what is 162/144 simplified fraction

2 min read 26-12-2024

The fraction 162/144 represents a part of a whole. To understand its simplest form, we need to simplify it. This means finding the greatest common divisor (GCD) of both the numerator (162) and the denominator (144), and then dividing both by that GCD. Let's break down the process.

Finding the Greatest Common Divisor (GCD)

The GCD is the largest number that divides evenly into both 162 and 144 without leaving a remainder. There are several ways to find the GCD. One common method is using prime factorization.

Prime Factorization Method

Prime factorization of 162: We can break 162 down into its prime factors: 2 x 3 x 3 x 3 x 3 = 2 x 3⁴
Prime factorization of 144: Similarly, 144's prime factorization is: 2 x 2 x 2 x 2 x 3 x 3 = 2⁴ x 3²

Now, we identify the common prime factors and their lowest powers:

Both numbers share two factors of 3 (3² = 9).
Both numbers share one factor of 2 (2¹ = 2).

Therefore, the GCD of 162 and 144 is 2 x 3² = 2 x 9 = 18.

Euclidean Algorithm Method

Another efficient method is the Euclidean Algorithm. This involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD.

Divide 162 by 144: 162 = 144 x 1 + 18
Divide 144 by the remainder 18: 144 = 18 x 8 + 0

Since the remainder is 0, the GCD is the last non-zero remainder, which is 18.

Simplifying the Fraction

Now that we've found the GCD (18), we can simplify the fraction:

162 ÷ 18 = 9 144 ÷ 18 = 8

Therefore, the simplified fraction is 9/8.

Representing the Fraction

The simplified fraction 9/8 is an improper fraction because the numerator (9) is larger than the denominator (8). This can also be expressed as a mixed number:

9 ÷ 8 = 1 with a remainder of 1.

So, 9/8 can also be written as 1 1/8.

Conclusion

By finding the greatest common divisor of 162 and 144, we successfully simplified the fraction 162/144 to its simplest form, which is 9/8 or 1 1/8. Both representations are correct, depending on the context. Understanding the process of finding the GCD is key to simplifying fractions effectively.