Simplifying the Variable Term

Oct 13, 2025 by miodeya.com 57 views

Introduction

In this article, we will delve into the world of mathematics and explore a numerical expression that represents the depth of a diver as they ascend from a depth of 200 feet. We will break down the expression, simplify it, and provide a clear understanding of the mathematical concepts involved.

The Numerical Expression

The numerical expression that represents the depth of the diver is given by:

-200 + 12 \frac{1}{3}(4.5)

This expression can be broken down into two parts: a constant term and a variable term. The constant term is -200, which represents the initial depth of the diver. The variable term is $12 \frac{1}{3}(4.5)$ , which represents the rate at which the diver ascends multiplied by the time it takes to ascend.

Simplifying the Variable Term

To simplify the variable term, we need to multiply the rate at which the diver ascends by the time it takes to ascend. The rate at which the diver ascends is $12 \frac{1}{3}$ feet per minute, and the time it takes to ascend is 4.5 minutes.

12 \frac{1}{3}(4.5) = 12 \frac{1}{3} \times 4.5

To multiply a mixed number by a decimal, we can convert the mixed number to an improper fraction. The mixed number $12 \frac{1}{3}$ can be converted to an improper fraction as follows:

12 \frac{1}{3} = \frac{37}{3}

Now, we can multiply the improper fraction by the decimal:

\frac{37}{3} \times 4.5 = \frac{37}{3} \times \frac{45}{10}

To multiply fractions, we multiply the numerators and multiply the denominators:

\frac{37}{3} \times \frac{45}{10} = \frac{37 \times 45}{3 \times 10}

Now, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor:

\frac{37 \times 45}{3 \times 10} = \frac{1655}{30}

We can further simplify the fraction by dividing the numerator and denominator by their greatest common divisor:

\frac{1655}{30} = \frac{331}{6}

Now, we can convert the improper fraction back to a mixed number:

\frac{331}{6} = 55 \frac{1}{6}

Simplifying the Numerical Expression

Now that we have simplified the variable term, we can substitute it back into the original numerical expression:

-200 + 12 \frac{1}{3}(4.5) = -200 + 55 \frac{1}{6}

To add a negative number and a positive number, we can add the absolute values of the numbers and then apply the negative sign:

-200 + 55 \frac{1}{6} = -200 + 55.17

Now, we can add the numbers:

-200 + 55.17 = -144.83

Conclusion

In this article, we have simplified a numerical expression that represents the depth of a diver as they ascend from a depth of 200 feet. We broke down the expression into two parts, simplified the variable term, and then substituted it back into the original expression. The final result is -144.83, which represents the depth of the diver after ascending for 4.5 minutes at a steady rate of $12 \frac{1}{3}$ feet per minute.

Mathematical Concepts

This article has demonstrated the following mathematical concepts:

Mixed numbers: We converted a mixed number to an improper fraction and then back to a mixed number.
Improper fractions: We converted a mixed number to an improper fraction and then simplified the fraction.
Multiplication of fractions: We multiplied two fractions together and then simplified the result.
Addition of negative and positive numbers: We added a negative number and a positive number together and then applied the negative sign.

Real-World Applications

This article has demonstrated the importance of simplifying numerical expressions in real-world applications. In this case, the numerical expression represents the depth of a diver as they ascend from a depth of 200 feet. The simplified expression can be used to calculate the depth of the diver at any given time, which is essential for safe diving practices.

Future Directions

Introduction

In our previous article, we simplified a numerical expression that represents the depth of a diver as they ascend from a depth of 200 feet. We broke down the expression into two parts, simplified the variable term, and then substituted it back into the original expression. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the initial depth of the diver?

A: The initial depth of the diver is 200 feet.

Q: What is the rate at which the diver ascends?

A: The rate at which the diver ascends is $12 \frac{1}{3}$ feet per minute.

Q: How long does it take for the diver to ascend?

A: It takes 4.5 minutes for the diver to ascend.

Q: What is the simplified expression for the depth of the diver?

A: The simplified expression for the depth of the diver is -144.83.

Q: What mathematical concepts are involved in simplifying the numerical expression?

A: The mathematical concepts involved in simplifying the numerical expression are mixed numbers, improper fractions, multiplication of fractions, and addition of negative and positive numbers.

Q: What is the importance of simplifying numerical expressions in real-world applications?

A: Simplifying numerical expressions is essential in real-world applications, such as safe diving practices, where the depth of the diver needs to be calculated accurately.

Q: Can you provide more examples of numerical expressions that can be simplified using the same techniques?

A: Yes, here are a few examples:

$-300 + 15 \frac{1}{2}(3.2)$
$-400 + 20 \frac{1}{4}(2.5)$
$-500 + 25 \frac{1}{6}(1.8)$

Q: How can I apply the techniques used in this article to other real-world problems?

A: You can apply the techniques used in this article to other real-world problems by identifying the mathematical concepts involved and using the same techniques to simplify the numerical expressions.

Q: What are some common mistakes to avoid when simplifying numerical expressions?

A: Some common mistakes to avoid when simplifying numerical expressions include:

Not converting mixed numbers to improper fractions
Not multiplying fractions correctly
Not adding negative and positive numbers correctly

Conclusion

In this article, we have answered some frequently asked questions related to the topic of simplifying numerical expressions. We have provided examples of numerical expressions that can be simplified using the same techniques and highlighted the importance of simplifying numerical expressions in real-world applications. We have also provided some common mistakes to avoid when simplifying numerical expressions.

Mathematical Concepts

This article has demonstrated the following mathematical concepts:

Mixed numbers: We converted mixed numbers to improper fractions and then back to mixed numbers.
Improper fractions: We converted mixed numbers to improper fractions and then simplified the fractions.
Multiplication of fractions: We multiplied two fractions together and then simplified the result.
Addition of negative and positive numbers: We added a negative number and a positive number together and then applied the negative sign.

Real-World Applications

Future Directions

In future articles, we can explore more complex numerical expressions and demonstrate how to simplify them using various mathematical techniques. We can also apply these techniques to real-world problems and demonstrate their importance in various fields such as engineering, physics, and computer science.