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When we talk about equivalent fractions, one key step to remember is multiplying the denominator. In our example, you start with the fraction \(\frac{2}{3}\) and need to express it with a new denominator of 21. To do this, check how the denominator changes from 3 to 21. You accomplish this by division — divide the new denominator by the original denominator: \(\frac{21}{3} = 7\). The result gives you the multiplier. This tells you that by multiplying 3 by 7, you get 21. So, you'll use this multiplier to adjust both the numerator and the denominator for creating an equivalent fraction.

Always remember, multiplication of the denominator assists in scaling the fraction up or down, but the fraction value as a whole must remain unchanged. This step is the foundation of finding equivalent fractions.

Having determined the multiplier through denominator multiplication, the next step is to adjust the numerator. This is important as it ensures that the new fraction is equivalent to the original. In the original fraction \(\frac{2}{3}\), the numerator is 2. Since your multiplier is 7, do the following calculation: \(2 \times 7 = 14\).

This operation adjusts the top half of your fraction proportionally to the change in the denominator. The aim is to maintain the overall value of the fraction. Since multiplying both the numerator and the denominator by the same number (7) keeps the fraction equivalent, the calculation confirms that the new numerator is 14.

It’s crucial to apply the same multiplier to both the numerator and denominator for the fraction to retain its original value.

Once an equivalent fraction is formed, it is sometimes necessary to simplify it. Simplification means rewriting the fraction with the smallest possible whole numbers while retaining its value. In our example, the resultant fraction is \(\frac{14}{21}\). To simplify this, find the greatest common divisor (GCD) of 14 and 21, which in this case is 7.

• Divide both 14 and 21 by their GCD: \(\frac{14 \div 7}{21 \div 7} = \frac{2}{3}\).
• The simplified form of \(\frac{14}{21}\) is indeed \(\frac{2}{3}\).

Though with the example provided, you find the equivalent fraction has come back to its original state during simplification, this process ensures that no matter the number you start with, the fraction is expressed in the simplest form. Simplified fractions are easier to interpret and compute in further mathematical operations.