Problem 86 - Elementary Algebra Within Reach

Question

Decide whether the statement is true or false. Justify your answer. Every percent can be written as a fraction.

Step-by-Step Solution

Answer

True. Every percent can be written as a fraction as it is essentially a number out of 100, making a percent directly expressible as a fraction.

Step 1: Understand Percentages

Percentages are a way of expressing a number as a fraction of 100. The word 'percentage' comes from the word 'cent,' which means 100. So, 1% means 1 out of 100.

Step 2: Conversion to Fractions

A percentage can be converted to a fraction by putting the percentage number over 100. For example, let's take 80%. In the form of a fraction, this would be written as 80/100.

Step 3: Simplification of Fractions

The fraction 80/100 just represents another representation of the original percent. We can further simplify such fractions to their lowest terms. In this case, simplifying 80/100 gives 4/5. Every percentage can be represented like this.

Key Concepts

Fraction Conversion

To convert a percentage to a fraction, the process is simple and straightforward. A percentage simply means "per hundred," so every percentage value can be expressed as part of 100.
For instance, when you see 50%, think of it as 50 out of every 100. To convert this into a fraction, you place the percentage value over 100. So, 50% converts to \( \frac{50}{100} \).
This fraction showcases the percentage in a mathematical format that allows for further manipulation and understanding within different mathematical contexts.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This means finding an equivalent fraction where both the numerator and the denominator are as small as possible while still maintaining the same value.
Using the previous example of \( \frac{50}{100} \), you simplify a fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
For \( \frac{50}{100} \), both 50 and 100 can be divided by their GCD, which is 50, resulting in the simplified fraction \( \frac{1}{2} \).
Simplifying makes fractions easier to work with and helps reveal relationships between different numbers.

Mathematical Representation

Mathematical representation allows us to express concepts like percentages, fractions, and decimals visually and analytically. Converting percentages to fractions, for example, provides a clear picture of proportional relationships.
By representing 75% as \( \frac{75}{100} \) or further simplified to \( \frac{3}{4} \), it's easier to see how parts relate to a whole.
Such mathematical expressions are crucial in comparing sizes, calculating actual figures from percentages, and simplifying complex numerical problems.
Understanding this representation aids in connecting various mathematical ideas and improving numerical literacy.