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Chapter 4: Problem 40

Write each fraction in simplest form. \(\frac{60}{150}\)

Short Answer

Expert verified
The simplest form of \( \frac{60}{150} \) is \( \frac{2}{5} \).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

First, we need to determine the greatest common divisor (GCD) of 60 and 150. The GCD is the largest number that divides both 60 and 150 without leaving a remainder. Through factoring or using the Euclidean algorithm, we find the GCD of 60 and 150 is 30.
02

Divide Both Numerator and Denominator by the GCD

Now, divide both the numerator (60) and the denominator (150) by their GCD, which is 30. \[\frac{60 \div 30}{150 \div 30} = \frac{2}{5}\]
03

Simplified Fraction

After dividing, we obtain the fraction \(\frac{2}{5}\). This fraction is in its simplest form because there are no common factors between the numerator and the denominator, other than 1.

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Key Concepts

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

Greatest Common Divisor
When simplifying fractions, finding the Greatest Common Divisor (GCD) is crucial. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. Knowing how to calculate it can greatly simplify our work.
  • Factoring Method: List all the factors of each number and identify the largest common factor.
  • Euclidean Algorithm: Use a process of successive division. For example, divide the larger number by the smaller one, then divide the divisor by the remainder repeatedly until the remainder is zero.
By identifying the GCD, such as 30 in the fraction \(\frac{60}{150}\), you find the key common point to bring the fraction to its simplest form.
Fraction Simplification
Fraction simplification involves the process of reducing a fraction to its simplest form. This means rewriting it so the numerator and denominator have no common divisors other than 1, making it easier to work with and understand negative relationships or operations.
To simplify a fraction:
  • Find the GCD: As discussed earlier, identifying this common divisor is the first step.
  • Divide Both Parts: Once you have the GCD, divide both the numerator and the denominator by this number.
  • Check for Simplicity: Ensure that no further simplification is possible.For example, after dividing 60 and 150 by their GCD, 30, you get \(\frac{2}{5}\). This is already simplified as the GCD of 2 and 5 is 1.
Numerator and Denominator
Understanding the roles of the numerator and denominator is fundamental in managing fractions. The numerator is the top part of a fraction, representing how many parts we have, while the denominator, at the bottom, shows how many parts make a whole.
In our fraction \(\frac{60}{150}\):
  • The numerator is 60, indicating 60 parts considered.
  • The denominator is 150, suggesting that the whole consists of 150 equal parts.
Knowing these roles helps in both setting up a fraction and performing operations like simplification. After simplification, \(\frac{2}{5}\) implies that you have 2 parts out of 5. This clearer picture better assists in comparing and evaluating mathematical statements.

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Most popular questions from this chapter

Perform the indicated operations. $$ -7 \frac{3}{10} \div(-100) $$

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. $$\frac{5}{6}, \frac{7}{9}$$

Solve. See the Concept Checks in this section. Round the mixed number \(5 \frac{11}{12}\) to the nearest whole number

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$3+\frac{1}{2}$$

Perform the indicated operations. $$ -4 \frac{2}{5} \cdot 2 \frac{3}{10} $$
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