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Chapter 1: Problem 31

Simplify each fraction by reducing it to its lowest terms. $$\frac{15}{18}$$

Short Answer

Expert verified
The simplified form of the fraction \(\frac{15}{18}\) is \(\frac{5}{6}\)

Step by step solution

01

Identify the numerator and the denominator

In the fraction \(\frac{15}{18}\), the numerator is 15 and the denominator is 18.
02

Find the Greatest Common Divisor (GCD)

The GCD of 15 and 18 is 3. This can be found by listing all the divisors of both numbers and pick the largest number that appears in both lists.
03

Divide the numerator and the denominator by the GCD

Divide the numerator 15 and the denominator 18 by the GCD 3 to get \(\frac{15 \div 3}{18 \div 3} = \frac{5}{6}\)

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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 a fundamental concept in mathematics that helps simplify fractions. It's the largest number that divides two or more numbers without leaving a remainder.
For finding the GCD of two numbers:
  • List all the positive divisors of each number.
  • Identify the largest number that appears in both lists.
In our example with the fraction \(\frac{15}{18}\), the positive divisors are:
  • Divisors of 15: 1, 3, 5, and 15.
  • Divisors of 18: 1, 2, 3, 6, 9, and 18.
The common divisors are 1 and 3, with 3 being the greatest. Thus, the GCD of 15 and 18 is 3. Knowing the GCD allows us to simplify the fraction effectively by reducing both the numerator and denominator by this factor.
Numerator and Denominator
In any fraction, understanding what the numerator and the denominator are is key to manipulating and simplifying them. The numerator is the top part of the fraction, while the denominator sits at the bottom.
Let's take \(\frac{15}{18}\) as an example:
  • Numerator: 15
  • Denominator: 18
The numerator tells us how many parts we have, whereas the denominator tells us into how many equal parts the whole is divided. When reducing fractions, these two important numbers need to be manipulated in a way that keeps their ratio equivalent, which is done by dividing both by their GCD. This process ensures that the value of the fraction doesn't change, even though its appearance does.
Reducing Fractions
Reducing fractions means simplifying them to their most basic form. This involves ensuring that the numerator and denominator have no common divisors other than 1. It makes fractions easier to understand and work with.
To reduce a fraction:
  • Find the GCD of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
For \(\frac{15}{18}\), our GCD was found to be 3. By dividing the numerator and denominator by 3, the fraction becomes \(\frac{5}{6}\).
This process does not change the value of the fraction. Instead, it merely redistributes it into smaller, more manageable parts, making it simpler to read and work with in further calculations.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d} ; \quad \frac{3}{7} \cdot \frac{2}{5}$$

Find this sum, indicated by a question mark. \(3(-3)=(-3)+(-3)+(-3)=?\)

Write each phrase as an algebraic expression. a loss of \(\frac{1}{3}\) of an investment of \(d\) dollars

In Palo Alto, California, a government agency ordered computer-related companies to contribute to a pool of money to clean up underground water supplies. (The companies had stored toxic chemicals in leaking underground containers.) The mathematical model $$ C=\frac{200 x}{100-x} $$ describes the cost, \(C,\) in tens of thousands of dollars, for removing \(x\) percent of the contaminants. Use this formula to solve. a. Find the cost, in tens of thousands of dollars, for removing \(50 \%\) of the contaminants. b. Find the cost, in tens of thousands of dollars, for removing \(80 \%\) of the contaminants. c. Describe what is happening to the cost of the cleanup as the percentage of contaminant removed increases.

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The quotient of \(-18\) and the sum of \(-15\) and 12
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