Problem 71 - College Algebra with Modeling and Visualization

Question

Use a calculator to evaluate the expression. Round your result to the nearest thousandth. $$ \frac{0.3+1.5}{5.5-1.2} $$

Step-by-Step Solution

Answer

0.419

Step 1: Calculate the Numerator

First, find the sum of the numbers in the numerator. Add 0.3 and 1.5. \[ 0.3 + 1.5 = 1.8 \]

Step 2: Calculate the Denominator

Find the difference of the numbers in the denominator. Subtract 1.2 from 5.5. \[ 5.5 - 1.2 = 4.3 \]

Step 3: Divide the Numerator by the Denominator

Now divide the result from Step 1 by the result from Step 2 to evaluate the expression.\[ \frac{1.8}{4.3} \approx 0.41860 \]

Step 4: Round the Result

Round the answer from Step 3 to the nearest thousandth. Look at the fourth decimal place to determine whether to round up or keep it the same. 0.41860 rounds to 0.419.

Key Concepts

Numerator and Denominator

In the world of rational expressions, the numerator and denominator are key components you need to understand. A rational expression is essentially a fraction consisting of a numerator—the number or expression above the fraction line—and a denominator—the number or expression below it.

• The numerator represents the part of the expression that you need to sum or consider first. In the exercise, this was represented by the expression \(0.3 + 1.5\). When you calculate, you first add these numbers together to get \(1.8\).
• The denominator, conversely, is the part of the expression that signals how the numerator should be divided. In the example, you perform \(5.5 - 1.2\) and get the result \(4.3\).

Understanding how to find the numerator and the denominator properly is crucial for successfully evaluating any rational expression.

Evaluating Expressions

Evaluating a rational expression involves simplifying and calculating it to find a numerical value. Once you have determined both the numerator and denominator:

• Solve the operations within the numerator and the denominator separately.
• Following the exercise example, once you have \(1.8\) as the numerator and \(4.3\) as the denominator, you need to divide them: \(\frac{1.8}{4.3}\).
• Use a calculator for better precision, especially when numbers result in decimals. For this exercise, the result was approximately \(0.41860\).

This step is crucial since evaluating entails working through each part logically and carefully to reach the correct answer.

Rounding Numbers

Rounding numbers is an essential skill, especially when dealing with decimal results. Once you have evaluated the expression, you might end up with a long decimal number. Here, rounding comes into play.

• Identify the decimal place to which you need to round. In this case, it's to the nearest thousandth place.
• Look at the digit immediately following the thousandth place to decide your rounding direction. For the answer \(0.41860\), the critical digit is the fourth one, which is \(6\).
• If this digit is 5 or greater, round up. If it is less than 5, round down. For \(0.41860\), you round up from \(0.4186\) to \(0.419\).

Proper rounding ensures you communicate your results efficiently and accurately, which is especially helpful in mathematics and sciences.