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Chapter 5: Problem 20

Simplify to lowest terms. $$\frac{-210}{294}$$

Short Answer

Expert verified
Answer: The simplified form of the fraction -210/294 is -5/7.

Step by step solution

01

Find the GCD of 210 and 294

To find the GCD of 210 and 294, we can use the Euclidean algorithm or prime factorization. Here, we use prime factorization: Prime factors of 210: 2 脳 3 脳 5 脳 7 Prime factors of 294: 2 脳 3 脳 7 脳 7 From this, we can see the common factors are 2, 3, and 7. Multiplying these common factors together, we find that the GCD of 210 and 294 is 2 脳 3 脳 7 = 42.
02

Divide the numerator and denominator by the GCD

Now that we have the GCD of 210 and 294, which is 42, we will divide both the numerator and the denominator by 42: $$\frac{-210}{294} = \frac{-210 \div 42}{294 \div 42}$$
03

Calculate the simplified fraction

Divide the numerator and denominator by 42: Numerator: -210 梅 42 = -5 Denominator: 294 梅 42 = 7 So, the simplified fraction is: $$\frac{-210}{294} = \frac{-5}{7}$$

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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, or GCD, is the largest number that divides two or more integers without leaving a remainder. It's a valuable tool in simplifying fractions by reducing them to their simplest form. To find the GCD, we can use methods like the Euclidean algorithm or prime factorization. In our example of simplifying \( \frac{-210}{294} \), we used prime factorization.

  • Prime factors of 210: \( 2 \times 3 \times 5 \times 7 \)
  • Prime factors of 294: \( 2 \times 3 \times 7 \times 7 \)
The common factors here are 2, 3, and 7. Multiplying these gives us the GCD: \( 2 \times 3 \times 7 = 42 \). The GCD helps simplify the fraction, making it easier to understand and work with.
Prime Factorization
Prime factorization breaks down a number into its basic building blocks: prime numbers. Prime numbers are numbers greater than 1 that only have two factors, 1 and themselves. By expressing numbers as products of primes, we can easily find common factors or divisors.

In our exercise, the numbers 210 and 294 were broken into their prime factors:

  • 210: \( 2 \times 3 \times 5 \times 7 \)
  • 294: \( 2 \times 3 \times 7 \times 7 \)
Using these factors, we identified common elements like 2, 3, and 7 to find the GCD. Prime factorization provides a systematic approach to problem-solving by highlighting relationships between numbers.
Simplifying Fractions
Simplifying fractions means reducing them to their lowest terms. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. We use the Greatest Common Divisor to achieve this simplification process.

For example, in transforming \( \frac{-210}{294} \) to its simplest form:

  • Identify the GCD, which is 42
  • Divide both the numerator and the denominator by 42
  • \(-210 \div 42 = -5\)
  • \(294 \div 42 = 7\)
Thus, \( \frac{-210}{294} \) simplifies to \( \frac{-5}{7} \). This process makes fractions easier to understand and compare, often improving clarity in mathematical expressions.

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