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Chapter 4: Problem 56

Reduce the given fraction to lowest terms. $$\frac{-100}{-46}$$

Short Answer

Expert verified
The reduced form of \( \frac{-100}{-46} \) is \( \frac{50}{23} \).

Step by step solution

01

Simplify the Negative Signs

Since both the numerator and the denominator of the fraction \( \frac{-100}{-46} \) are negative, the fraction itself is positive. Thus, we can rewrite the fraction as \( \frac{100}{46} \).
02

Find the Greatest Common Divisor (GCD)

To reduce the fraction, we need to find the greatest common divisor (GCD) of 100 and 46. By performing the Euclidean algorithm, we find:1. 100 divided by 46 is 2 with a remainder of 8 (since \( 100 = 46 \times 2 + 8 \)).2. Then, divide 46 by 8 to get a quotient of 5 with a remainder of 6 (since \( 46 = 8 \times 5 + 6 \)).3. Next, divide 8 by 6 to get a quotient of 1 with a remainder of 2 (since \( 8 = 6 \times 1 + 2 \)).4. Finally, divide 6 by 2 with a quotient of 3 and a remainder of 0 (since \( 6 = 2 \times 3 + 0 \)).The process stops here, and the GCD is 2.
03

Divide by the GCD

To reduce the fraction \( \frac{100}{46} \) to its simplest form, divide both the numerator and the denominator by their GCD, which is 2:\[ \frac{100}{46} = \frac{100 \div 2}{46 \div 2} = \frac{50}{23} \]

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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 an essential tool in mathematics for simplifying fractions. It is the largest positive integer that can evenly divide two or more numbers.
To truly simplify a fraction, we need to consider the GCD of the numerator and denominator. This ensures that both parts of the fraction are divided by the same value, bringing it to its simplest form.
The GCD helps us understand the shared factors between numbers, which is key to reducing complexity in calculations.
Euclidean Algorithm
The Euclidean algorithm is a systematic method of finding the greatest common divisor (GCD) of two numbers.
This ancient and efficient algorithm hinges on the principle that the GCD of two numbers also divides their remainder.
Here's how you can apply it:
  • Divide the larger number by the smaller one.
  • Note the remainder after division.
  • Replace the larger number with the smaller number, and the smaller number with the remainder.
  • Repeat the process until the remainder becomes zero.
  • At this point, the last non-zero remainder is the GCD.
By following these steps, the Euclidean algorithm efficiently reduces the problem, making it ideal for dealing with large numbers when simplifying fractions.
Reducing Fractions
Reducing fractions is the process of simplifying them to their lowest terms.
By doing this, we express the fraction in the simplest way without changing its value.
Here's how to reduce a fraction step by step:
  • Identify the numerator and the denominator of the fraction.
  • Find their greatest common divisor (GCD).
  • Divide both the numerator and the denominator by the GCD.
For example, consider the fraction \( \frac{100}{46} \):
  • Its GCD is 2, as discovered through the Euclidean algorithm.
  • By dividing both the numerator and the denominator by 2, we get the fraction \( \frac{50}{23} \), which is in simplest terms.
Remember that reducing fractions not only simplifies mathematical expressions but also makes comparisons between fractions more manageable.

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