/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Convert each fraction to a decim... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 39

Convert each fraction to a decimal. Round to five decimal places. \(\frac{3}{9}\)

Short Answer

Expert verified
The fraction \( \frac{3}{9} \) converts to a decimal as 0.33333.

Step by step solution

01

Understand the problem

The goal is to convert the fraction \( \frac{3}{9} \) into its decimal form and round the result to five decimal places.
02

Simplify the fraction

To simplify the fraction, divide the numerator (3) and the denominator (9) by their greatest common divisor (GCD), which is 3: \[ \frac{3 \div 3}{9 \div 3} = \frac{1}{3} \] Now the fraction is \( \frac{1}{3} \).
03

Convert the fraction to a decimal

Divide the numerator by the denominator: \( 1 \div 3 = 0.33333\ldots \). The decimal repeats indefinitely, so it is \( 0.\overline{3} \).
04

Round the decimal

Since the repeating decimal \( 0.\overline{3} \) would continue as \( 0.33333\ldots \), we simply take the first five decimals after the decimal point. Thus, it remains \( 0.33333 \).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is the process of reducing the fraction to its simplest form. This means finding an equivalent fraction where the numerator and the denominator are as small as possible. You achieve simplification by finding the greatest common divisor (GCD) of the numerator and denominator and then dividing both by this number.
  • For example, to simplify the fraction \(\frac{3}{9}\), we find that the GCD of 3 and 9 is 3.
  • We divide both top and bottom by 3 to get \(\frac{3 \div 3}{9 \div 3} = \frac{1}{3}\).
This equivalent fraction, \(\frac{1}{3}\), is much simpler to work with. The art of fraction simplification helps in making calculations easier and more intuitive. It鈥檚 an essential step before diving into converting fractions to decimals.
Repeating Decimals
Decimals can be either terminating or repeating. A repeating decimal is a decimal fraction that after some initial digits begins to represent a sequence of numbers which repeats indefinitely. Identifying and understanding repeating decimals is crucial in mathematical conversions and calculations.
  • When you convert \(\frac{1}{3}\) to a decimal, you perform the division \(1 \div 3\).
  • You'll find that the result is 0.33333... where the number 3 repeats indefinitely. This is denoted as \(0.\overline{3}\).
Recognizing repeating decimals is important because even though they seem infinite, they follow a predictable pattern. Handling these correctly is vital when rounding or performing further calculations with the number.
Rounding Decimals
Rounding decimals is a way to simplify numbers by reducing the number of digits right after the decimal point. Rounding can make numbers easier to work with, especially in real-life applications where precision to many decimal places is often unnecessary.
  • In the exercise of converting \(\frac{3}{9}\) to its decimal form, after simplification to \(\frac{1}{3}\), we found the repeating decimal \(0.\overline{3}\) or \(0.33333\ldots\).
  • When rounding this to five decimal places, we consider only the first five digits after the decimal point.
  • Thus, the rounded decimal for this fraction is simply 0.33333.
The skill of rounding is useful as it helps in expressing long decimal fractions in a more manageable form, while preserving as much accuracy as needed.

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Most popular questions from this chapter

Perform each operation. $$ \frac{3}{5} \cdot\left(\frac{1}{2}+1.75\right) $$

Convert each decimal to a proper fraction or a mixed number. \(4.01 \frac{1}{27}\)

Perform each operation and simplify. \(41.8 \cdot 0.19\)

Perform the following operations. \(7+0.15 \div \frac{3}{30}\)

Perform the following operations. \(0.5 \frac{1}{4}+(0.3)^{2}\)
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