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Chapter 5: Problem 5

Reduce each rational number to its lowest terms. \(\frac{24}{42}\)

Short Answer

Expert verified
\(\frac{4}{7}\)

Step by step solution

01

Find the Greatest Common Divisor (GCD)

First, find the GCD of 24 and 42. This can be done using the Euclidean algorithm, or by listing the factors of both numbers and finding the largest one they share. The GCD of 24 and 42 is 6.
02

Divide by the GCD

Next, divide both the numerator and the denominator by the GCD. This gives \(\frac{24}{6}\) / \(\frac{42}{6}\).
03

Simplify

Lastly, carry out the division to simplify the fraction. This results in \(\frac{4}{7}\), which is the fraction in its lowest terms.

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Key Concepts

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

Greatest Common Divisor
Understanding the Greatest Common Divisor (GCD) is a fundamental concept in math, especially when working with fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

Finding the GCD can be done in several ways:
  • Listing factors: Find all factors of each number, then choose the largest one they have in common.
  • Prime factorization: Break down each number into its prime factors. Multiply the common prime factors to get the GCD.
  • Euclidean algorithm: Use this method to iteratively apply the division algorithm until you reach a remainder of zero.
For the numbers 24 and 42, we use the Euclidean algorithm to find that their GCD is 6.

By calculating the GCD, you lay the groundwork for simplifying fractions, which brings us to the next concept.
Fractions
Fractions represent a part of a whole, and they are written in the form of a numerator over a denominator, like \(\frac{a}{b}\).

Every fraction consists of two parts:
  • Numerator: The top number, representing how many parts we have.
  • Denominator: The bottom number, indicating the total number of equal parts in the whole.
A fraction shows the division of the numerator by the denominator. For instance, \(\frac{24}{42}\) suggests that 24 parts of something are being divided into 42 equal parts.

Understanding this division is crucial in simplifying fractions, which makes it easier to work with them in mathematical operations.
Simplification of Fractions
Simplifying fractions involves reducing them to their lowest terms. This means performing an operation that results in the smallest possible numbers in the numerator and denominator while keeping the value of the fraction unchanged.

To simplify a fraction like \(\frac{24}{42}\):
  • Find the GCD of the numerator and denominator.
  • Divide both the numerator and the denominator by the GCD.
  • Perform the division until you cannot simplify the fraction any more.
In our example, after finding that the GCD is 6, you divide both 24 and 42 by 6. This results in the fraction \(\frac{4}{7}\), which is in its simplest form.

Remember, simplifying fractions not only makes calculations easier but also helps in clearly understanding and comparing different fractions.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-4, r=-2\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{2}, r=2\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{40}\), when \(a_{1}=6, r=-1\).

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(0.0004,-0.004,0.04,-0.4, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-1000, r=0.1\)
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