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Chapter 5: Problem 7

Reduce each rational number to its lowest terms. \(\frac{60}{108}\)

Short Answer

Expert verified
The rational number \( \frac{60}{108} \) reduces to its lowest terms as \( \frac{5}{9} \).

Step by step solution

01

Identify the Numerator and Denominator

The numerator of our rational number is 60 and the denominator is 108.
02

Find the Greatest Common Divisor (GCD)

Calculate the GCD of 60 and 108. We can do this using the Euclidean algorithm, by continuously dividing the larger number by the smaller number and replacing the larger number with the remainder, until the remainder is 0. When the remainder is 0, the smaller number will be the GCD. Here, the GCD of 60 and 108 is 12.
03

Divide the Numerator and Denominator by the GCD

We then divide both the numerator and the denominator by the GCD, which gives us \(\frac{60}{12}\) over \(\frac{108}{12}\), which simplifies to \(\frac{5}{9}\).

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Key Concepts

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

Lowest Terms
When expressing a fraction or a rational number, it's important to present it in its simplest form, often referred to as its 'lowest terms'. This means that the numerator and the denominator have no common divisors other than 1. Simplifying to lowest terms makes the number easier to understand and work with, especially when performing arithmetic operations or comparing different rational numbers. For example, simplifying the fraction \(\frac{60}{108}\) to \(\frac{5}{9}\) means there are no common factors between 5 and 9, offering the most reduced form of the fraction.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD), also known as the greatest common factor or highest common factor, is the largest positive integer that divides two numbers without leaving a remainder. Finding the GCD is a critical step in simplifying fractions to their lowest terms. The GCD of the numerator and denominator of a fraction, when divided into both, reduces the fraction to its simplest form. For example, with the fraction \(\frac{60}{108}\), the GCD is 12. Dividing both the top and bottom by 12 simplifies the fraction effectively.
Euclidean Algorithm
The Euclidean algorithm is an efficient method for computing the greatest common divisor (GCD) of two numbers, which is vital in the simplification of fractions. It involves a series of divisions, replacing the larger number with the remainder of the prior division, and continuing this process until a remainder of 0 is obtained. The last non-zero remainder is then the GCD. For instance, when finding the GCD of 60 and 108, as in our exercise, we would perform a series of divisions \(108 = 60 \times 1 + 48, 60 = 48 \times 1 + 12, 48 = 12 \times 4 + 0\) to find that the GCD is in fact 12.
Numerator and Denominator
In any rational number, expressed as a fraction, there are two main components: the numerator and the denominator. The numerator, found above the fraction bar, represents the number of parts we have. The denominator, below the fraction bar, indicates the total number of equal parts the whole is divided into. In the fraction \(\frac{60}{108}\), 60 is the numerator and 108 is the denominator. When simplifying a fraction, we perform operations on both the numerator and the denominator to reduce the fraction to its lowest terms while maintaining the value of the rational number.

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Most popular questions from this chapter

State the associative property of multiplication and give an example.

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(18,6,2, \frac{2}{3}, \ldots\)

Explain how to divide integers.

Use a calculator with a square root key to find a decimal approximation for each square root. Round the number displayed to the nearest \(\mathbf{a}\). tenth, b. hundredth, c. thousandth. \(\sqrt{3176}\)

Express each terminating decimal as a quotient of integers. If possible, reduce to lowest terms. \(0.725\)
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