Problem 78 - Elementary Algebra

Question

For exercises \(75-80\), rewrite the fraction as an equivalent fraction with the given denominator. $$ \frac{5}{6} ; 42 $$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{35}{42}\).

Step 1: Understand the Problem

Given a fraction \(\frac{5}{6}\) and a new denominator 42, the objective is to find an equivalent fraction with 42 as the denominator.

Step 2: Determine the Multiplication Factor

To find the equivalent fraction, first determine what number must be multiplied to 6 to get 42. This can be found by dividing 42 by 6: \[ \frac{42}{6} = 7 \]

Step 3: Apply the Multiplication Factor

Both the numerator and the denominator of \(\frac{5}{6}\) must be multiplied by the same number to keep the fraction equivalent. So, multiply both the numerator and the denominator by 7: \[ \frac{5 \times 7}{6 \times 7} = \frac{35}{42} \]

Step 4: Write the Final Answer

The equivalent fraction with 42 as the denominator is \(\frac{35}{42}\).

Key Concepts

Fractions

Fractions are a way of representing parts of a whole. They consist of two parts: a numerator and a denominator. The numerator is the top number and shows how many parts we have. The denominator is the bottom number and shows how many equal parts the whole is divided into. For example, in \(\frac{5}{6}\), 5 is the numerator, and 6 is the denominator. Understanding fractions is essential in many math concepts, including finding equivalent fractions, adding, subtracting, multiplying, and dividing them. Equivalent fractions may look different, but they represent the same value. For instance, \(\frac{5}{6}\) is equivalent to \(\frac{35}{42}\). This means the same part of a whole is represented in each case.

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. Finding the LCM is helpful in tasks such as adding, subtracting, and comparing fractions. For example, to find the LCM of two denominators, you can list the multiples of each and find the smallest number they share. In our exercise, to change \(\frac{5}{6}\) to an equivalent fraction with a denominator of 42, we find that 42 is a multiple of 6. By dividing 42 by 6, we discover the multiplication factor: \(\frac{42}{6} = 7\). This factor helps us convert the fraction to an equivalent one with the new denominator.

Numerator and Denominator

The terms numerator and denominator are crucial in understanding fractions. The numerator is the top number of the fraction and represents how many parts of the whole we have or are considering. The denominator is the bottom number and indicates into how many equal parts the whole is divided. For example, in the fraction \(\frac{5}{6}\), 5 is the numerator and 6 is the denominator. To find an equivalent fraction with a different denominator:

• First, determine what number to multiply the denominator by to get the new denominator.
• Second, multiply both the numerator and the denominator by this number.

Using our example:
\(\frac{5}{6}\) and the new denominator is 42. We multiply both parts of the fraction by 7 (since \(\frac{42}{6} = 7\)). This makes the fraction \(\frac{35}{42}\), which is an equivalent fraction to the original \(\frac{5}{6}\).