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Chapter 1: Problem 1

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{6}{8}\)

Short Answer

Expert verified
Answer: The simplest form of the fraction 6/8 is 3/4.

Step by step solution

01

Find the GCD of the numerator and the denominator

To find the GCD of 6 (numerator) and 8 (denominator), consider their factors. Factors of 6 are 1, 2, 3, and 6. Factors of 8 are 1, 2, 4, and 8. The highest common factor between these two sets is 2.
02

Divide the numerator and the denominator by their GCD

Divide both the numerator (6) and the denominator (8) by the GCD (2) to simplify the fraction to its lowest terms: \(\frac{6}{2} = 3\), \(\frac{8}{2} = 4\).
03

Write the simplified fraction

By dividing both numerator and denominator by their GCD, we get the simplified fraction: \(\frac{3}{4}\).

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor (GCD) is a very useful concept in fraction simplification. It is the largest number that can divide two or more numbers without leaving a remainder.
To find the GCD of two numbers, begin by listing out all their factors.
  • Factors are numbers that can multiply together to give the original number.
  • For example, the factors of 6 are 1, 2, 3, and 6.
  • Similarly, the factors of 8 are 1, 2, 4, and 8.
By comparing these lists, you can see that the biggest number that appears in both lists is 2. Hence, 2 is the GCD of 6 and 8.
This figure plays a crucial role in simplifying fractions, as dividing both the numerator and denominator by the GCD results in a fraction in its simplest form.
Numerator and Denominator
When dealing with fractions, it's essential to understand the roles of the numerator and the denominator.
The numerator is the top part of the fraction, and it represents how many parts of the whole are being considered.
In our example, 6 is the numerator, showing that we are working with 6 parts out of a possible 8.
The denominator is the bottom part of the fraction, which indicates the total number of equal parts the whole is divided into. In this case, 8 shows that the whole is divided into 8 parts.
  • Fractions are essentially a division of these two numbers, meaning that the fraction \( \frac{6}{8} \) is 6 divided by 8.
  • Understanding the roles of the numerator and denominator helps when simplifying fractions.
By performing this division and then simplifying, we can make any fraction easier to work with in calculations.
Simplified Fraction
A simplified fraction is one where the numerator and the denominator have no common factors other than 1. This ensures that the fraction is in its simplest form, making it more intuitive and easier to understand.
To simplify a fraction:
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator of the fraction by their GCD.
In the fraction \( \frac{6}{8} \), after finding the GCD to be 2, both the 6 and the 8 are divided by 2. This gives the simplified fraction \( \frac{3}{4} \).
This process is crucial because simplified fractions are more convenient to use in further mathematical calculations, such as addition, subtraction, multiplication, or division of fractions.

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Most popular questions from this chapter

For the following problems, perform each indicated operation. \(\frac{16}{20}+\frac{1}{20}+\frac{2}{20}\)

For the following problems, perform each indicated operation. \(\frac{24}{75} \div \frac{8}{15}\)

For the following problems, perform each indicated operation. \(\frac{3}{7} \cdot \frac{14}{18} \cdot \frac{6}{2}\)

For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places. \(\frac{7}{11}\)

For the following problems, convert each percent to a decimal. $$ 0.00005 \% $$
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