6394 divided by 42 as a fraction

elenas club

2 min read

·

13-02-2025

8

0

Share

Understanding how to express division as a fraction is a fundamental math skill. This article will guide you through solving 6394 divided by 42 as a fraction, explaining the process clearly and concisely. We'll cover the steps involved and offer some helpful tips.

Converting Division to a Fraction

The core concept is straightforward: division can always be represented as a fraction. The dividend (the number being divided) becomes the numerator (the top number), and the divisor (the number you're dividing by) becomes the denominator (the bottom number).

Therefore, 6394 divided by 42 can be written as the fraction:

6394/42

Simplifying the Fraction

This fraction, however, can be simplified. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers evenly.

Finding the GCD can be done through several methods, including prime factorization or using the Euclidean algorithm. For this example, let's use prime factorization:

• Prime factorization of 6394: 2 x 7 x 456 + 2 = 2 x 7 x 2 x 228 +2 = 2 x 7 x 2 x 2 x 114 +2= 2 x 7 x 2 x 2 x 2 x 57 = 2³ x 7 x 57
• Prime factorization of 42: 2 x 3 x 7

Now, let's identify the common factors: both 6394 and 42 share factors of 2 and 7.

Therefore, the GCD of 6394 and 42 is 2 x 7 = 14

Performing the Simplification

To simplify the fraction, we divide both the numerator and the denominator by the GCD (14):

6394 ÷ 14 = 456.714... 42 ÷ 14 = 3

This reveals that the fraction 6394/42 simplifies to 456.714.../3 which is not a whole number. There must be an error in the prime factorization of 6394. Let's re-examine.

Correct Prime Factorization of 6394: It appears there was an error in the initial prime factorization. Let's use a calculator or online tool to find the prime factors. The correct prime factorization of 6394 is 2 x 7 x 456 + 2. Further prime factorization yields: 2 x 7 x 2 x 228 = 2³ x 7 x 2 x 114 = 2⁴ x 7 x 2 x 57 = 2⁵ x 7 x 3 x 19

Let's re-attempt simplification using the Euclidean algorithm to ensure accuracy.

Euclidean Algorithm:

1. Divide 6394 by 42: 6394 = 42 * 152 + 10
2. Divide 42 by the remainder 10: 42 = 10 * 4 + 2
3. Divide 10 by the remainder 2: 10 = 2 * 5 + 0

The last non-zero remainder is 2. Therefore, the GCD of 6394 and 42 is 2.

Now, let's simplify the fraction:

6394 ÷ 2 = 3197 42 ÷ 2 = 21

Therefore, the simplified fraction is 3197/21

Conclusion

Through a step-by-step process involving the identification of the greatest common divisor and subsequent simplification, we have determined that 6394 divided by 42 expressed as a fraction is 3197/21. This simplified fraction accurately represents the division problem. Remember to always double-check your work, especially when dealing with larger numbers, to ensure accuracy.