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Chapter 4: Problem 23

Write each fraction in simplest form. \(\frac{20}{30}\)

Short Answer

Expert verified
The simplest form of \(\frac{20}{30}\) is \(\frac{2}{3}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor (GCD) of the numerator (20) and the denominator (30). The GCD is the largest number that divides both numbers without leaving a remainder. The factors of 20 are 1, 2, 4, 5, 10, 20. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The greatest common factor is 10.
02

Divide Numerator and Denominator by GCD

Now, divide both the numerator and the denominator by the GCD we found in Step 1. This means dividing 20 by 10 and 30 by 10: \[ \frac{20}{30} = \frac{20 \div 10}{30 \div 10} = \frac{2}{3} \]
03

Verify the Simplified Fraction

Check that the simplified fraction \(\frac{2}{3}\) is in its simplest form. Since 2 and 3 have no common factors other than 1, \(\frac{2}{3}\) is indeed the simplest form of \(\frac{20}{30}\).

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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 key concept in simplifying fractions. It helps us to identify the largest number that can divide both the numerator and the denominator without leaving a remainder. Here's how you can find the GCD step by step:

  • List the factors of each number. For instance, for 20, the factors are 1, 2, 4, 5, 10, and 20. For 30, we have 1, 2, 3, 5, 6, 10, 15, and 30.
  • Identify the common factors between both lists of factors. From the example above, 1, 2, 5, and 10 are common factors of 20 and 30.
  • Select the largest common factor, which is the greatest common divisor. In our case, it's 10.
Finding the GCD is crucial because it allows us to reduce fractions to their simplest form smoothly. By using the GCD to divide both the numerator and denominator, we ensure that the fraction is simplified completely.
Numerator and Denominator
In any fraction, the numerator is the top number and the denominator is the bottom number. These two components play different roles but work together to represent a part of a whole.

  • The numerator tells you how many parts of the whole you have. Using our example, 20 is the numerator in \(\frac{20}{30}\), signifying 20 parts.
  • The denominator shows how many equal parts the whole is divided into. Here, 30 is the denominator, meaning the whole is divided into 30 parts.
When simplifying fractions, it's important to apply the same operation (typically division by the GCD) to both the numerator and the denominator. Maintaining this equality ensures you do not change the overall value of the fraction. By dividing both by their GCD, 10 in the example, \(\frac{20}{30}\) becomes \(\frac{2}{3}\).
Simplest Form
The simplest form of a fraction is when the numerator and denominator have no common factors other than 1. This means the fraction is fully reduced and cannot be simplified any further. Simplifying a fraction involves a few key steps:

  • Calculate the GCD of the numerator and denominator.
  • Divide both the numerator and denominator by their GCD.
  • Verify the result to ensure no further simplification is possible.
In our example, \(\frac{20}{30}\) simplifies to \(\frac{2}{3}\) because 2 and 3 have no common factors other than 1. It's important to simplify fractions not just for ease in calculations, but also to enhance clarity and understanding of their accurate mathematical value.

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Most popular questions from this chapter

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{47}{15}$$

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. $$\frac{1}{4}, \frac{2}{14}$$

Solve. A student asked you to check his work below. Is it correct? If not, where is the error? \(3 \frac{2}{3} \cdot 1 \frac{1}{7} \stackrel{?}{=} 3 \frac{2}{21}\)

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{901}{123}$$

Simplify. See Section 1.7. $$3^{4}$$
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