Problem 74 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 30. $$\frac{1}{6}$$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{5}{30}\).

Step 1: Understand the Problem

We need to convert the fraction \(\frac{1}{6}\) into an equivalent fraction with a denominator of 30.

Step 2: Determine the Factor

To find an equivalent fraction with a denominator of 30, determine what number you need to multiply 6 by to get 30. This is done by dividing 30 by 6.

Step 3: Calculate the Multiplier

Divide 30 by 6 to find the multiplier: \(30 \div 6 = 5\). This tells us that we need to multiply both the numerator and the denominator of \(\frac{1}{6}\) by 5.

Step 4: Multiply the Numerator and Denominator

Multiply both the numerator and the denominator of \(\frac{1}{6}\) by 5. This results in: \(\frac{1 \times 5}{6 \times 5} = \frac{5}{30}\).

Step 5: Verify the Solution

Check if the fraction \(\frac{5}{30}\) is equivalent to \(\frac{1}{6}\) by simplifying \(\frac{5}{30}\). In this case, \(\frac{5}{30}\) simplifies back to \(\frac{1}{6}\). This confirms our solution is correct.

Key Concepts

Denominator

In mathematics, the denominator is the bottom number in a fraction, which states how many equal parts the whole is divided into. For example, in the fraction \(\frac{1}{6}\), the number 6 is the denominator. The denominator is pivotal when comparing or equating fractions because it represents the division of the whole.

• When manipulating fractions, particularly when creating equivalent fractions, you often change the denominator to a common number—this facilitates easy comparison and calculation between fractions.
• Always remember, changing the denominator of a fraction requires multiplying or dividing both the numerator and the denominator by the same number to maintain the fraction's value.
• In our exercise, the fraction \(\frac{1}{6}\) needed to have a denominator of 30. By recognizing 30 as the desired denominator, we established the goal for our equivalent fraction calculation.

The key is ensuring that any changes do not alter the original value of the fraction while making arithmetic operations like addition or multiplication more straightforward.

Numerator

The numerator is the top part of a fraction, representing how many parts we are considering out of the whole. In the fraction \(\frac{1}{6}\), 1 is the numerator. It essentially tells us "how many" of the parts described by the denominator are being counted.

• When converting to an equivalent fraction, the numerator and denominator both need to be multiplied by the same number, ensuring the fraction’s value remains unchanged.
• In the given exercise, multiplying the numerator 1 by 5 alongside the denominator was crucial. This step maintains the fraction's integrity, transitioning it from \(\frac{1}{6}\) to \(\frac{5}{30}\).
• The act of multiplying only impacts the count of the parts taken, not the size of those parts relative to the whole.

Understanding the numerator's role in fraction operations like simplification, comparison, and calculation helps in various mathematical scenarios including algebra and calculus.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This process ensures that the fraction is expressed as compactly as possible, yet retaining its original value.

• Simplification involves finding the greatest common divisor (GCD) of the numerator and denominator, then dividing both by this number.
• For instance, although the initial equivalent fraction form was \(\frac{5}{30}\), you simplify it back to \(\frac{1}{6}\) by recognizing that both 5 and 30 can be divided by 5, their GCD.
• It’s essential for calculating and communicating fractions clearly, especially when performing addition, subtraction, or comparison between different fractions.

By reducing fractions, you simplify calculations, making it easier to comprehend and solve complex mathematical problems swiftly. Simplification transforms a bulky-looking fraction into a neat, recognizable form that is both easier to understand and apply.