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Chapter 13: Problem 43

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$3 \frac{7}{10}$$

Short Answer

Expert verified
The decimal form of \(3 \frac{7}{10}\) is 3.7.

Step by step solution

01

Convert the Mixed Number to an Improper Fraction

First, we need to convert the mixed number \(3 \frac{7}{10}\) into an improper fraction. A mixed number consists of a whole number and a fraction. To convert it, multiply the whole number (3) by the denominator of the fraction (10) and then add the numerator of the fraction (7). This gives us: \(3 \times 10 + 7 = 37\). The improper fraction is \(\frac{37}{10}\).
02

Divide to Find the Decimal

Now, we need to convert the improper fraction \(\frac{37}{10}\) into a decimal by dividing the numerator (37) by the denominator (10). Performing the division gives \(37 \div 10 = 3.7\).
03

Check for Repeating Decimal

Check the decimal to see if it repeats. In this case, 3.7 is a finite decimal, meaning it does not repeat. Therefore, there is no need to add a bar over any digits.

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Key Concepts

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

Mixed Numbers
A mixed number is a mathematical term that combines a whole number with a fraction. For example, if you've seen something like \(3 \frac{7}{10}\), that's a mixed number.
This representation tells us that we have 3 whole parts and 7 tenths of another part. To work with mixed numbers, often we need to convert them into improper fractions first. Here's why.
  • Understanding Mixed Numbers: Mixed numbers make it easy to visualize quantities, as they show both whole parts and fractional parts clearly.
  • Conversion to Improper Fractions: For calculations, converting a mixed number into an improper fraction is crucial.
  • Ease of Calculation: Converting to improper fractions means we can perform operations like addition, subtraction, multiplication, and division with ease.
When you want to convert a mixed number to an improper fraction, simply multiply the whole number by the denominator then add the numerator. In our example, with \(3 \times 10 + 7\), we get \(\frac{37}{10}\).
This makes it easier to perform further operations.
Improper Fractions
Improper fractions come into play when the numerator is greater than the denominator.
This means the fraction is actually representing a value greater than one.
In our example, \(\frac{37}{10}\) is an improper fraction.
  • Conversion: Any improper fraction can be converted back into a mixed number by dividing the numerator by the denominator.
  • Understanding Improper Fractions: They often look unusual but simplify mathematical functions as they do not require division into parts.
  • Applications: Improper fractions are used in calculations like division of fractions, converting to decimal numbers, and more.
To convert an improper fraction into a decimal, you simply divide the numerator by the denominator.
For instance, with \(\frac{37}{10}\), dividing 37 by 10 gives the decimal 3.7.
This is the representation of the mixed number \(3 \frac{7}{10}\) as a decimal.
Repeating Decimals
Repeating decimals are special kinds of decimals where one or more digits repeat infinitely.
They are denoted by placing a bar over the repeating section of numbers.
For example, in a decimal like 0.333..., the digit 3 repeats endlessly.
  • Finite vs Repeating: Not all divisions lead to repeating decimals; some, like our example, end after a few digits.
  • Identifying Repeating Decimals: You can see a repeating pattern by dividing the numerator by the denominator and checking if the remainder repeats.
  • Notation: The bar over the repeating section helps in identifying and communicating repeating decimals clearly.
In our calculation of \(\frac{37}{10}\), we find that 3.7 is a finite decimal as it stops without repeating.
This indicates that the division was exact without any repeating sequence in the result.

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