Problem 19 - Elementary Algebra Within Reach

Question

Convert the percent to a fraction. $$130 \%$$

Step-by-Step Solution

Answer

The fraction equivalent of 130% is \( \frac{13}{10} \).

Step 1: Write the percentage as a fraction

Write the percentage as a fraction of 100. This involves simply putting the number 130 over 100 because the term 'per cent' means 'per 100'. So, \(130\% \) becomes the fraction \(\frac{130}{100}\).

Step 2: Simplify the resulting fraction

Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor. Dividing both 130 and 100 by 10, a common factor of both, we get \(\frac{130}{100} = \frac{13}{10}\).

Key Concepts

Percentage

When dealing with percentages, we are essentially talking about parts per hundred. The word 'percentage' itself comes from 'per cent', which is Latin for 'per hundred'. This is why when we express something as a percentage, it is relative to 100. For example, if we say 50%, it means 50 parts out of a hundred, or half.

To convert a percentage to a fraction, you take the number representing the percentage and write it over 100. This makes it very straightforward. If the percentage is more than 100%, like in the exercise where we have 130%, it implies that we have more than a whole. Such fractions can also be proper or improper, leading to mixed numbers or whole numbers when simplified further.

Fraction Simplification

Simplifying fractions is the process of making them as 'simple' as possible. What does 'simple' mean in this context? It means having the numerator and denominator as small as possible. To simplify a fraction, you need to find the greatest common divisor (GCD) of both the numerator and the denominator — that's the largest number that can evenly divide both.

Once you find this GCD, divide both components of the fraction by it. The result will be a fraction where the numerator and denominator have no common factors other than 1. In our problem with the fraction \(\frac{130}{100}\), the number 10 is the GCD, so when we divide both the numerator and the denominator by 10, the fraction simplifies to \(\frac{13}{10}\).

Greatest Common Divisor

The GCD, or Greatest Common Divisor, is paramount in fraction simplification. It's the largest integer that divides into both the numerator and the denominator without leaving a remainder. Finding the GCD is a step that sometimes puzzles students, but there are a few methods like prime factorization, using the Euclidean algorithm, or simply listing out the factors of both numbers until you find the largest shared one.

In many cases, particularly with smaller numbers or where one number is a multiple of the other, you can often find the GCD by inspection. For instance, noticing that both 130 and 100 can be divided by 10 can lead you directly to the GCD without more complicated methods.

Numerator and Denominator

Any fraction is a way of expressing division between two numbers: the numerator and the denominator. The numerator, located on top of the fraction, represents the number of parts you're working with, while the denominator, at the bottom, tells you how many parts make up a whole. In the case of converting a percent to a fraction, the original percentage becomes the numerator, and the denominator is 100 because a percent is a part per hundred.

It's essential to understand the roles of the numerator and the denominator when simplifying fractions. The goal is to reduce both to their smallest possible numbers while ensuring they convey the same proportion, and recognizing these parts in context can make dealing with fractions much more intuitive.