Problem 68 - Prealgebra

Question

Suppose a gallon of regular gas costs \(\$ 3.99,\) and 54 cents of this goes to pay state gas taxes. What fractional part of the cost of a gallon of gas goes to state taxes? Write your answer in lowest terms.

Step-by-Step Solution

Answer

The fractional part of the cost that goes to state taxes is \(\frac{18}{133}\).

Step 1: Understand the Problem

The task is to find the fraction of the gas price represented by state gas taxes. We need to express the portion of the gas price attributed to taxes as a fraction of the total gas price.

Step 2: Identify the Relevant Values

Identify the total cost of the gas as \(3.99\) dollars and the amount paid in state gas taxes as \(0.54\) dollars.

Step 3: Set Up the Fraction

Set up the fraction with the state gas taxes as the numerator and the total cost as the denominator. That gives us a fraction of \(\frac{0.54}{3.99}\).

Step 4: Simplify the Fraction

To simplify the fraction \(\frac{0.54}{3.99}\), divide both the numerator and the denominator by their greatest common divisor. Convert the decimal to a fraction: \(\frac{54}{399}\). Find the greatest common divisor of 54 and 399, which is 3, and divide them, resulting in \(\frac{54 \div 3}{399 \div 3} = \frac{18}{133}\).

Step 5: Verify Simplification

Check if \(\frac{18}{133}\) is in its lowest terms by ensuring there are no common prime factors in the numbers 18 and 133 beyond 1. The fraction is indeed simplified to its lowest terms.

Key Concepts

Understanding 'Fraction of a Number'

When you're finding a fraction of a number, you're essentially determining how much of that number is represented by the fraction. This is a fundamental concept in mathematics, especially when dealing with parts of a whole. To find the fraction of a number:

• Identify the base number (the total or whole quantity).
• Determine the part of the base number that you're focusing on.
• Set up your fraction with the part as the numerator and the whole as the denominator.

The numerator represents the part you are examining, while the denominator shows the whole. In this example, we looked at the state gas taxes as part of the total price of gas. We used 0.54 as the part (numerator) and 3.99 as the whole (denominator) to set up our fraction.

Finding the Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is a key concept when simplifying fractions. It's the largest number that can divide both the numerator and the denominator without leaving a remainder. Here's how you can find the GCD:

• List the factors for the numerator and the denominator.
• Identify the common factors between them.
• The greatest common factor among these is your GCD.

Using the GCD to simplify fractions involves dividing both the numerator and the denominator by this common factor.
In our example, the GCD of 54 and 399 was found to be 3. Dividing each by 3, the fraction \[\frac{54}{399}\] simplifies to \[\frac{18}{133}\]. By simplifying, we ensure the fraction is in its simplest or lowest form, which makes it easier to work with.

Decimal to Fraction Conversion

Converting decimals to fractions is a crucial skill in math, often necessary when working with fractional numbers. Here's a simple guide on how to convert a decimal to a fraction:

• Count how many decimal places are in the number. This will determine the denominator.
• Place the decimal number over "1" followed by the same number of zeros as there are decimal places.
• Simplify the resulting fraction by finding the GCD of the numerator and denominator.

For instance, the decimal 0.54 translates to the fraction \[\frac{54}{100}\], initially. However, considering our problem's context, it needed adjusting for comparability, which led to using \[\frac{54}{399}\].
By understanding and practicing this conversion, you can handle more complex math problems effectively, ensuring all fractions used are precise and equivalent to their decimal forms.