Problem 57 - Prealgebra

Question

Write as equivalent fractions with denominator \(20 .\) a. \(\frac{1}{4}\) b. \(\frac{3}{5}\) c. \(\frac{9}{10}\) d. \(\frac{1}{10}\)

Step-by-Step Solution

Answer

The equivalent fractions are: \(\frac{5}{20}\), \(\frac{12}{20}\), \(\frac{18}{20}\), \(\frac{2}{20}\).

Step 1: Understand the Problem

We need to convert each given fraction to an equivalent fraction with a common denominator of 20.

Step 2: Convert Fraction a

To convert the fraction \( \frac{1}{4} \) to have a denominator of 20, find a number to multiply both the numerator and denominator by so that the denominator becomes 20. The least common multiple of 4 and 20 is 20, so multiply both by 5:\[\frac{1}{4} \times \frac{5}{5} = \frac{5}{20}\]

Step 3: Convert Fraction b

For the fraction \( \frac{3}{5} \), the common denominator is already 20. Multiply both the numerator and denominator by 4:\[\frac{3}{5} \times \frac{4}{4} = \frac{12}{20}\]

Step 4: Convert Fraction c

The fraction \( \frac{9}{10} \) is converted by making the denominator 20. Multiply both by 2:\[\frac{9}{10} \times \frac{2}{2} = \frac{18}{20}\]

Step 5: Convert Fraction d

For the fraction \( \frac{1}{10} \), multiply both the numerator and denominator by 2 to get the denominator to 20:\[\frac{1}{10} \times \frac{2}{2} = \frac{2}{20}\]

Step 6: Verify and Conclude

Ensure each conversion results in an equivalent fraction with a denominator of 20:- \(\frac{1}{4} = \frac{5}{20} \)- \(\frac{3}{5} = \frac{12}{20} \)- \(\frac{9}{10} = \frac{18}{20} \)- \(\frac{1}{10} = \frac{2}{20} \)Each fraction has been correctly converted to an equivalent fraction with a denominator of 20.

Key Concepts

Common Denominator

A common denominator is a significant concept when working with fractions, especially for adding, subtracting, or comparing them. In this exercise, we are focused on rewriting each fraction with a common denominator of 20.
When fractions have the same denominator, they are easier to work with. For example, if you are given \( \frac{1}{4} \) and you want to compare it with \( \frac{3}{5} \), finding a common denominator, like 20, simplifies this task.

• To transform \( \frac{1}{4} \) to have the denominator 20, multiply the numerator and denominator by 5, because \( 4 \times 5 = 20 \).
• Similarly, for \( \frac{3}{5} \), you multiply both by 4, resulting in \( \frac{12}{20} \).

By converting each fraction to have the same denominator, you simplify the math and create equivalent expressions that are easy to compare or combine.

Multiplying Fractions

Multiplying fractions is a straightforward operation. To multiply two fractions, you multiply the numerators together and the denominators together.
This principle helps in adjusting fractions to have a specific denominator, as shown in this problem.

• For instance, with \( \frac{1}{4} \), multiplying by \( \frac{5}{5} \) gives us \( \frac{5}{20} \).
• For \( \frac{3}{5} \), multiplying by \( \frac{4}{4} \) provides \( \frac{12}{20} \).

The whole number you multiply by, in these cases, is technically 1 (like \( \frac{5}{5} \)), so the value of the fraction does not change, just its appearance.
Multiplying by these special forms of 1 ensures that the fraction is equivalent and has the desired denominator.

Least Common Multiple

The least common multiple (LCM) is a crucial element when finding a common denominator for fractions. The LCM of two numbers is the smallest multiple that both numbers can divide without leaving a remainder.
When working with \( \frac{1}{4} \), \( \frac{3}{5} \), \( \frac{9}{10} \), and \( \frac{1}{10} \), you want each to have a denominator of 20.

• The LCM of 4 and 20 is 20.
• The LCM of 5 and 20 is also 20.
• Likewise, for 10 and 20, it is 20.

Determining these LCMs is simple and requires minimal calculation, making it easier to alter the denominators of fractions for consistent equivalence.

Numerator and Denominator

In fractions, the numerator and denominator represent parts of a whole. The numerator is the top number and shows how many parts you have. The denominator is the bottom number and reveals the number of parts the whole is divided into.
For instance, in \( \frac{1}{4} \), 1 is the numerator, and 4 is the denominator.

• The numerator is adjusted when finding an equivalent fraction, as shown by multiplying it by the same number as the denominator.
• The denominator determines how many pieces the whole is cut into, so adjusting the denominator allows the fractions to speak the same 'language' or scale.

Understanding these roles makes it easier to perform fraction operations and see their equivalence, crucial for comparing and adjusting fractions.