Problem 21 - Elementary Algebra Within Reach

Question

Convert the percent to a fraction. $$1.4 \%$$

Step-by-Step Solution

Answer

The fraction equivalent of 1.4% is \(7/500\).

Step 1: Express as a Fraction out of 100

1.4% would be expressed as a fraction out of 100, which is \(1.4/100\).

Step 2: Simplify the Fraction

The fraction \(1.4/100\) can be further simplified to \(7/500\). We obtain the simplified fraction by dividing the numerator and denominator both by 0.2.

Key Concepts

Expressing Percents as Fractions

Percentages and fractions are two different ways of expressing proportions of a whole. With percents, we express the proportion out of 100, but when we convert to fractions, we express this proportion as a ratio of two numbers - the numerator over the denominator.

Let's look at the process of converting a percent to a fraction with the example of converting 1.4%. The first step is to write the percentage as a fraction with the denominator of 100, as the word percent means per one hundred. Thus, 1.4% can be written as \(\frac{1.4}{100}\). This represents the percentage as a fraction of a whole.

When dealing with decimals in the numerator, the next step is to convert it into a whole number without changing the value of the fraction. This is usually done by multiplying both the numerator and the denominator by 10 for each decimal place. For 1.4, we would multiply by 10 to get \(\frac{14}{1000}\), but we can do this process in our head and skip to the simplified fraction in step 2.

Simplifying Fractions

Once we have a fraction, the next objective is to simplify it, making it easier to understand and work with. Simplifying a fraction involves finding a common factor for the numerator and the denominator and then dividing them both by this factor.

In our example, the fraction \(\frac{1.4}{100}\) needs to be simplified. First, we convert the decimal in the numerator to a whole number to simplify our calculations. Knowing that dividing or multiplying both terms of the fraction by the same number doesn't change its value, we find a common factor. In this case, we can divide the numerator and denominator each by 0.2 - since 1.4 divided by 0.2 equals 7, and 100 divided by 0.2 equals 500, the simplified fraction is \(\frac{7}{500}\).

To ensure a fraction is in its simplest form, we continue to divide both the numerator and the denominator by their greatest common factor until no more common factors aside from 1 remain.

Fractional Notation

Fractional notation is a concise way to represent the division of two numbers. A fraction has two main components: the numerator, which is the number above the line, and the denominator, which is the number below the line.

The numerator indicates how many parts we are considering, while the denominator gives the total number of parts that make up a whole. For example, in the fraction \(\frac{7}{500}\), 7 is the numerator and 500 is the denominator, indicating that we have 7 parts out of a total of 500.

It is essential that fractions are always presented in their simplest form to ensure clarity and ease of understanding. This helps in further mathematical processes, like adding, subtracting, or comparing fractions. The fractional notation is immensely useful for expressing ratios, rates, and proportions in a wide array of contexts, from academic to everyday life.