Problem 14 - Elementary Algebra

Question

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{5}{7} ; 42 $$

Step-by-Step Solution

Answer

The equivalent expression is \( \frac{30}{42} \).

Step 1 - Understand the Goal

Rewrite the fraction \( \frac{5}{7} \) so that it has a denominator of 42.

Step 2 - Find the Multiplication Factor

Determine what number 7 must be multiplied by to get 42. This is calculated by $$\frac{42}{7} = 6$$.

Step 3 - Adjust the Numerator

Multiply the numerator (5) by the same factor (6) to keep the fraction equivalent. \( 5 \times 6 = 30 \).

Step 4 - Rewrite the Fraction

Combine the new numerator and the given denominator to create the equivalent fraction: \( \frac{30}{42} \).

Key Concepts

Denominators

In math, when working with fractions, it is important to understand the role of the denominator. The denominator is the bottom number in a fraction. It shows into how many equal parts the whole is divided. For example, in the fraction \( \frac{5}{7} \), the denominator is 7, which means the whole is divided into 7 equal parts.
Sometimes you may need to rewrite a fraction with a different denominator. This process is known as finding an equivalent fraction. By finding equivalent fractions, you can compare, add, or subtract fractions more easily. In our example, we want to rewrite \( \frac{5}{7} \) to have a denominator of 42. Remember, while changing the denominator, the value of the fraction itself remains unchanged.

Multiplication Factor

When changing the denominator of a fraction, we use a multiplication factor. The multiplication factor is simply the number you multiply the original denominator by to get the new denominator. To find the multiplication factor, divide the new denominator by the old denominator. For our example, we need to change the denominator from 7 to 42.
We calculate the multiplication factor by \( \frac{42}{7} = 6 \). This factor ensures that the fractions will be equivalent, meaning they represent the same part of a whole, even though they look different.
The idea is to multiply both the numerator and the denominator by this same multiplication factor to maintain the fraction's value. It's like scaling the fraction up or down while keeping its essence the same.

Numerators

A fraction's numerator is the top number, indicating how many parts of the whole you're considering. In our original fraction, \( \frac{5}{7} \), the numerator is 5. When you change the denominator using a multiplication factor, you must also adjust the numerator.
To keep the fraction equivalent, multiply the numerator by the same factor you used for the denominator. For our exercise, with a multiplication factor of 6, we calculate the new numerator as follows:
\( 5 \times 6 = 30 \).
Now, using our new numerator and the specified denominator, we rewrite the fraction as \( \frac{30}{42} \). This way, we ensured that the original fraction \( \frac{5}{7} \) and the new fraction \( \frac{30}{42} \) represent the same value.