Problem 70 - 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

The result is approximately 0.419.

Step 1: Evaluate the Numerator

First, calculate the expression in the numerator of the fraction. The numerator is given by \(0.3 + 1.5\). We add these numbers: \(0.3 + 1.5 = 1.8\).

Step 2: Evaluate the Denominator

Next, calculate the expression in the denominator of the fraction. The denominator is given by \(5.5 - 1.2\).Subtract these numbers: \(5.5 - 1.2 = 4.3\).

Step 3: Divide the Numerator by the Denominator

Now, divide the evaluated numerator by the evaluated denominator. Use the results from Step 1 and Step 2:\(\frac{1.8}{4.3} = 0.4186046511627907...\)Enter this calculation in your calculator.

Step 4: Round the Result

Round the result from Step 3 to the nearest thousandth. The result was approximately 0.4186046511627907%. Rounding to the nearest thousandth, the value becomes 0.419.

Key Concepts

Numerator and Denominator Evaluation

When dealing with fractions, it's essential to understand both the numerator and the denominator. The numerator is the top part of a fraction, showing how many parts of the whole you have. The denominator, on the other hand, is the bottom part, indicating the number of equal parts the whole is divided into.

In our given exercise, the numerator expression was \(0.3 + 1.5\). To evaluate it, simply add the two numbers, which gives us \(1.8\). The denominator was \(5.5 - 1.2\). Subtracting these numbers, we get \(4.3\).

• Numerator: Top part of a fraction (e.g., \(1.8\) in this case).
• Denominator: Bottom part of a fraction (e.g., \(4.3\) here).

Proper evaluation of both these components is crucial, as errors in addition or subtraction can lead to incorrect results in your fraction calculations. Always double-check your arithmetic to ensure accuracy.

Fraction Division

Dividing one number by another is at the heart of understanding fractions. Once you've evaluated both the numerator and denominator, you divide the two to get your fraction's value.

Using our example, to find the fraction value, we divide \(1.8\) by \(4.3\). Upon division, you get a result of approximately \(0.4186046511627907\). This value might seem a bit daunting at first due to its length, but it just shows how precise division results can be when using a calculator. This step shows the beauty of fractions — representing a relationship between two numbers as a single value.

• Ensure both numerator and denominator are correctly evaluated.
• Use a calculator for accuracy, as manual division can lead to small mistakes.

This division gives the exact decimal representation of your fraction — up to as many decimal places as you need. However, knowing how to handle this result and when to stop is just as crucial.

Rounding Decimal Numbers

Rounding is a key skill when working with decimals, providing a means to simplify numbers for easier interpretation or reporting. The result of dividing \(1.8\) by \(4.3\) was a lengthy decimal: \(0.4186046511627907\). To make real-world usage easier, we often round numbers to a certain decimal place.

Our exercise specifically asked to round to the nearest thousandth. The thousandth place is the third digit after the decimal point. Here, the number is \(0.418\). Look at the next digit (which is \(6\)) to decide if \(8\) will stay the same or round up. Since \(6\) is greater than \(5\), \(8\) rounds up to \(9\). Thus, the rounded result is \(0.419\).

• Locate the place value you need to round to (e.g., thousandth).
• Check the next digit to adjust the rounding correctly.

Proper rounding helps in maintaining the necessary precision while avoiding information overload from overly long decimal numbers. Being skilled at this makes handling fractions and decimals much simpler.