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Chapter 1: Problem 54

Simplify each fraction, if possible. $$ \frac{175}{490} $$

Short Answer

Expert verified
\(\frac{175}{490}\) simplifies to \(\frac{5}{14}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

First, identify the greatest common divisor (GCD) of both the numerator (175) and the denominator (490). This is the largest number that can divide both 175 and 490 without leaving a remainder. Begin by listing the factors of each number. \(175 = 1, 5, 7, 25, 35, 175\) and \(490 = 1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490\). The common factors are \(1, 5, 7, 35\). Thus, the GCD is 35.
02

Divide Both the Numerator and Denominator by the GCD

Now that we have determined that the GCD is 35, divide both the numerator (175) and the denominator (490) by the GCD to simplify the fraction. Calculate \( \frac{175}{35} = 5 \) and \( \frac{490}{35} = 14 \).
03

Write the Simplified Fraction

After dividing both the numerator and denominator by the GCD, the simplified form of the fraction is \( \frac{5}{14} \). Ensure there is no further common divisor other than 1 for both 5 and 14, which confirms it's simplified completely.

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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 concept of the Greatest Common Divisor (GCD) can greatly simplify the process of fraction simplification. The GCD of two numbers is the highest number that divides both numbers without leaving a remainder. It's like finding the biggest piece that perfectly fits into both numbers. For example, when simplifying the fraction \( \frac{175}{490} \), the first step is identifying the GCD of 175 and 490.
  • Factors of 175: 1, 5, 7, 25, 35, 175
  • Factors of 490: 1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490
The biggest common number in both lists is 35. That's our GCD. Finding the GCD can be done by listing the factors as shown or using the Euclidean algorithm for large numbers. By discovering the GCD, you set a solid foundation for the simplification process.
Numerator and Denominator Division
Once you have the GCD, the next step is to divide both the numerator and the denominator by this number. This step requires you to handle basic division accurately. In the fraction \( \frac{175}{490} \), we found the GCD to be 35.Now, divide both parts:
  • Numerator: \( \frac{175}{35} = 5 \)
  • Denominator: \( \frac{490}{35} = 14 \)
After doing the math, you're left with a new, smaller fraction, \( \frac{5}{14} \), that is equivalent to the original. This step is essentially about scaling down the fraction proportionally. If you do this right, you maintain the value of the fraction while making it simpler to work with or understand.
Simplified Fraction
A simplified fraction is the most reduced form of the original fraction, ensuring that the numerator and denominator have no common divisors other than one. For \( \frac{175}{490} \), after dividing by the GCD, we ended up with \( \frac{5}{14} \). In this simplified form, 5 and 14 have no common factors other than 1, which makes them coprime. To verify, you can list the factors again or simply check:
  • Factors of 5: 1, 5
  • Factors of 14: 1, 2, 7, 14
The only common factor is 1, which means further simplification isn't possible. Simplifying fractions to their lowest terms like this helps make calculations easier and more intuitive in further mathematical operations, especially when comparing or adding fractions.

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Most popular questions from this chapter

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