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Chapter 7: Problem 8

Convert the percent to a fraction. $$62 \frac{1}{2} \%$$

Short Answer

Expert verified
The fraction is \(\frac{5}{8}\).

Step by step solution

01

Convert the Mixed Number to an Improper Fraction

The percent is given as a mixed number: 62 \(\frac{1}{2}\) \% . Convert this to an improper fraction. Multiply the whole number part by the denominator and add the numerator: 62 \times 2 + 1 = 125. Therefore, 62 \(\frac{1}{2}\) is equal to \(\frac{125}{2}\).
02

Divide by 100 to Convert to a Fraction

Since the value is a percent, divide the improper fraction by 100 to convert the percent to a fraction. This gives: \(\frac{125}{2}\) \div 100. Dividing by 100 is the same as multiplying by \(\frac{1}{100}\), so perform the multiplication: \(\frac{125}{2}\) \times \(\frac{1}{100}\) = \(\frac{125}{200}\).
03

Simplify the Fraction

Simplify the fraction \(\frac{125}{200}\). Find the greatest common divisor (GCD) of 125 and 200, which is 25. Divide both the numerator and the denominator by their GCD: \(\frac{125 \div 25}{200 \div 25} = \(\frac{5}{8}\).\)

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Key Concepts

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

Mixed Number Conversion
When dealing with mixed numbers, it's essential to convert them into improper fractions for easier calculations. A mixed number like 62 1/2 consists of a whole number (62) and a fraction (1/2). To convert it, multiply the whole number part by the denominator of the fraction and add the numerator. Here's the step-by-step process:

For 62 1/2:
  • Multiply 62 by 2 (the denominator): 62 * 2 = 124
  • Add the numerator (1): 124 + 1 = 125
So, 62 1/2 becomes the improper fraction 125/2.
Improper Fraction
An improper fraction has a numerator that鈥檚 equal to or larger than its denominator. After converting the mixed number 62 1/2 to 125/2, we have an improper fraction. Improper fractions are often useful in mathematical operations because they can be directly manipulated, unlike mixed numbers. In our example:

  • The numerator is 125.
  • The denominator is 2.
Once we have the improper fraction, we can perform further operations, such as converting it into a percent or simplifying it.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD). For the fraction 125/200:

  • First, identify the GCD of 125 and 200, which is 25.
  • Divide both the numerator and the denominator by 25.
  • This gives us: 125 梅 25 = 5 and 200 梅 25 = 8.
So, 125/200 simplified is 5/8. Simplifying fractions helps to make numbers easier to work with and understand.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that can divide both the numerator and the denominator without leaving a remainder. To find the GCD of 125 and 200:

  • List the factors of both numbers. Factors of 125: 1, 5, 25, 125; Factors of 200: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200.
  • The highest common factor is 25.
Finding the GCD can be done efficiently using the Euclidean algorithm, but listing factors works well for smaller numbers. Knowing the GCD allows us to simplify fractions quickly and accurately.

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Most popular questions from this chapter

Solve the problem using a percent equation. \(225 \%\) of what number is \(18 ?\)

Solve the problem using a percent equation. \(23 \%\) of what number is \(34.5 ?\)

Brian has been saving money to buy a 61-in. Samsung Projection HDTV. He has saved \(\$ 1440\) so far, but this is only \(60 \%\) of the total cost of the television. What is the total cost?

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