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Chapter 4: Problem 25

Reduce the given fraction to lowest terms. $$\frac{69}{24}$$

Short Answer

Expert verified
The reduced fraction is \( \frac{23}{8} \).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{69}{24} \) to its lowest terms, we first need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. In this case, the GCD of 69 and 24 is 3.
02

Divide Numerator and Denominator by GCD

Now that we know the GCD is 3, we will divide both the numerator and the denominator by 3. So, the fraction \( \frac{69}{24} \) becomes: \[\frac{69 \div 3}{24 \div 3} = \frac{23}{8}.\]
03

Verify the Result

The resulting fraction after dividing both terms by their GCD is \( \frac{23}{8} \). We can confirm that the fraction is in its simplest form because 23 and 8 have no common factors other than 1. Therefore, \( \frac{23}{8} \) is indeed the fraction reduced to 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 key when it comes to reducing fractions. The GCD of two numbers is the largest positive number that divides both numbers evenly, without leaving a remainder. To find the GCD, you can list the factors of each number and identify the greatest number common to both lists. For example, when dealing with numbers 69 and 24, we list their factors:
  • 69: 1, 3, 23, 69
  • 24: 1, 2, 3, 4, 6, 8, 12, 24
The largest common factor is 3. Thus, 3 is the GCD. Knowing the GCD allows us to effectively reduce fractions to their simplest form by dividing the numerator and denominator by this common factor.
Numerator and Denominator
To reduce a fraction, it's essential to clearly understand its components: the numerator and the denominator. The numerator is the top part of the fraction, representing the number of parts we have. The denominator is the bottom part, telling us into how many parts the whole is divided.
For the fraction \( \frac{69}{24} \):
  • Numerator = 69
  • Denominator = 24
Reducing a fraction involves dividing both the numerator and the denominator by their GCD. Here, since our GCD is 3, we divide both parts by 3 to simplify the fraction.
Thus, the fraction \( \frac{69}{24} \) becomes \( \frac{23}{8} \) after dividing both terms by 3, significantly simplifying it.
Simplest Form
Reaching the simplest form of a fraction means expressing it in the lowest possible terms, where the numerator and denominator have no common factors other than 1. This is also known as reducing fractions. Once a fraction is simplified, it cannot be reduced any further.
In our example, \( \frac{69}{24} \) was reduced using their GCD, 3, resulting in the fraction \( \frac{23}{8} \). Checking whether a fraction is in its simplest form is straightforward:
  • Check for common factors between numerator and denominator.
  • If none exist beyond 1, the fraction is simplest.
For \( \frac{23}{8} \), 23 and 8 have no common factors besides 1, confirming it's in simplest form. This makes \( \frac{23}{8} \) the final reduced version of our initial fraction.

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Most popular questions from this chapter

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