Problem 5 - Fundamentals of Mathematics

Question

Determine if the pairs of fractions are equivalent. $$ \frac{3}{12}, \frac{1}{4} $$

Step-by-Step Solution

Answer

The fractions \( \frac{3}{12} \) and \( \frac{1}{4} \) are equivalent.

Step 1: Simplify the First Fraction

To determine if two fractions are equivalent, we should first simplify both fractions to their lowest terms. We'll start with the first fraction, \( \frac{3}{12} \). To simplify it, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 3 and 12 is 3. Dividing both the numerator and the denominator by 3 gives:\[\frac{3 \div 3}{12 \div 3} = \frac{1}{4}\]

Step 2: Compare with the Second Fraction

Now that we have simplified the first fraction to \( \frac{1}{4} \), we compare it to the second fraction \( \frac{1}{4} \). Since both fractions are now exactly \( \frac{1}{4} \), they are equivalent.

Step 3: Conclusion

Since both fractions simplify to \( \frac{1}{4} \), we can conclude that the fractions \( \frac{3}{12} \) and \( \frac{1}{4} \) are equivalent.

Key Concepts

Simplifying Fractions

To understand if two fractions are equivalent, an important step is simplifying each fraction. Simplifying means making the fraction as simple as possible by reducing it to its smallest possible numbers—also known as its "lowest terms."

• Take the fraction \( \frac{3}{12} \) as an example.
• We look for any common factors shared by both the numerator (the top number) and the denominator (the bottom number).

By dividing both the numerator and the denominator by the common factor, we can simplify the fraction. Most often, this factor is not immediately obvious, so finding the greatest one is key!
Reducing fractions makes comparison much easier as we'll see further ahead in this topic.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is crucial for simplifying fractions. It is the largest number that divides both the numerator and denominator without leaving a remainder. Finding it helps in reducing a fraction effectively.

• For the fraction \( \frac{3}{12} \), the GCD of 3 and 12 is 3.
• Dividing both by their GCD simplifies the fraction to \( \frac{1}{4} \).

To discover the GCD:1. List out the factors of both numbers.
2. Identify the largest factor common to both lists.
This process enables us to express the fraction in its simplest form, making calculations and comparisons easier and more accurate.

Lowest Terms

The term "lowest terms" refers to a fraction that cannot be simplified any further. A fraction in its lowest terms has only 1 as a common divisor for its numerator and its denominator other than itself.

• Consider the fraction \( \frac{1}{4} \) derived from \( \frac{3}{12} \).
• There are no numbers other than 1 that divide both 1 and 4.

This shows that \( \frac{1}{4} \) is in its lowest terms. When fractions are in their lowest terms, comparing them is straightforward, as any two reduced fractions can be easily matched for equivalency. This process of reducing to lowest terms by finding the GCD is essential for determining if fractions like \( \frac{3}{12} \) and \( \frac{1}{4} \) are equivalent.

Numerator and Denominator Comparison

Understanding fractions means recognizing the role of both the numerator and the denominator. When comparing fractions, ensure both parts have been simplified to the "lowest terms".
After simplifying, we easily compare numerators and denominators:

• For \( \frac{3}{12} \) simplified to \( \frac{1}{4} \), it is clear both are exactly the same as \( \frac{1}{4} \).
• Comparing simplified forms is straightforward, as numbers become smaller and clearer.

When both fractions result in identical numerators and denominators this means they are equivalent. It proves that despite different appearances, both fractions share the exact same value!