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Chapter 5: Problem 11

Reduce each rational number to its lowest terms. \(\frac{308}{418}\)

Short Answer

Expert verified
The fraction \(\frac{308}{418}\) reduced to its lowest terms is \(\frac{14}{19}\).

Step by step solution

01

Identify the Greatest Common Divisor

Firstly, you need to identify the greatest common divisor of the numbers 308 and 418. The GCD of these two numbers is 22, which is the largest number that can divide both 308 and 418 without leaving a remainder.
02

Divide by the GCD

Next, divide both the numerator and the denominator of the fraction by the GCD. This gives us: \(\frac{308}{22} = 14\) and \(\frac{418}{22} = 19\). Consequently, the fraction \(\frac{308}{418}\) simplifies to \(\frac{14}{19}\).
03

Confirm that the Fraction is in Lowest Terms

Lastly, confirm that \(\frac{14}{19}\) is in its lowest terms. Since 14 and 19 are prime to each other (meaning they have no common divisors other than 1), we can confirm that this fraction is 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
To simplify a fraction, the greatest common divisor (GCD), also known as the greatest common factor (GCF), is a crucial concept. It refers to the largest integer that divides two numbers without leaving a remainder. For example, finding the GCD helps reduce fractions to their lowest terms.

Here's how you can find the GCD:
  • List the factors: Find all positive factors of the numerator and denominator.
  • Identify the largest common factor: Look for the biggest factor common to both lists.
For instance, the GCD of 308 and 418 is 22. It divides both numbers completely, making it the optimal choice for simplifying the fraction.
Fractions
Fractions represent parts of a whole and consist of a numerator (the top number) and a denominator (the bottom number). Understanding fractions is fundamental in math, especially when simplifying them.
To simplify a fraction, you aim to rewrite it in a simpler form, without changing its value. This involves the GCD, which helps remove the extra factors.
Fractions can be simplified in these steps:
  • Find the GCD of the numerator and denominator.
  • Divide both by this GCD to reduce them accordingly.
For example, with the fraction \(\frac{308}{418}\), you find the GCD is 22 and divide both: \(\frac{308}{22} = 14\) and \(\frac{418}{22} = 19\). This leaves you with \(\frac{14}{19}\).
Lowest Terms
A fraction is in its lowest terms when the numerator and denominator have no common factors other than 1. This means that the fraction is as simplified as possible, ensuring clarity and accuracy in mathematical operations.

Why is simplifying fractions to their lowest terms important?
  • It makes fractions easier to understand and compare.
  • It simplifies calculations, especially involving multiple fractions.
In our example, \(\frac{14}{19}\) is in its lowest terms. This is because the numbers 14 and 19 are coprime; they don鈥檛 share any divisors except for 1. Identifying when a fraction is in its simplest form is a valuable skill in mathematics.

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Most popular questions from this chapter

Give an example of a number that is an irrational number and a real number.

How does doubling a number affect its square root?

Express each number in scientific notation. 220,000,000

Express each number in decimal notation. \(1 \times 10^{5}\)

Express each terminating decimal as a quotient of integers. If possible, reduce to lowest terms. \(0.59\)
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