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Chapter 7: Problem 13

Convert the given percent to a fraction, and simplify the result. $$ 86 \% $$

Short Answer

Expert verified
The simplified fraction is \( \frac{43}{50} \).

Step by step solution

01

Understand the Percent

A percent is a ratio that compares a number to 100. So, 86% means 86 out of 100.
02

Write Percent as a Fraction

Convert 86% to a fraction by placing 86 over 100. So, we have \( \frac{86}{100} \).
03

Simplify the Fraction

To simplify \( \frac{86}{100} \), find the greatest common divisor (GCD) of 86 and 100. The GCD is 2. Divide both the numerator and the denominator by 2 to simplify: \( \frac{86 \div 2}{100 \div 2} = \frac{43}{50} \).
04

Verify the Simplification

Ensure the fraction \( \frac{43}{50} \) is in its simplest form. Since 43 is a prime number and does not divide 50, the fraction cannot be simplified further.

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Key Concepts

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

Percent to Fraction Conversion
Converting a percent to a fraction might seem tricky at first, but it's quite simple once you understand how it works. Percents are essentially fractions with a denominator of 100. This is because the term "percent" means "per one hundred." Thus, when you're working with percents, you're dealing with a part of a whole, represented as out of 100. To convert a percent to a fraction, you just need to remove the percent sign (%) and place the number over 100. For example, converting 86% to a fraction:
  • First, remove the percentage sign to get the number 86.
  • Next, write it as a fraction over 100: \( \frac{86}{100} \).
That's it! You've successfully converted the percent to a fraction.
Simplifying Fractions
Simplifying fractions is an important skill that helps make working with fractions much easier. A fraction is simplified when its numerator and denominator have no common factors other than 1. To simplify a fraction, you'll need to divide both the numerator and the denominator by their greatest common divisor (GCD). For instance, if you have the fraction \( \frac{86}{100} \), the goal is to reduce it to its simplest form.
  • First, identify the common factors of both numbers. For \( 86 \) and \( 100 \), they have a common factor of 2.
  • Divide both 86 and 100 by 2 to get \( \frac{43}{50} \).
This fraction, \( \frac{43}{50} \), is now in its simplest form because 43 and 50 share no common factors.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that divides two numbers without leaving a remainder. Finding the GCD is a crucial step in simplifying fractions because it allows you to determine how much you can reduce the fraction by. To find the GCD of two numbers:
  • List the factors of both numbers. For instance, the factors of 86 are 1, 2, 43, and 86, and the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
  • Identify the largest factor that both numbers share, which in this case is 2.
By dividing the numerator and denominator of your fraction by this number, you simplify the fraction to its simplest form. In our example, the GCD of 86 and 100 is 2, which helps simplify \( \frac{86}{100} \) to \( \frac{43}{50} \). This makes understanding and working with the fraction more straightforward.

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