/*! 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 22 Replace the question mark by \(&... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 22

Replace the question mark by \(<,>,\) or \(=\), whichever is correct. \(\frac{1}{3} ? 0.33\)

Short Answer

Expert verified
\( \frac{1}{3} > 0.33 \)

Step by step solution

01

Convert Fraction to Decimal

Convert the fraction \(\frac{1}{3}\) to its decimal form. \(\frac{1}{3} = 0.3333...\). This decimal repeats infinitely.
02

Compare Decimals

Compare the decimal form of \(\frac{1}{3} \) which is 0.3333... with 0.33. Since the decimal 0.3333... is slightly larger than 0.33, determine the correct inequality.
03

Write the Correct Symbol

Since 0.3333... is greater than 0.33, replace the question mark with the symbol \( > \).

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Key Concepts

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

Fraction to Decimal Conversion
Converting fractions to decimals is a common task in mathematics. It helps simplify comparisons and calculations. To convert a fraction to a decimal, we divide the numerator (top number) by the denominator (bottom number). For example, let's convert the fraction \(\frac{1}{3}\).
Divide 1 by 3, and you will get 0.3333..., where the digit 3 repeats infinitely. This is called a repeating decimal. Knowing how to convert fractions to decimals is essential for understanding and comparing numbers effectively.
Repeating Decimals
Repeating decimals occur when a decimal number has one or more digits that repeat infinitely. It often happens when converting fractions whose denominators are not factors of 10. In our example, \(\frac{1}{3}\) converts to 0.3333..., and the digit 3 repeats forever.
Here are some key points about repeating decimals:
  • They are represented with a bar over the repeating digits, like this: \(\frac{1}{3} = 0.\bar{3}\).
  • Understanding how to work with repeating decimals is crucial for accurate comparisons and calculations.
Knowing that \(\frac{1}{3} = 0.3333...\) helps clarify the magnitude of the number compared to other decimals.
Decimal Comparison
Comparing decimals involves determining which number is greater, smaller, or if they are equal. To compare two decimals, start by aligning their decimal points and compare digits from left to right. Let's compare 0.3333... and 0.33.
Following these steps:
  • Align the decimal points: 0.3333... and 0.3300...
  • Compare digit by digit: 3 is equal to 3, 3 is equal to 3, but the additional 3s in 0.3333... make it larger than 0.33.
Thus, 0.3333... is greater than 0.33. This understanding helps us write the inequality \( \frac{1}{3} > 0.33 \). Effective decimal comparison is vital in mathematics for accurate and meaningful results.

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