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Chapter 4: Problem 22

Reduce the given fraction to lowest terms. $$\frac{34}{30}$$

Short Answer

Expert verified
The fraction \(\frac{34}{30}\) simplifies to \(\frac{17}{15}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To reduce the fraction \(\frac{34}{30}\), we need to find the greatest common divisor of the numerator (34) and the denominator (30). We first list the factors of 34, which are 1, 2, 17, and 34. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The common factors are 1 and 2, with 2 being the greatest.
02

Divide Numerator and Denominator by the GCD

Once we have the greatest common divisor, we divide both the numerator and the denominator by it to simplify the fraction. So, we divide 34 and 30 by 2: \(\frac{34 \div 2}{30 \div 2} = \frac{17}{15}\).
03

Verify the Simplified Fraction

Now, let's check if \(\frac{17}{15}\) can be simplified further. The factors of 17 are 1 and 17, while the factors of 15 are 1, 3, 5, and 15. Since there are no common factors other than 1, the fraction \(\frac{17}{15}\) is in its simplest form.

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Key Concepts

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

Greatest Common Divisor
When we talk about the greatest common divisor, or GCD for short, we refer to the largest number that can divide two numbers without leaving a remainder. This is crucial for reducing fractions. Let's break it down with an example: suppose you have a fraction \( \frac{34}{30} \). To simplify it, you need to find the GCD of 34 and 30.

First, list the factors of each number:
  • Factors of 34: 1, 2, 17, 34
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Take note of the common factors: 1 and 2. Out of these, 2 is the greatest common divisor. Finding the GCD is crucial because it ensures we simplify fractions to their lowest terms efficiently.
Simplifying Fractions
Simplifying fractions means expressing a fraction in its simplest form, where the numerator and denominator have no common factors other than 1. Once you find the GCD of the numerator and denominator, simplifying is straightforward.

Let's use our previous example where the GCD of 34 and 30 was found to be 2. You divide both the numerator (34) and the denominator (30) by their GCD. So, \[ \frac{34}{30} = \frac{34 \div 2}{30 \div 2} = \frac{17}{15} \]

This simplifies the fraction \( \frac{34}{30} \) to \( \frac{17}{15} \). Remember, a fraction is fully simplified when no number other than 1 can divide both terms.
Factors of Numbers
Factors are numbers you multiply together to get another number. Understanding factors is key to many areas of math, particularly simplifying fractions. Each number has a set of factors. Finding them involves identifying all numbers that divide the given number evenly.

For the fraction \( \frac{34}{30} \), we looked at:
  • Factors of 34: 1, 2, 17, 34
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
These lists help us identify common factors, and ultimately, the greatest common divisor. Always ensure to list factors thoroughly to avoid errors in simplification tasks.

Knowing how to efficiently find factors not only helps in simplifying fractions but also provides a foundational math skill that is useful in solving more complex problems later on.

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