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Chapter 4: Problem 21

Reduce the given fraction to lowest terms. $$\frac{66}{57}$$

Short Answer

Expert verified
The reduced form of \( \frac{66}{57} \) is \( \frac{22}{19} \).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{66}{57} \), we need to find the GCD of both the numerator (66) and the denominator (57). The prime factorization of 66 is \( 2 \times 3 \times 11 \) and for 57 it is \( 3 \times 19 \). The common factor between the two numbers is 3. Therefore, the GCD of 66 and 57 is 3.
02

Divide Numerator and Denominator by the GCD

Next, we divide both the numerator and the denominator of the fraction \( \frac{66}{57} \) by the GCD we found in Step 1. So, we have:- Numerator: \( \frac{66}{3} = 22 \)- Denominator: \( \frac{57}{3} = 19 \)
03

Write the Reduced Fraction

The simplified or reduced form of the given fraction is obtained by using the results from Step 2. Thus, \( \frac{66}{57} \) simplifies to \( \frac{22}{19} \), which is in its simplest form as the numerator and denominator have no common factors other than 1.

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor (GCD) is a key concept when simplifying fractions. It is the largest number that divides both the numerator and denominator without leaving a remainder. To find the GCD, one effective method is prime factorization, where we express each number as a product of its prime factors. For example, in the fraction \( \frac{66}{57} \), we break down 66 into \( 2 \times 3 \times 11 \) and 57 into \( 3 \times 19 \). The common prime factor here is 3, so the GCD is 3. Identifying the GCD is an essential step in reducing fractions to their simplest form, as it ensures you are dividing by the largest possible number.
Prime Factorization
Prime factorization involves breaking down a number into its prime components, which are numbers that have only two divisors: 1 and themselves. Understanding prime factorization helps find the GCD, thereby simplifying fractions. In the example of 66 and 57 from our fraction \( \frac{66}{57} \), we saw that 66 turns into \( 2 \times 3 \times 11 \) and 57 into \( 3 \times 19 \).
  • Prime numbers are fundamental to understanding and simplifying complex numbers.
  • Once prime factors of both numerator and denominator are identified, locating their common factors becomes easy.
  • This allows us to determine the greatest common divisor efficiently.
Prime factorization is a powerful tool not just in arithmetic simplifications but also in understanding the structure of numbers themselves.
Reduced Fractions
A reduced fraction is a fraction that cannot be simplified any further. This means its numerator and denominator have no common divisors other than 1. Reducing fractions makes them easier to work with and can clarify their proportional relationships. Once you have identified the GCD through methods like prime factorization, simplifying the fraction \( \frac{66}{57} \) by dividing both numbers by the GCD of 3 gives you \( \frac{22}{19} \).
  • Reduced fractions are the simplest form of any fraction.
  • The process ensures the fraction is succinctly expressing the same value as the original.
  • Understanding reduced fractions is useful for comparing and performing arithmetic operations on fractions.
Remember, the easiest way to confirm a fraction is fully reduced is to check whether 1 is the only common factor between the numerator and denominator.

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