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Chapter 6: Problem 49

Convert to decimal notation. \(\frac{18}{102}\)

Short Answer

Expert verified
0.1764705882

Step by step solution

01

Simplify the fraction

First, simplify the fraction \(\frac{18}{102}\). Find the greatest common divisor (GCD) of 18 and 102. The GCD of 18 and 102 is 6. Now, divide both the numerator and the denominator by their GCD: \(\frac{18 \div 6}{102 \div 6} = \frac{3}{17}\).
02

Convert the simplified fraction to a decimal

Now convert the simplified fraction \(\frac{3}{17}\) to a decimal. Divide the numerator by the denominator: \(\frac{3}{17} = 0.1764705882\) (rounded to 10 decimal places).

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Key Concepts

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

Fraction Simplification
Fractions can often be simplified to make calculations easier. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator are as small as possible. To simplify, we need to find common factors of both the numerator and the denominator.

In this exercise, the fraction \(\frac{18}{102}\) needs to be simplified. We look for the greatest common factor that both numbers share. This step helps in making further calculations, like converting the fraction to decimal, easier and more accurate. By reducing fractions, we ensure that all calculations are done with the simplest numbers.

Simplification: \(\frac{18}{102} = \frac{3}{17}\)
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is vital in simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

We use the GCD to reduce fractions to their simplest form. For \(\frac{18}{102}\), the GCD is 6. This means 6 is the largest number that can evenly divide both 18 and 102.

Steps to find the GCD:
  • List the factors of both numbers. For example, for 18: \(1, 2, 3, 6, 9, 18\), and for 102: \(1, 2, 3, 6, 17, 34, 51, 102\).
  • Identify the largest common factor. Here, it is 6.

Knowing the GCD helps us divide both the numerator and the denominator, making the fraction simpler and the calculation more straightforward.
Converting Fractions to Decimals
One common mathematical task is converting a fraction to decimal notation. This helps in various practical applications, like measurements or financial calculations.

For conversion, we divide the numerator by the denominator. This gives us a decimal form of the fraction. For example, in \(\frac{3}{17}\), divide 3 by 17, resulting in 0.1764705882, rounded to 10 decimal places.

Steps to Convert:
  • Simplify the fraction if necessary. For \(\frac{18}{102}\), we've simplified it to \(\frac{3}{17}\).
  • Use long division to divide the numerator by the denominator.
  • The quotient is the decimal form of the fraction.

Conversion gives us a decimal, which can be easier to work with in certain contexts, offering more insight into the fraction's value. This process is an essential skill in mathematics.

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