Problem 78 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator \(10 .\) $$\frac{1}{2}$$

Step-by-Step Solution

Answer

The equivalent fraction of \(\frac{1}{2}\) with denominator 10 is \(\frac{5}{10}\).

Step 1: Understanding Equivalent Fractions

An equivalent fraction is a different fraction that represents the same quantity or value. When two fractions are equivalent, they are equal.

Step 2: Identify the Required Denominator

The problem asks for the fraction \(\frac{1}{2}\) to have a denominator of 10. Thus, we need to convert \(\frac{1}{2}\) into a fraction with 10 as the denominator.

Step 3: Determine the Multiplier for the Denominator

Analyze what number we can multiply the denominator 2 by to make it 10. Since 2 times 5 equals 10, the multiplier is 5.

Step 4: Multiply the Numerator and Denominator by the Same Number

To keep the fraction equivalent, multiply both the numerator and denominator of \(\frac{1}{2}\) by 5. This gives us \(\frac{1 \times 5}{2 \times 5} = \frac{5}{10}\).

Step 5: Verify the Equivalent Fraction

Check that \(\frac{5}{10}\) is an equivalent fraction to \(\frac{1}{2}\). Simplifying \(\frac{5}{10}\) results in \(\frac{1}{2}\), verifying they are equivalent.

Key Concepts

Converting Fractions

Converting fractions is a fundamental concept in mathematics that allows you to change the appearance of a fraction without altering its value. This process involves rewriting a fraction so that it has a different denominator, often specified in a problem, while maintaining its equivalence.

To convert fractions successfully, one must find a common or desired denominator. For instance, when converting \(\frac{1}{2}\) to have a denominator of 10, you must determine how you can transform the original denominator of 2 into 10.

This leads us to the process of determining the appropriate multiplier, which we will discuss in more detail later. Remember, the essence of converting fractions is to find a new fraction that keeps the same value but has a denominator that fits the requirement.

Multiplying Fractions

Multiplying fractions is a key step in converting to equivalent fractions with a different denominator. Once you identify the multiplier needed to adjust the denominator, you apply it uniformly to both the numerator and the denominator of the fraction.

Consider \(\frac{1}{2}\) needing a denominator of 10. Calculate what number multiplied by 2 gives 10, which is 5 in this case. Then, to keep the fractions equivalent, multiply both the numerator (which is 1) and the denominator (which is 2) by this multiplier.

This multiplication results in \(\frac{1 \times 5}{2 \times 5} = \frac{5}{10}\). This ensures the fraction maintains its original value while complying with the demand for a new denominator.

Denominator Adjustment

Adjusting the denominator is a crucial skill for achieving equivalent fractions. It involves carefully choosing a multiplier for the existing denominator in order to reach the targeted denominator.

For example, in the exercise of converting \(\frac{1}{2}\) to have 10 as the denominator, you need to determine a multiplication factor: 2 must be multiplied by 5 to get 10. This factor is not chosen randomly but is precisely calculated to ensure accuracy in the new equivalent fraction.

Denominator adjustment requires careful calculation but results in a valid equivalent fraction when done correctly. It ensures that both the original and converted fractions reflect the same portion or quantity, in this case, turning \(\frac{1}{2}\) into \(\frac{5}{10}\), which is verified through simplification back to \(\frac{1}{2}\).