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Chapter 0: Problem 179

Write each percent as a fraction. Give answers in lowest terms. See Example 13. $$ 7.5 \% $$

Short Answer

Expert verified
The fraction is \( \frac{3}{40} \).

Step by step solution

01

Convert Percent to Fraction

The first step is to convert the given percentage into a fraction by dividing it by 100. Thus, 7.5% can be written as \( \frac{7.5}{100} \).
02

Remove Decimal from Numerator

To remove the decimal from the numerator, multiply both the numerator and the denominator by 10. Therefore, \( \frac{7.5}{100} \) becomes \( \frac{7.5 \times 10}{100 \times 10} = \frac{75}{1000} \).
03

Simplify the Fraction to Lowest Terms

To simplify the fraction, find the greatest common divisor (GCD) of 75 and 1000, which is 25. Divide both the numerator and the denominator by 25: \( \frac{75 \div 25}{1000 \div 25} = \frac{3}{40} \).

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Key Concepts

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

fraction simplification
Fraction simplification means reducing a fraction to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor. Simplifying fractions makes them easier to understand and compare.

For example, in the exercise, the fraction \(\frac{75}{1000}\) was simplified. The fraction is transformed by dividing both the top (75) and the bottom (1000) by their GCD (25). So, \(\frac{75 \div 25}{1000 \div 25} = \frac{3}{40}\). This means \(\frac{3}{40}\) is the simplest form of \(\frac{75}{1000}\).

Always remember:
  • Identify the GCD of the numerator and the denominator.
  • Divide both by the GCD.
  • Your fraction is now in its simplest form.
Practicing this regularly makes simplifying fractions second nature.
greatest common divisor (GCD)
The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD is essential for fraction simplification.

To find the GCD of two numbers:
  • List the factors of both numbers.
  • Identify the largest common factor.
Let's see an example. For 75 and 1000, we list the factors:
75: 1, 3, 5, 15, 25, 75
1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
The largest common factor is 25. Thus, 25 is the GCD of 75 and 1000. Using it, we can simplify \(\frac{75}{1000}\) to \(\frac{3}{40}\).
decimal to fraction
Converting a decimal to a fraction is straightforward. Follow a few simple steps:

First, express the decimal as a fraction with a denominator of a power of 10. For example, 7.5 becomes \(\frac{7.5}{1}\). To remove the decimal, multiply both the numerator and denominator by 10.

So, \(\frac{7.5 \times 10}{1 \times 10} = \frac{75}{10}\).
Convert the fraction to its lowest terms by simplifying. It follows:

Next, divide both the numerator and denominator by their GCD (5 in this case): \(\frac{75 \div 5}{10 \div 5} = \frac{15}{2}\). This can be simplified further if needed.
With regular practice, converting decimals to fractions becomes easier.

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