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Chapter 8: Problem 64

Convert to fraction notation. $$ 34.2 \% $$

Short Answer

Expert verified
34.2% as a fraction is \( \frac{171}{500} \).

Step by step solution

01

- Understand Percentage

A percentage is a way of expressing a number as a fraction of 100. The symbol % means 'per hundred.' For example, 34.2% means 34.2 out of 100.
02

- Write as a Fraction Over 100

To convert 34.2% into fraction notation, start by expressing it as 34.2 out of 100. This gives us the fraction \[ \frac{34.2}{100} \].
03

- Multiply to Eliminate Decimal

Since 34.2 is a decimal, we multiply the numerator and the denominator by 10 to get rid of the decimal point: \[ \frac{34.2 \times 10}{100 \times 10} = \frac{342}{1000} \].
04

- Simplify the Fraction

The fraction \( \frac{342}{1000} \) can be simplified by finding the greatest common divisor (GCD) of 342 and 1000. The GCD of 342 and 1000 is 2. Divide both the numerator and the denominator by 2: \[ \frac{342 \div 2}{1000 \div 2} = \frac{171}{500} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Notation
Fraction notation is a way to represent numbers by showing the relation between a part and a whole. A fraction has a numerator (the top number) and a denominator (the bottom number). The numerator represents how many parts we have, while the denominator represents the total number of equal parts.
For example, in the fraction \(\frac{34.2}{100}\), 34.2 is the numerator and 100 is the denominator.
Understanding fraction notation is essential because it helps us simplify and perform various mathematical operations much more easily.
Decimal Elimination
When dealing with percentages like 34.2%, we often encounter decimals. To convert this percentage into fraction notation, we need to eliminate the decimal point from the numerator.
To do this, we multiply both the numerator and the denominator by the same power of 10. This shifts the decimal point to the right in the numerator. For instance, in the fraction \(\frac{34.2}{100}\), multiplying the numerator and the denominator by 10 eliminates the decimal, transforming it into \(\frac{342}{1000}\).
This process ensures that we work with whole numbers, making subsequent calculations easier and cleaner.
Simplifying Fractions
Once the decimal has been eliminated, our next step is to simplify the fraction to its lowest terms. Simplifying involves reducing both the numerator and the denominator by their greatest common divisor (GCD).
For example, with the fraction \(\frac{342}{1000}\), the GCD of 342 and 1000 is 2. We divide both the numerator and the denominator by 2 to get \(\frac{171}{500}\).
By simplifying fractions, we make them easier to interpret and work with, ensuring our final answer is in the simplest form.

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