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Chapter 5: Problem 42

Write the fractions as decimals. Explain how to memorize the decimal forms for these fractions with denominator of 3 a. \(\frac{1}{3} \quad\) b. \(\frac{2}{3}\)

Short Answer

Expert verified
The decimal form of \(\frac{1}{3}\) is 0.333 (repeating), and for \(\frac{2}{3}\) is 0.666 (repeating).

Step by step solution

01

Understand the Fraction to Decimal Conversion

To convert a fraction to a decimal, divide the numerator (the top number) by the denominator (the bottom number). For a fraction \(\frac{a}{b}\), perform the division \a \div \b\.
02

Convert \(\frac{1}{3}\) to a Decimal

Perform the division: \1 \div \ 3 \ = 0.3333...\ (repeating). Therefore, \(\frac{1}{3}\) as a decimal is approximately 0.333, but it goes on infinitely as 0.3333... (repeating).
03

Memorize \(\frac{1}{3}\)'s Decimal Form

Memorize that \(\frac{1}{3}\) equals approximately 0.333 (repeating). Visualize the fraction as a pie divided into three equal pieces, where each piece is 0.333 (repeating) of the pie.
04

Convert \(\frac{2}{3}\) to a Decimal

Perform the division: \2 \div \ 3 \ = 0.6666...\ (repeating). Therefore, \(\frac{2}{3}\) as a decimal is approximately 0.666, but it goes on infinitely as 0.6666... (repeating).
05

Memorize \(\frac{2}{3}\)'s Decimal Form

Memorize that \(\frac{2}{3}\) equals approximately 0.666 (repeating). Visualize the fraction as two-thirds of a pie, where each third is 0.333 (repeating), making two-thirds 0.666 (repeating).

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Key Concepts

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

repeating decimals
When we talk about converting fractions to decimals, sometimes the division never ends. This is especially common with certain fractions, resulting in repeating decimals.
The numbers after the decimal point go on forever, forming a repeating pattern.
For example, when converting \(\frac{1}{3} \) to a decimal, you get 0.3333... with the digit '3' repeating endlessly. Similarly, \(\frac{2}{3}\) converts to 0.6666... with an infinite string of '6'.
These repeating blocks are usually represented with an overline: 0.\bar{3}\frac{1}{3}\bar{6}.
Understanding repeating decimals is crucial when working with fractions, as they help make sense of infinite sequences in a more manageable form.
numerator and denominator
Every fraction has two main parts: the numerator and the denominator.
The numerator is the top number and the denominator is the bottom number.
In the fraction \(\frac{1}{3}\), '1' is the numerator and '3' is the denominator.
To convert any fraction to a decimal, you simply divide the numerator by the denominator.
Think of the numerator as the dividend and the denominator as the divisor.
Following this logic, for \(\frac{2}{3}\), you divide 2 (numerator) by 3 (denominator) to get the decimal form.
This basic understanding of numerator and denominator helps to perform the conversions easily and accurately.
memorizing decimal forms
Though it might sound challenging, memorizing some common decimal forms of fractions can simplify math problems.
Knowing that \(\frac{1}{3}\) equals 0.333 (repeating) and \(\frac{2}{3}\) equals 0.666 (repeating) can save time.
One useful trick is to use visual aids, like imagining a pie split into equal parts.
For \(\frac{1}{3}\), picture a pie cut into three pieces, each piece representing 0.333... of the whole pie.
For \(\frac{2}{3}\), think of two-thirds of the pie, which helps visualize why it's 0.666...
With repetition and visualization, these forms stick in memory, assisting in quicker mental math.
division in fractions
Understanding division is fundamental in converting fractions to decimals.
The fraction itself is a division: the numerator divided by the denominator.
For instance, with \(\frac{1}{3}\), you're calculating 1 divided by 3.
This division can be executed long-hand if needed, but understanding the concept is key.
When performing this division for \(\frac{1}{3}\), you鈥檒l find that the result is an endless chain of 3s: 0.3333...
Likewise, \(\frac{2}{3}\) is equivalent to dividing 2 by 3, resulting in 0.6666...
Recognizing that fractions are divisions at their core makes the conversion process intuitive and straightforward.

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