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Chapter 7: Problem 10

Convert the given percent to a fraction, and simplify the result. $$ 8 \frac{5}{8} \% $$

Short Answer

Expert verified
The fraction is \(\frac{69}{800}\).

Step by step solution

01

Convert Mixed Number to Improper Fraction

First, convert the mixed number percentage \(8 \frac{5}{8}\%\) to an improper fraction. To do this, multiply the whole number by the denominator of the fraction and add the numerator. This gives \(8 \times 8 + 5 = 64 + 5 = 69\). Thus, the improper fraction is \(\frac{69}{8}\%\).
02

Convert Percentage to Fraction

To convert the percentage (now \(\frac{69}{8}\%\)) to a fraction, divide by 100. Dividing \(\frac{69}{8}\) by 100 is equivalent to multiplying \(\frac{69}{8}\) by \(\frac{1}{100}\): \[\frac{69}{8} \times \frac{1}{100} = \frac{69}{800}.\]
03

Simplify the Fraction

Check if \(\frac{69}{800}\) can be simplified. Since 69 and 800 have no common factors other than 1, the fraction is already in its simplest form.

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Key Concepts

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

Mixed Numbers
Mixed numbers combine a whole number with a fraction, and they often need conversion to solve mathematical problems involving percentages. In this exercise, you saw the mixed number \(8 \frac{5}{8}\%\) which involves converting a fraction with a whole number to an improper fraction. To convert a mixed number like \(8 \frac{5}{8}\), follow these simple steps:
  • Multiply the whole number (8) by the denominator of the fraction (8).
  • Add the result to the numerator, giving \(8 \times 8 + 5 = 69\).
  • This converts the mixed number to the improper fraction \(\frac{69}{8}\).
Understanding this conversion makes further calculations, such as those involving percentages, much more manageable.
Simplifying Fractions
Simplifying fractions is a skill needed to make fractions as simple as possible. This involves reducing the fraction to its lowest terms. For the fraction \(\frac{69}{800}\), simplifying means checking for the greatest common divisor (GCD) between the numerator and the denominator. Here's how:
  • List the factors of both the numerator (69) and the denominator (800).
  • Since the only common factor here is 1, as they have no other shared factors, \(\frac{69}{800}\) is already in its simplest form.
Simplifying makes a fraction easier to interpret and often more useful in diverse mathematical applications.
Fractions and Percentages
Fractions and percentages are closely related. A percent represents a ratio out of 100, whereas a fraction is just a part of a whole. Converting a percent like \(8 \frac{5}{8}\%\) into a fraction involves a few key steps:
  • Convert the mixed number to an improper fraction (earlier, \(\frac{69}{8}\)).
  • Since a percentage is essentially a fraction over 100, convert by dividing the improper fraction by 100.
  • Instead of actual division, multiply by \(\frac{1}{100}\), transforming \(\frac{69}{8}\%\) to \(\frac{69}{800}\).
Understanding the conversion provides a solid foundation for working with both percentages and fractions in various mathematical contexts. Whether in financial calculations or data analysis, this process is an essential skill.

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Most popular questions from this chapter

How much simple interest is earned if $4,000 is invested at 7.25% per year for 6 years?

A business owner takes out a 2-month loan at a 8.5% per year simple interest rate. At the end of the 2-month loan period, the interest owed is $85. What was the principal amount borrowed?

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