = 6x ²y

The prime factorization of 6 is 2 × 3. 

Therefore, 6 = 2 × 3

Similarly, the prime factorization of x 2 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 = 3xy ²

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 2 × 3 × x × x × y/3 × x × y × y

Now, canceling out common factors (3, x, and y), we get: 

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: a ^m/a ⁿ = a ^m-n where a is not equal to 0. 

Example: x ⁵/x ²  = x ^5-2 = x ³

The final result = x ³

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.

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

2. Use prime factorization to break down a large number.

3. Don’t forget to reduce numbers and variables.

4. Use the exponent rule carefully while applying.

5. Should always check that the given factor is already in a simple form.

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 2/3 cup of sugar, then while cooking for 2 people, we need to divide 2/3 by 2. So, 2/3 ÷ 2 = 2/6. Now, reducing the fraction further, we get 1/3. Now we know that we must add 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 6/12 of the pizza, which means 6/12 is equal to 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 30/60. It is ½ of the distance. So you covered half the distance. 

4. In shopping: If an item costs 50₹. You get that at a cost of 25₹, then the fraction is 25/50, it is equal to 1/2. So you get a reduction of 50% on the item.

5. Time management: If you take 20 minutes out of 60 minutes, then the fraction is 20/60. Keep reducing these fractions we get ⅓. Which means you take one-third of the time.