Evaluating the Given Proportion

Oct 13, 2025 by miodeya.com 13 views

Introduction

In mathematics, proportions are used to compare the relationships between different quantities. A proportion is a statement that two ratios are equal. In this article, we will determine whether the proportion $\frac{6 \text{ tiles }}{5 \text{ ft}} = \frac{42 \text{ tiles }}{35 \text{ ft}}$ is true or not true.

Understanding Proportions

A proportion is a mathematical statement that two ratios are equal. It is written in the form $\frac{a}{b} = \frac{c}{d}$ , where $a$ and $b$ are the quantities being compared, and $c$ and $d$ are the corresponding quantities in the second ratio. To evaluate a proportion, we need to determine whether the two ratios are equal.

Evaluating the Given Proportion

Let's evaluate the given proportion $\frac{6 \text{ tiles }}{5 \text{ ft}} = \frac{42 \text{ tiles }}{35 \text{ ft}}$ . To do this, we need to compare the two ratios.

Step 1: Compare the Numerators

The numerators of the two ratios are 6 and 42. To compare them, we need to determine whether they are equal or not.

The numerator of the first ratio is 6.
The numerator of the second ratio is 42.

Since 6 is not equal to 42, we can conclude that the numerators are not equal.

Step 2: Compare the Denominators

The denominators of the two ratios are 5 and 35. To compare them, we need to determine whether they are equal or not.

The denominator of the first ratio is 5.
The denominator of the second ratio is 35.

Since 5 is not equal to 35, we can conclude that the denominators are not equal.

Step 3: Compare the Ratios

Since the numerators and denominators are not equal, we can conclude that the ratios are not equal.

Conclusion

Based on the evaluation of the proportion, we can conclude that the proportion $\frac{6 \text{ tiles }}{5 \text{ ft}} = \frac{42 \text{ tiles }}{35 \text{ ft}}$ is not true.

Why the Proportion is Not True

The proportion is not true because the two ratios are not equal. The numerators and denominators are not equal, which means that the ratios are not equal.

Real-World Applications

Proportions are used in many real-world applications, such as:

Architecture: Proportions are used to design buildings and structures.
Engineering: Proportions are used to design machines and mechanisms.
Finance: Proportions are used to calculate interest rates and investment returns.

Conclusion

In conclusion, the proportion $\frac{6 \text{ tiles }}{5 \text{ ft}} = \frac{42 \text{ tiles }}{35 \text{ ft}}$ is not true. The two ratios are not equal, which means that the proportion is not valid. Proportions are used in many real-world applications, and understanding how to evaluate them is essential in mathematics and other fields.

Final Answer

The final answer is:

Introduction

In our previous article, we evaluated the validity of a proportion and determined that it was not true. In this article, we will answer some frequently asked questions (FAQs) about proportions.

Q: What is a proportion?

A: A proportion is a mathematical statement that two ratios are equal. It is written in the form $\frac{a}{b} = \frac{c}{d}$ , where $a$ and $b$ are the quantities being compared, and $c$ and $d$ are the corresponding quantities in the second ratio.

Q: How do I evaluate a proportion?

A: To evaluate a proportion, you need to compare the two ratios. You can do this by comparing the numerators and denominators of the two ratios.

Compare the numerators: Are they equal?
Compare the denominators: Are they equal?
If the numerators and denominators are equal, then the ratios are equal, and the proportion is true.
If the numerators and denominators are not equal, then the ratios are not equal, and the proportion is not true.

Q: What are some real-world applications of proportions?

A: Proportions are used in many real-world applications, such as:

Architecture: Proportions are used to design buildings and structures.
Engineering: Proportions are used to design machines and mechanisms.
Finance: Proportions are used to calculate interest rates and investment returns.
Science: Proportions are used to describe the relationships between different quantities in scientific experiments.

Q: How do I simplify a proportion?

A: To simplify a proportion, you can divide both sides of the equation by the greatest common divisor (GCD) of the numerators and denominators.

For example, consider the proportion $\frac{12}{16} = \frac{3}{4}$ . To simplify this proportion, you can divide both sides by the GCD of 12 and 16, which is 4.

$\frac{12}{16} = \frac{3}{4}$

$\frac{12 \div 4}{16 \div 4} = \frac{3 \div 4}{4 \div 4}$

$\frac{3}{4} = \frac{3}{4}$

Q: What is the difference between a proportion and a ratio?

A: A proportion is a mathematical statement that two ratios are equal. A ratio is a comparison of two quantities.

For example, consider the ratio $\frac{3}{4}$ . This is a comparison of the quantities 3 and 4.

A proportion is a statement that two ratios are equal, such as $\frac{3}{4} = \frac{6}{8}$ .

Q: How do I convert a proportion to a ratio?

A: To convert a proportion to a ratio, you can divide both sides of the equation by the denominator.

For example, consider the proportion $\frac{3}{4} = \frac{6}{8}$ . To convert this proportion to a ratio, you can divide both sides by 4.

$\frac{3}{4} = \frac{6}{8}$

$\frac{3 \div 4}{4 \div 4} = \frac{6 \div 4}{8 \div 4}$

$\frac{3}{1} = \frac{6}{2}$

Conclusion

In conclusion, proportions are an important concept in mathematics that are used to compare the relationships between different quantities. We have answered some frequently asked questions (FAQs) about proportions and provided examples of how to evaluate and simplify proportions.

Final Answer

The final answer is:

Q: What is a proportion? A: A proportion is a mathematical statement that two ratios are equal.
Q: How do I evaluate a proportion? A: To evaluate a proportion, you need to compare the two ratios by comparing the numerators and denominators.
Q: What are some real-world applications of proportions? A: Proportions are used in many real-world applications, such as architecture, engineering, finance, and science.
Q: How do I simplify a proportion? A: To simplify a proportion, you can divide both sides of the equation by the greatest common divisor (GCD) of the numerators and denominators.
Q: What is the difference between a proportion and a ratio? A: A proportion is a mathematical statement that two ratios are equal, while a ratio is a comparison of two quantities.
Q: How do I convert a proportion to a ratio? A: To convert a proportion to a ratio, you can divide both sides of the equation by the denominator.