Problem 34 - Prealgebra

Question

Write each of the following fractions as an equivalent fraction with denominator 6. $$\frac{65}{78}$$

Step-by-Step Solution

Answer

The equivalent fraction with denominator 6 is \( \frac{5}{6} \).

Step 1: Find the Greatest Common Divisor (GCD)

To express \( \frac{65}{78} \) with the denominator 6, first reduce it to its simplest form by finding the Greatest Common Divisor (GCD) of 65 and 78. The GCD is the largest number that divides both numbers without leaving a remainder. In this case, the GCD is 13.

Step 2: Simplify the Fraction

Now divide both the numerator and the denominator of \( \frac{65}{78} \) by their GCD, 13. This simplifies the fraction to the smallest equivalent form: \( \frac{65 \div 13}{78 \div 13} = \frac{5}{6} \).

Step 3: Adjust the Denominator

Notice that the simplified fraction \( \frac{5}{6} \) already has the desired denominator of 6. Therefore, \( \frac{65}{78} \) is equivalent to \( \frac{5}{6} \) without further adjustments.

Key Concepts

Simplification of fractions

When it comes to fractions, simplification is a handy tool that makes numbers more manageable and comparisons easier. Simplifying a fraction means reducing it to its smallest, simplest form.
To simplify a fraction, you need to find a common factor between the numerator (the top number) and the denominator (the bottom number).
The simplest form of a fraction is achieved when both the numerator and the denominator cannot be divided by any number other than 1.

• Start by identifying a number that divides both the numerator and the denominator without leaving a remainder.
• This number is known as the Greatest Common Divisor (GCD).
• Once identified, divide both parts of the fraction by this GCD.

In our exercise, simplifying \( \frac{65}{78} \) resulted in \( \frac{5}{6} \), showing us the smallest form of the original fraction.

Equivalent fractions

Equivalent fractions are like different representations of the same portion or amount. They may look different but actually express the same value.
This concept is especially important when comparing or adding fractions.
To find equivalent fractions:

• Multiply or divide both the numerator and the denominator by the same number.
• This does not change the overall value of the fraction.
• For example, \( \frac{1}{2} \) is equivalent to \( \frac{2}{4} \) and \( \frac{3}{6} \).

In our exercise, simplifying \( \frac{65}{78} \) down to \( \frac{5}{6} \) was key to find an equivalent fraction with a specific denominator. Being able to adjust fractions this way is very useful in both mathematical problems and real-life situations.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest integer that divides two numbers without leaving a remainder. Finding the GCD is fundamental when working with fractions, especially for simplification.
The GCD helps us reduce fractions by dividing both numerator and denominator by this greatest factor.

• To find the GCD, list all factors of each number.
• Identify the largest factor that appears in both lists.
• This factor is the GCD.

In the given exercise, the GCD of 65 and 78 was identified as 13. Dividing both numerator and denominator by 13 simplified \( \frac{65}{78} \) to \( \frac{5}{6} \). Knowing how to find the GCD fast-tracks the process of simplifying fractions efficiently.