Question

Find the simplest form of \[\dfrac{69}{92}\].

Answer 1

Hint: Assume the simplest form of the given fraction as ‘E’. Now, use the method of prime factorization to write the numbers present in the numerator and denominator as the product of their prime factors. Check if there are any common factors or not, if there are common factors then cancel them and take the product of remaining factors to get the answer. If there are no common factors then we will say that we will say that the given fraction is already in simplified form.

Complete step-by-step answer:
Here, we have been provided with the fraction \[\dfrac{69}{92}\] and we are asked to write it in its simplest form. That means we have to cancel the common factors of both the numerator and denominator if present. To find whether they have common factors or not we use the prime factorization method to write the numbers as a product of their primes.
Now, in mathematics, prime factorization is a method of writing a number as the product of its prime factors. Generally, a large composite number is written as the product of its prime to find its root or used in calculations.
Let us come to our question, we have the fraction \[\dfrac{69}{92}\]. Let us assume this fraction ass E, so we have,
\[\Rightarrow E=\dfrac{69}{92}\]
Here, the numerator 69 can be written as \[69=3\times 23\] as the product of its prime factors. Also, the denominator 92 can be written as \[92=2\times 2\times 23\] as the product of its primes. So, we have,
\[\Rightarrow E=\dfrac{3\times 23}{2\times 2\times 23}\]
Clearly, we can see that 23 is the common factor, so cancelling 23 from both the numerator and denominator, we get,
\[\Rightarrow E=\dfrac{3}{2\times 2}\]
Now, on common factors are there, so simplifying the above fraction we get,
\[\Rightarrow E=\dfrac{3}{4}\]
Hence, \[\dfrac{3}{4}\] is the simplified form of the given expression.

Note: One must know that process of finding the prime factors of a number because we use them in finding L.C.M., H.C.F., square root, cube root of numbers provided. Here, if you want to determine the H.C.F. then you may see that we have cancelled the factor 23 and other than that there are no common factors, so the H.C.F. is 23. It would be difficult for us to select numbers randomly and check if they divide the provided numbers in the fraction or not and that is why the method of prime factorization is used.