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Chapter 3: Problem 62

Write each fraction in lowest terms. $$ \frac{140}{196} $$

Short Answer

Expert verified
\(\frac{5}{7}\)

Step by step solution

01

Find the Greatest Common Divisor (GCD)

To simplify the fraction, first determine the greatest common divisor (GCD) of the numerator (140) and the denominator (196). Start by writing out the prime factorizations: 140: \(2^2 \times 5 \times 7\)196: \(2^2 \times 7^2\)The common factors are \(2^2\) and \(7\). Therefore, the GCD is \(2^2 \times 7 = 28\).
02

Divide Numerator and Denominator by the GCD

To reduce the fraction to its lowest terms, divide both the numerator and the denominator by their GCD, which is 28:\(\frac{140}{196} = \frac{140 \div 28}{196 \div 28} = \frac{5}{7}\).
03

Write the Simplified Fraction

After dividing, the fraction reduced to its lowest terms is \(\frac{5}{7}\).

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Key Concepts

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

Greatest Common Divisor (GCD)
To simplify a fraction, the first step is to find the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder. Finding the GCD helps in reducing the fraction to its simplest form. For instance, to simplify the fraction \(\frac{140}{196}\), we need to find the GCD of 140 and 196. A good method to find the GCD is through prime factorization.
Prime Factorization
Prime factorization involves breaking down a number into its basic building blocks - prime numbers. For our example with \(\frac{140}{196}\), we'll break down both 140 and 196 into their prime components. Here's how you can do it:
  • 140 can be broken down into \(2^2 \times 5 \times 7\).
  • 196 is broken down into \(2^2 \times 7^2\).
By identifying the common factors, we can determine the greatest common divisor (GCD). The common prime factors between 140 and 196 are \(2^2\) and 7. Thus, the GCD of 140 and 196 is \(2^2 \times 7 = 28\).
Fraction Reduction
Once the GCD is found, the next step is to divide both the numerator and the denominator by this number. This process reduces the fraction to its lowest terms. For our fraction \(\frac{140}{196}\), we divide both 140 and 196 by the GCD (28):
\( \frac{140}{196} = \frac{140 \div 28}{196 \div 28} = \frac{5}{7} \)
This simplifies the fraction to \(\frac{5}{7}\). Reducing fractions makes them easier to understand and work with, especially in further calculations or real-life applications.

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