Problem 76 - Elementary Algebra

Question

For exercises \(75-80\), rewrite the fraction as an equivalent fraction with the given denominator. $$ \frac{2}{7} ; 56 $$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{16}{56}\).

Step 1: Identify the current and new denominators

The original fraction is \(\frac{2}{7}\). The target denominator is 56.

Step 2: Find the multiplication factor

Determine the factor by which the original denominator (7) must be multiplied to get the new denominator (56). This is found by dividing the new denominator by the old denominator: \(\frac{56}{7} = 8\).

Step 3: Adjust the numerator

Multiply both the numerator and the denominator of the original fraction by this factor (8): \(\frac{2 \times 8}{7 \times 8} = \frac{16}{56}\).

Step 4: Write the equivalent fraction

Thus, the equivalent fraction of \(\frac{2}{7}\) with a denominator of 56 is \(\frac{16}{56}\).

Key Concepts

fractions

Fractions are a way to represent parts of a whole. They consist of two numbers separated by a slash. The number on top is called the numerator, and the number on the bottom is called the denominator. For example, in the fraction \(\frac{2}{7}\), 2 is the numerator, and 7 is the denominator.
Fractions can describe any part of a whole. If you have a fraction \(\frac{2}{7}\), it means you have 2 parts out of a total 7.Understanding fractions is crucial, as they are used in various aspects of mathematics and everyday life, like cooking or measuring.
Not all fractions look the same, but they can represent the same amount. These are called equivalent fractions.

denominator

The denominator of a fraction is the number at the bottom. It shows how many equal parts the whole is divided into. For example, in \(\frac{2}{7}\), 7 indicates the whole is divided into 7 parts.
The denominator plays a critical role in understanding the size of the fraction. Changing the denominator can change the fraction's value.
To create an equivalent fraction, we often change the denominator to a number we want, like 56 in this example. The goal is to make a new fraction that represents the same part of the whole.

multiplication factor

To find an equivalent fraction with a different denominator, we need a tool called the multiplication factor.
This factor helps us adjust both the numerator and the denominator, so the value of the fraction remains the same.
For instance, if we need the denominator to be 56 for \(\frac{2}{7}\), we find the factor by dividing 56 by 7, which gives us 8.
So, we multiply both the numerator and the denominator by 8: \(\frac{2 \times 8}{7 \times 8} = \frac{16}{56}\). This keeps the fraction balanced and equivalent to the original.
The multiplication factor is essential for calculating equivalent fractions without changing their value.