Problem 35 - Precalculus

Question

Say whether the statement is TRUE or FALSE. (In Exercises \(37-40\), do not use a calculator or table; use instead the approximations \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$\frac{13}{14}>\frac{15}{16}$$

Step-by-Step Solution

Answer

The statement is FALSE.

Step 1: Convert Fractions to Decimal Approximations

The fraction \( \frac{13}{14} \) can be approximated by performing the division 13 divided by 14. Similarly, \( \frac{15}{16} \) can be approximated by dividing 15 by 16.- \( \frac{13}{14} \approx 0.9286 \) (as \( 13 \div 14 \approx 0.9286 \))- \( \frac{15}{16} \approx 0.9375 \) (as \( 15 \div 16 \approx 0.9375 \))

Step 2: Compare the Decimal Values

Now compare the decimal approximations of the two fractions.- Check if \( 0.9286 > 0.9375 \).- Since \( 0.9286 < 0.9375 \), the inequality is not true.

Step 3: Determine the Truth of the Statement

Based on the comparison, determine whether the original statement \( \frac{13}{14} > \frac{15}{16} \) is true or false.Since \( 0.9286 < 0.9375 \), the original statement is FALSE.

Key Concepts

Decimal Approximation

When working with fractions, it's often helpful to convert them into decimal numbers. This makes it easier to compare their sizes and understand their values. The process of decimal approximation involves dividing the numerator by the denominator to get a decimal equivalent.
- For example, to find the decimal approximation of \( \frac{13}{14} \), divide 13 by 14. The result is approximately \( 0.9286 \).
- Similarly, for \( \frac{15}{16} \), dividing 15 by 16 gives around \( 0.9375 \).
Using decimal approximations helps us easily see which of the two fractions represents a larger or smaller quantity. Keep in mind, though, decimal approximations may sometimes overlook the small decimals that still matter in precise calculations.

Inequality Evaluation

Once you have decimal approximations, the next step is to evaluate the inequality. This means comparing the calculated decimal values to see which one is greater, or if they are equal.
- To test the inequality \( \frac{13}{14} > \frac{15}{16} \), we compare \( 0.9286 \) and \( 0.9375 \).
- Since \( 0.9286 \) is less than \( 0.9375 \), the statement \( \frac{13}{14} > \frac{15}{16} \) is false.
Inequality evaluation is a crucial step in verifying mathematical statements. By using decimal approximations, it becomes intuitive and straightforward to assess which number is larger.

Mathematical Reasoning

Mathematical reasoning is the thought process involved in solving problems and proving statements. It's about drawing logical connections based on mathematical rules and principles.
- In this exercise, mathematical reasoning allows us to conclude that \( \frac{13}{14} \) is not greater than \( \frac{15}{16} \) after evaluating their decimal approximations.
- You can think of mathematical reasoning as the bridge between performing calculations (like getting decimal approximations) and making conclusions (like determining truthfulness of an inequality).
By using both calculations and logical thinking, mathematical reasoning helps ensure that our conclusions are based on solid foundations and accurate interpretations of the data available.