Problem 56 - Prealgebra

Question

Which of the fractions \(\frac{4}{9}, \frac{10}{15}, \frac{8}{12},\) and \(\frac{6}{12}\) do not reduce to \(\frac{2}{3} ?\)

Step-by-Step Solution

Answer

\(\frac{4}{9}\) and \(\frac{6}{12}\) do not reduce to \(\frac{2}{3}\).

Step 1: Analyze Fraction \(\frac{4}{9}\)

To reduce \(\frac{4}{9}\), identify the greatest common divisor (GCD) of the numerator 4 and the denominator 9. The GCD is 1, so the fraction \(\frac{4}{9}\) is already in its simplest form. Since it does not simplify to \(\frac{2}{3}\), \(\frac{4}{9}\) does not reduce to \(\frac{2}{3}\).

Step 2: Analyze Fraction \(\frac{10}{15}\)

Determine the GCD of the numbers 10 and 15. The GCD is 5. Simplify \(\frac{10}{15}\) by dividing both the numerator and the denominator by 5: \(\frac{10}{15} = \frac{2}{3}\). This fraction reduces to \(\frac{2}{3}\).

Step 3: Analyze Fraction \(\frac{8}{12}\)

Find the GCD for 8 and 12, which is 4. Divide both the numerator and denominator by 4 to simplify: \(\frac{8}{12} = \frac{2}{3}\). This fraction simplifies to \(\frac{2}{3}\).

Step 4: Analyze Fraction \(\frac{6}{12}\)

Calculate the GCD of 6 and 12. The GCD is 6. Simplifying \(\frac{6}{12}\) by dividing both terms by 6 gives: \(\frac{6}{12} = \frac{1}{2}\). This fraction does not reduce to \(\frac{2}{3}\).

Key Concepts

Greatest Common Divisor

When you want to reduce a fraction to its simplest form, understanding the greatest common divisor (GCD) is key. The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. To find the GCD, use these easy steps:

• List the factors of each number.
• Identify the common factors they share.
• The largest of these common factors is the GCD.

For instance, in the fraction \(\frac{10}{15}\), the factors of 10 are 1, 2, 5, and 10, while the factors of 15 are 1, 3, 5, and 15. The GCD is 5, making this the number used to simplify the fraction.Breaking fractions down using the GCD ensures you find their simplest form accurately. This is a fundamental concept in fraction reduction, paving the way for clearer and exact calculations.

Simplifying Fractions

Simplifying a fraction means rewriting it in its lowest possible terms. A fraction is in simplest form when the only number that divides both the numerator and the denominator is 1. Here's a quick guide to do this:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and denominator by this GCD.
• Rewrite the fraction with these new terms.

For example, to simplify \(\frac{8}{12}\), we find that the GCD is 4. Dividing both 8 and 12 by 4 gives us \(\frac{2}{3}\), which is the simplified form.This process helps make fractions easier to work with and understand. It can also reveal equivalent fractions and assist in comparing sizes of different fractions effectively.

Equivalent Fractions

Equivalent fractions look different but represent the same value or portion of a whole. Understanding equivalence is crucial for solving problems that involve comparing or simplifying fractions. Creating equivalent fractions is straightforward:

• Multiply or divide both the numerator and denominator by the same non-zero number.

For example, \(\frac{2}{3}\) is equivalent to \(\frac{4}{6}\) because multiplication by 2 doesn't change its value.To determine if two fractions like \(\frac{10}{15}\) and \(\frac{2}{3}\) are equivalent, check if they simplify to the same form. By dividing both 10 and 15 by 5, we also get \(\frac{2}{3}\), proving they are equivalent.This concept aids in understanding fraction comparisons and simplifications across various mathematical problems.