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Chapter 5: Problem 30

Express each rational number as a decimal. \(\frac{5}{16}\)

Short Answer

Expert verified
The decimal form of \( \frac{5}{16} \) is 0.3125.

Step by step solution

01

Perform the division

Divide the numerator (5) by the denominator (16) using long division or a calculator.
02

Write the result as a decimal

The result of the division represents the rational number as a decimal. Write this down.

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Key Concepts

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

Rational Numbers
Rational numbers are numbers that can be expressed as a fraction where both the numerator and denominator are integers. These numbers represent a ratio between the two integers, with the numerator as the upper number and the denominator as the lower one. Rational numbers are part of the real number system. They can be finite decimals, infinite repeating decimals, or whole numbers.

  • A fraction like \( \frac{5}{16} \) is rational because 5 and 16 are both integers, and the denominator is not zero.
  • The decimal representation of rational numbers may terminate or repeat.
  • For example, \( \frac{1}{2} \) is 0.5, a terminating decimal, while \( \frac{1}{3} \) is 0.333..., a repeating decimal.

Recognizing and converting rational numbers into decimal form through division helps you see the exact value a fraction represents.
Long Division
Long division is a method used to divide two numbers and is especially handy when dividing a smaller number by a larger number, as with fractions. In our example of dividing 5 by 16, long division can help us find the decimal equivalent of \( \frac{5}{16} \).

Here's how long division works:
  • Start by determining how many times the divisor (16) can fit into the dividend (5). Since 16 does not fit into 5, put a 0 before the decimal point in the quotient.
  • Next, add a decimal point and a zero to make it 50. Now consider how many times 16 goes into 50, which is 3 times, because 16 times 3 is 48.
  • The remaining number is 2 after subtracting 48 from 50. Bring down another 0 to make it 20. Ask again how many times 16 fits into 20, which again is 1.
  • Continue this process to get more decimal places, resulting in 0.3125, a terminating decimal.

This technique, while lengthy, allows you to convert fractions into their decimal forms without a calculator. It is very useful for understanding how division of integers translates to decimal numbers.
Numerator and Denominator
In a fraction, the numerator and denominator are foundational concepts to understand. The numerator is the top number which indicates how many parts of the whole are being considered, while the denominator is the bottom number, showing how many equal parts the whole is divided into. For instance, in the fraction \( \frac{5}{16} \), 5 is the numerator, and 16 is the denominator.

  • The denominator determines the size of each part.
  • The numerator tells you how many of these parts you have.
  • When doing division for conversion into decimal form, the numerator is the number you're dividing, and the denominator is the number by which you divide.

Understanding these roles is critical when performing operations like long division to convert fractions into decimals. This basic knowledge underpins your ability to work comfortably with fractions and decimals alike.

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(0.0007,-0.007,0.07,-0.7, \ldots\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(6,-6,6,-6, \ldots\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(15,30,60,120, \ldots\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(-9,-5,-1,3, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=3, r=-2\)
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