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Chapter 4: Problem 26

Reduce the given fraction to lowest terms. $$\frac{18}{96}$$

Short Answer

Expert verified
\(\frac{18}{96}\) reduces to \(\frac{3}{16}\).

Step by step solution

01

Find the Greatest Common Divisor (GCD)

The first step is to identify the greatest common divisor (GCD) of the numerator and the denominator. For \(18\) and \(96\), list the factors.Factors of \(18\) are: \(1, 2, 3, 6, 9, 18\).Factors of \(96\) are: \(1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96\).The greatest common divisor of \(18\) and \(96\) is \(6\).
02

Divide Both the Numerator and Denominator by the GCD

Now divide both the numerator and the denominator by their GCD, \(6\).\(\frac{18 \div 6}{96 \div 6} = \frac{3}{16}\)
03

Verify the Reduction

To ensure the fraction is reduced, check that \(3\) and \(16\) have no common factors other than \(1\). The factors of \(3\) are \(1, 3\) and of \(16\) are \(1, 2, 4, 8, 16\). Since there are no common factors other than \(1\), \(\frac{3}{16}\) 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 (GCD)
The greatest common divisor (GCD) is a cornerstone concept in reducing fractions. It is the largest number that can perfectly divide both the numerator and the denominator without leaving any remainder. To find the GCD, list all the factors of both numbers involved in the fraction and identify the largest one they have in common.
  • For instance, with the numbers 18 and 96, we determine their factors: - Factors of 18: 1, 2, 3, 6, 9, 18. - Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.
  • The greatest factor that appears in both lists is 6. Therefore, 6 is the GCD of 18 and 96.
Knowing how to quickly find the GCD can significantly simplify the process of reducing fractions.
Factors of Numbers
Understanding the concept of factors is vital for finding the GCD and for simplifying fractions. A factor is a number that divides into another number without leaving a remainder.
Let's see the steps to identify factors:
  • Start with the number 1 and proceed upward.
  • Test each number to see if it divides evenly (i.e., without a remainder) into the number you are interested in.
  • Continue this process up to the number itself.
For example:
  • To find the factors of 18, check divisibility starting from 1 to 18.
  • Record the numbers that divide 18 without a remainder: 1, 2, 3, 6, 9, 18.
Understanding factors is key to reducing fractions as it helps in efficiently finding the GCD.
Simplifying Fractions
Simplifying fractions means reducing them to their lowest terms. This involves dividing both the numerator and denominator by their greatest common divisor (GCD). This process maintains the value of the fraction while presenting it in a simpler form.
Here's the step-by-step method to simplify a fraction like \( \frac{18}{96} \):
  • First, find the GCD of 18 and 96, which is 6.
  • Next, divide both the numerator (18) and the denominator (96) by 6.
  • Performing this division gives you a new fraction: \( \frac{3}{16} \).
Lastly, to verify the fraction is fully simplified, check that the new numerator and denominator have no common factors other than 1 (as demonstrated with 3 and 16). This ensures \( \frac{3}{16} \) is indeed the simplest form.

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Most popular questions from this chapter

Solve the equation and simplify your answer. $$-\frac{3}{4} x-\frac{2}{3}=-\frac{2}{3} x-\frac{1}{2}$$

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Solve the equation and simplify your answer. $$x+\frac{5}{2}=-\frac{9}{8}$$

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