Problem 13 - Elementary Algebra

Question

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{3}{7} ; 56 $$

Step-by-Step Solution

Answer

The equivalent expression is \ \( \frac{24}{56} \).

Step 1: Identify the Given Expression and New Denominator

The given expression is \ \( \frac{3}{7} \) and the new denominator to achieve is 56.

Step 2: Determine the Multiplication Factor

To change the denominator from 7 to 56, determine the factor needed. Divide 56 by 7. \ \( 56 \div 7 = 8 \). So, we need to multiply the numerator and the denominator of the given fraction by 8.

Step 3: Multiply Both Numerator and Denominator

Multiply 3 by 8 and 7 by 8 to get the equivalent fraction with a denominator of 56: \ \( \frac{3 \cdot 8}{7 \cdot 8} = \frac{24}{56} \).

Step 4: Rewrite the Expression

Rewrite the original fraction \ \( \frac{3}{7} \) as an equivalent fraction with a denominator of 56: \ \( \frac{24}{56} \).

Key Concepts

Denominator

The denominator is the number below the fraction line. It shows how many equal parts the whole is divided into. For example, in the fraction \(\frac{3}{7}\), 7 is the denominator. When we want to change the denominator, we are changing the number of parts we are dealing with, while keeping the total value the same.
Changing the denominator involves finding a common multiple or an equivalent number to scale the fraction up or down.

Multiplication Factor

The multiplication factor is the number used to multiply both the numerator and the denominator to find an equivalent fraction. In our given exercise, we needed to change the denominator from 7 to 56. To do this, we find the multiplication factor by dividing the new denominator by the old denominator:
\[ 56 \div 7 = 8 \]
This means the multiplication factor is 8. We use this factor to multiply both the top number (numerator) and the bottom number (denominator) of the fraction. This keeps the value of the fraction the same but changes its form.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. They are found by multiplying or dividing both the numerator and denominator by the same number.
For instance, \(\frac{3}{7}\) and \(\frac{24}{56}\) are equivalent fractions. Even though they look different, they have the same value because:
\[ \frac{3}{7} = \frac{3 \cdot 8}{7 \cdot 8} = \frac{24}{56} \]
Multiplying the numerator and denominator by the same number does not change the overall value of the fraction, it simply scales it up or down.

Numerator

The numerator is the number above the fraction line. It tells us how many parts of the whole we have. In the original fraction \(\frac{3}{7}\), 3 is the numerator. When converting to an equivalent fraction, the numerator changes in proportion to the multiplication factor used for the denominator.
For example, to match the new denominator of 56 using the factor of 8, we multiply the numerator by 8:
\[ 3 \cdot 8 = 24 \]
This gives us the new fraction \(\frac{24}{56}\), which is equivalent to the original fraction.