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Chapter 7: Problem 55

Convert each percent to a fraction. $$37.5 \%$$

Short Answer

Expert verified
37.5% is equal to \( \frac{3}{8} \) as a simplified fraction.

Step by step solution

01

Understand Percent Notation

A percent represents a number out of 100. For example, 37.5% means 37.5 out of 100. This can be converted into a fraction by using the formula where the percent becomes the numerator and 100 becomes the denominator.
02

Write the Percent as a Fraction

Turn 37.5% into a fraction by writing it as \( \frac{37.5}{100} \).
03

Remove the Decimal

To remove the decimal, multiply the numerator and the denominator by 10 to convert 37.5 into a whole number. This gives us \( \frac{375}{1000} \).
04

Simplify the Fraction

Now simplify \( \frac{375}{1000} \) by finding the greatest common divisor (GCD) of 375 and 1000, which is 125. Dividing both the numerator and the denominator by 125, we get \( \frac{3}{8} \).

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Key Concepts

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

Percent Notation
Percent notation is a way of expressing numbers as parts of 100. It's important because it helps us quickly understand proportions relative to a whole. When we say 37.5%, it means 37.5 out of 100. This is quite handy for comparing values, especially in statistics and everyday calculations like discounts or interest rates.
  • The percent sign (%) conveniently indicates 'per hundred'.
  • To convert from percent to fraction, remember that the number preceding % is your numerator, and 100 is your denominator.
For example, 37.5% can be changed to a fraction as\( \frac{37.5}{100}\). After converting, we aim to simplify it, but the essence remains that it's still a part of a whole, showing proportion in a clear and simple way.
Simplifying Fractions
Simplifying fractions is the process of reducing the fraction to its simplest form. This means making the numerator and denominator as small as possible while still keeping the same value. For example, converting 37.5% into a fraction gives us \( \frac{375}{1000}\) first.
  • We simplify by dividing the numerator and denominator by their greatest common divisor (GCD).
  • This process ensures that both parts of the fraction are the smallest possible whole numbers.
Continuing with our example, \( \frac{375}{1000}\) simplifies to \( \frac{3}{8}\) when divided by the GCD. This fraction represents the original percent value in the most concise way, ensuring clarity and simplicity.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is the largest whole number that divides both the numerator and denominator of a fraction without leaving a remainder. Identifying the GCD is key to simplifying fractions efficiently.
  • To find the GCD, list all the divisors of both numbers.
  • Choose the highest number that appears in both lists.
For \( \frac{375}{1000}\), the GCD is 125. This means 125 is the largest number that can evenly divide both 375 and 1000. Once the GCD is found, use it to divide the numerator and the denominator, and this gives us the simplified form of the fraction. By using the GCD, we ensure that the fraction is reduced to its simplest form, making it easier to understand and work with in mathematical calculations.

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