Problem 79 - Elementary Algebra

Question

For exercises \(75-80\), rewrite the fraction as an equivalent fraction with the given denominator. $$ \frac{4}{21} ; 63 $$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{12}{63}\).

Step 1: Understand the Problem

The goal is to rewrite the fraction \(\frac{4}{21}\) so that it has a new denominator of 63.

Step 2: Determine the Multiplier

To convert the denominator of the fraction from 21 to 63, find the factor by which 21 must be multiplied to get 63. Since \(\frac{63}{21} = 3\), the numerator and the denominator must be multiplied by 3.

Step 3: Multiply the Numerator and Denominator

Multiply both the numerator and the denominator of \(\frac{4}{21}\) by 3: \[ \frac{4 \times 3}{21 \times 3} = \frac{12}{63} \]

Step 4: Verify the Solution

Check that the new fraction \(\frac{12}{63}\) is equivalent to \(\frac{4}{21}\). Both fractions reduce to the same simplest form: \(\frac{4}{21} = \frac{4}{3 \times 7} = \frac{4}{21}\) and \(\frac{12}{63} = \frac{12}{3 \times 21} = \frac{4}{21}\). Thus, the solution is correct.

Key Concepts

fraction multiplication

When dealing with fractions, one key operation you'll frequently encounter is fraction multiplication. This skill allows you to change the fractional representation while keeping the value the same. Fraction multiplication involves multiplying the numerators and denominators of two fractions to get a new fraction. For example, if you want to convert \(\frac{4}{21}\) to a fraction with 63 as the denominator, you must multiply both the numerator and the denominator by 3. Hence, \(\frac{4 \times 3}{21 \times 3} = \frac{12}{63}\). This step ensures that the fraction remains equivalent but represented using a different denominator. Remember, always multiply both parts of the fraction by the same number to maintain the value.

denominator

The denominator of a fraction is the number below the line, indicating into how many equal parts the whole is divided. When converting between equivalent fractions, understanding the denominator's role is crucial. To find a new fraction with a different denominator, you typically multiply or divide both the numerator and the denominator by the same number. In our example, we had to rewrite \(\frac{4}{21}\) with a denominator of 63. We identified that multiplying 21 (the original denominator) by 3 gives us 63. Therefore, we also multiplied the numerator by 3, which brought us to \(\frac{12}{63}\). This way, the fractions are equivalent because the value they represent doesn't change. This process is fundamental in simplifying and converting fractions.

simplifying fractions

Simplifying fractions is the process of reducing them to their simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. In the example provided, \(\frac{12}{63}\) had to be checked to ensure it was equivalent to the original fraction \(\frac{4}{21}\). To verify this, both fractions were reduced to their simplest form. For \(\frac{12}{63}\), we divide the numerator and the denominator by their greatest common divisor (GCD), which is 3. Thus, \(\frac{12 \div 3}{63 \div 3} = \frac{4}{21}\). Similarly, for the given fraction, \(\frac{4}{21}\), the GCD is already 1, meaning it is simplified. Simplifying fractions ensures you and others can easily understand and compare them without extra computations.