Question

\(\frac{6}{42}\)

Short answer

\(\frac{6}{42} = \frac{1}{7}\) when simplified.

Step-by-step solution

Identify the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{6}{42}\), we need to determine the greatest common divisor (GCD) of the numerator (6) and the denominator (42). The GCD is the largest positive integer that divides both numbers without leaving a remainder. The factors of 6 are 1, 2, 3, and 6. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The largest factor they both share is 6.

Simplify the Fraction

Divide both the numerator and the denominator by their GCD, 6. This results in: \[ \frac{6 \div 6}{42 \div 6} = \frac{1}{7} \] So, \(\frac{6}{42}\) simplifies to \(\frac{1}{7}\).

Key concepts

Greatest Common Divisor

When simplifying fractions, the greatest common divisor (GCD) is very important. It's the largest number that divides both the numerator and the denominator without any remainder. For instance, consider the fraction \( \frac{6}{42} \). First, we find the factors of both numbers:

• The factors of 6 are 1, 2, 3, and 6.
• The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

We can see that the largest factor they share is 6. Calculating the GCD is crucial since it tells us how much we can "shrink" both numbers, simplifying the fraction properly. Using the greatest common divisor ensures that we reduce the fraction to its simplest form.

Numerator and Denominator

Understanding the terms "numerator" and "denominator" is key to working with fractions effectively. In the fraction \( \frac{6}{42} \), the number 6 is the numerator, and the number 42 is the denominator.The numerator indicates how many parts of a whole we have, while the denominator shows the total number of equal parts the whole is divided into.Here are some key points about numerators and denominators:

• The numerator is always on top.
• The denominator is always on the bottom.
• A larger denominator means smaller parts of the whole.

Knowing these terms helps in the step-by-step reduction of fractions and strengthens overall fraction comprehension.

Simplifying Fractions

Simplifying fractions involves reducing them to their smallest possible form, using the greatest common divisor (GCD). Here, we'll demonstrate how to simplify \( \frac{6}{42} \).To do this, find the GCD of the numerator (6) and the denominator (42). As previously calculated, the GCD is 6.Next, divide both the numerator and the denominator by this GCD:\[\frac{6 \div 6}{42 \div 6} = \frac{1}{7} \]And there you have it, the fraction \( \frac{6}{42} \) simplifies to \( \frac{1}{7} \). This means we have reduced the fraction to its simplest form, where the only common factor of the numerator and the denominator is 1. Hence,

• Simplifying fractions means fewer parts to consider.
• It makes numerical comparisons and calculations more straightforward.

Awareness of this process is essential, especially when working with fractions in various math problems.