Problem 57 - Elementary Algebra

Question

Determine whether the statements for the following problems are true or false. $$\frac{20+4.3}{16}<5$$

Step-by-Step Solution

Answer

Answer: The statement is true.

Step 1: Simplify the numerator of the fraction

Begin by simplifying the numerator of the fraction: $$(20 + 4.3) = 24.3$$

Step 2: Simplify the fraction

Now, we simplify the fraction by dividing the numerator by the denominator: $$\frac{24.3}{16} = 1.51875$$

Step 3: Compare to the right side of the inequality

Now, compare the result of the fraction with the right side of the inequality (5): $$1.51875 < 5$$ Since the inequality holds, the statement is .

Step 4: Provide the answer

The statement is true since \(1.51875 < 5\).

Key Concepts

Fraction Simplification

When it comes to fractions, simplifying them can make comparisons much easier. Simplifying a fraction involves reducing its expression so that both the numerator and the denominator are coprime or finding the decimal equivalent when there's no simpler form.

Here's a simple roadmap to simplify a fraction:

• First, if possible, find the greatest common divisor (GCD) of both the numerator and denominator, and divide both by that number.
• If your fraction results in a decimal, like in our exercise, you can compute this division directly to get a clear numerical value.

Reducing or expressing in decimal form helps us easily determine relationships, especially in contexts like inequalities.

Numerator and Denominator

Each fraction has two key components: the numerator and the denominator. In our particular problem, the fraction \(\frac{20+4.3}{16}\) has its numerator as \(20 + 4.3\) and the denominator as 16.

• The numerator, \(20 + 4.3\), represents the total portion or quantity we are referring to.
• The denominator, 16, stands for the number of equal parts the numerator is divided into.

When working with fractions, clearly identifying these components helps in performing operations like addition, subtraction, or multiplication. In inequalities or equations, understanding the basic arithmetic of the numerator and denominator is crucial.

Comparing Numbers

Comparing numbers is straight-forward, yet essential to solving inequality problems. After converting fractions to decimals or reducing them, we can easily compare one number to another.

In this exercise, we simplify \(\frac{24.3}{16}\) to 1.51875. This decimal representation helps in recognizing the smaller or larger value by straightforward comparison when set against another number, like 5 in our case.

Helpful Tips:

• Always ensure all numbers you compare are in the same form, either decimals or fractions.
• If needed, round your numbers appropriately to match the precision required in the problem.

Practice comparing different numbers will refine your skills in assuming the correct relational operator (\(<\), \(>\), \(=\)) between them.

Truth Value in Inequalities

Understanding the truth value of inequalities is key in determining whether statements are correct or not.

In our exercise, we are asked if \(1.51875 < 5\). Since 1.51875 is indeed less than 5, the statement is proven true.

Considerations for Evaluating Truth Values:

• Compute calculations precisely to avoid errors that may lead to misinterpretation of inequalities.
• Understand the implications of each inequality symbol: \(<\) means "less than," \(>\) signifies "greater than," and \(=\) indicates equality.

Being adept at judging these symbols helps not only in math problems, but also in real-world decision-making when placing values in a contextual relationship.