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Chapter 1: Problem 20

Given: \(\quad \frac{1}{2}=0.5\) and \(0.5=50 \% ;\) Transitive Property of Equality

Short Answer

Expert verified
\(\frac{1}{2} = 50\%\) by the transitive property.

Step by step solution

01

Understand the Given Information

We are given a set of equalities: \( \frac{1}{2} = 0.5 \) and \( 0.5 = 50\% \). These are conversions between fractions, decimals, and percentages.
02

Identify the Transitive Property of Equality

The transitive property of equality tells us that if \( a = b \) and \( b = c \), then \( a = c \). This means if one element equals a second, and the second equals a third, the first must equal the third.
03

Apply the Transitive Property

Apply the transitive property to our given equations. We have \( \frac{1}{2} = 0.5 \) and \( 0.5 = 50\% \). By the transitive property, \( \frac{1}{2} = 50\% \).

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Key Concepts

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

Fraction Decimal Conversion
Fractions and decimals are two ways to represent the same part-whole relationship in numbers. A decimal is simply another representation of a fraction, often making calculations a bit easier. To convert a fraction to a decimal, you need to divide the numerator (top number) by the denominator (bottom number).
For example, to convert the fraction \( \frac{1}{2} \) into a decimal, you divide 1 by 2:
  • 1 divided by 2 equals 0.5, so \( \frac{1}{2} = 0.5 \).
This is the decimal expression of the fraction. Fractions can be expressed as exact decimals or repeating decimals depending on their divisors. Understanding this conversion is fundamental to solving problems involving mixed forms of numbers.
Percentage Conversion
The concept of percentages represents parts per hundred, a very useful way of expressing fractions in a more universal format. To convert a decimal to a percentage, you multiply the decimal number by 100.
When considering our example, converting 0.5 to percentage:
  • Multiply 0.5 by 100: \( 0.5 \times 100 = 50 \). Therefore, 0.5 is 50%.
This percentage form is intuitive, especially in contexts involving growth, discounts, or statistics. It allows people to quickly understand proportions and compare them to a known scale of 100.
Mathematical Reasoning
Mathematical reasoning involves logical thinking and applying known principles to derive conclusions. One such principle is the Transitive Property of Equality. This states that if one value is equal to a second, and the second is equal to a third, then the first is equal to the third.
In the given exercise, we have:
  • \( \frac{1}{2} = 0.5 \)
  • and \( 0.5 = 50\% \)
By applying the Transitive Property, it follows that \( \frac{1}{2} = 50\% \). This is a practical application of mathematical reasoning, showing that understanding properties and relationships among numbers helps solve equations and identify equalities across different representations.

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