Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ \left|\frac{0.2-1.4}{1.6-2.4}\right| $$

Step-by-Step Solution

Answer

The given expression, \(\left|\frac{0.2-1.4}{1.6-2.4}\right|\), evaluates as follows: calculate the numerator \(-1.2\), the denominator \(-0.8\), and then the fraction \(\frac{-1.2}{-0.8} = 1.5\). Taking the absolute value gives the final result: \(|1.5| = 1.5\).

Step 1: Calculate the numerator

First, let's calculate the numerator of the fraction: \(Numerator = 0.2 - 1.4\) Now, subtract the two numbers. \(Numerator = -1.2\)

Step 2: Calculate the denominator

Now, let's calculate the denominator of the fraction: \(Denominator = 1.6 - 2.4\) Now, subtract the two numbers. \(Denominator = -0.8\)

Step 3: Calculate the fraction

Now, let's calculate the fraction: \(\frac{Numerator}{Denominator} = \frac{-1.2}{-0.8}\) Now, divide the numerator by the denominator. \(\frac{-1.2}{-0.8} = 1.5\)

Step 4: Calculate the absolute value

Finally, let's calculate the absolute value of the fraction: \(\left|\frac{-1.2}{-0.8}\right| = |1.5|\) The absolute value of 1.5 is 1.5 since it is already a positive number. Therefore, the result of the given expression is 1.5.

Key Concepts

Numerator and Denominator

When dealing with fractions, it's essential to understand what the numerator and denominator are. These two parts make up every fraction:

• The numerator is the top part of the fraction. It represents how many parts we are considering.
• The denominator is the bottom part of the fraction. It indicates the total number of equal parts the whole is divided into.

In the exercise, the expression involves subtracting numbers to find the numerator and the denominator:

• Numerator: Calculated as \(0.2 - 1.4\), which results in \(-1.2\).
• Denominator: Calculated as \(1.6 - 2.4\), giving \(-0.8\).

Understanding these components is crucial, especially when simplifying or evaluating fractions.

Fractions

Fractions represent parts of a whole and are useful in many mathematical expressions. With fractions:

• A positive numerator and a negative denominator (or vice versa) result in a negative fraction.
• Both a negative numerator and a denominator result in a positive fraction since two negatives make a positive.

In our exercise, the fraction \(\frac{-1.2}{-0.8}\) is addressed:- The negativity of both parts cancels each other, resulting in a positive outcome.- By dividing \(-1.2\) by \(-0.8\), the result is \(1.5\).
This step highlights how basic divisions and sign rules can simplify complex expressions.

Step by Step Solution

Breaking problems into steps makes them manageable and easier to understand. This method involves calculating smaller parts one at a time:1. **Calculate the Numerator**: - Subtract \(1.4\) from \(0.2\) to find \(-1.2\).2. **Calculate the Denominator**: - Subtract \(2.4\) from \(1.6\) to get \(-0.8\).3. **Compute the Fraction**: - Divide the numerator \(-1.2\) by the denominator \(-0.8\), resulting in \(1.5\). 4. **Find the Absolute Value**: - Consider the absolute value, \(\left|1.5\right|\), which remains \(1.5\) because it’s already positive.
Taking each step carefully ensures a clear understanding of how to solve similar problems. Remember, when working with fractions and absolute values, each step builds on the previous one, leading to the correct final answer.