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Chapter 6: Problem 57

Write each fraction in simplest form. See Section 4.2. $$ \frac{10}{12} $$

Short Answer

Expert verified
The simplest form of \(\frac{10}{12}\) is \(\frac{5}{6}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify a fraction, we need to find the greatest common divisor of the numerator and the denominator. The numerator is 10, and the denominator is 12. We need to find the largest number that divides both 10 and 12 without leaving any remainder. The divisors of 10 are 1, 2, 5, and 10, while the divisors of 12 are 1, 2, 3, 4, 6, and 12. The common divisors are 1 and 2. Among these, the greatest is 2.
02

Divide Both Terms by the GCD

Now that we have identified that the greatest common divisor is 2, we will divide both the numerator and the denominator by 2. This will give us the simplest form of the fraction.\[\frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}\]
03

Check the Simplified Fraction

To make sure we have simplified the fraction correctly, we should check that there is no common divisor between the new numerator and the denominator except for 1. In this case, 5 and 6 have no common divisors other than 1, confirming \(\frac{5}{6}\) is in its simplest form.

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Key Concepts

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

Greatest Common Divisor
The concept of the greatest common divisor (GCD) is essential in simplifying fractions. When we have a fraction, such as \(\frac{10}{12}\), our goal is to find the largest number that can divide both the numerator and the denominator without leaving any remainder. This number is known as the greatest common divisor.

In this given example, we need to find the GCD of 10 and 12:
  • Divisors of 10: 1, 2, 5, 10
  • Divisors of 12: 1, 2, 3, 4, 6, 12
  • Common divisors: 1 and 2
The greatest of these common divisors is 2, which makes it our GCD. Finding the GCD is vital because it helps us reduce the fraction to its simplest form by dividing both the numerator and the denominator by this number.
Numerator and Denominator
To understand fractions, it's important to clearly define the terms numerator and denominator. In any fraction, such as \(\frac{10}{12}\), the number on top is called the numerator, and the number below the line is called the denominator.

The numerator (10 in this case) represents how many parts of the whole we have, whereas the denominator (12 here) indicates into how many total parts the whole is divided.
  • Numerator (10): the parts we are focused on
  • Denominator (12): the total parts that make up the whole
Understanding numerators and denominators is key to performing operations like simplification, as they help determine how we reduce the fraction using the greatest common divisor.
Fraction Simplification
Simplifying a fraction means rewriting it in the simplest or lowest terms. After identifying the greatest common divisor, the next step is to divide both the numerator and the denominator by this number. This process does not change the value of the fraction, only its appearance.

For the fraction \(\frac{10}{12}\), we found the GCD to be 2. We can simplify as follows:
  • Divide the numerator by the GCD: \(10 \div 2 = 5\)
  • Divide the denominator by the GCD: \(12 \div 2 = 6\)
  • Thus, the fraction simplifies to \(\frac{5}{6}\)
It's also crucial to double-check the simplified fraction to ensure no common divisor exists between the new numerator and the denominator other than 1. In this case, 5 and 6 don't share common divisors other than 1, confirming \(\frac{5}{6}\) is indeed simplified.

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