Fill In The Box To Make The Proportion True: ${ \frac{3}{8} = \frac{\square}{48} }$

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Introduction

Proportions are a fundamental concept in mathematics, used to compare the relationships between different quantities. In this article, we will focus on solving proportions, specifically the one given in the problem: ${ \frac{3}{8} = \frac{\square}{48} }$. We will break down the solution step by step, using clear and concise language to ensure that readers understand the concept.

Understanding Proportions

A proportion is a statement that two ratios are equal. It can be written in the form: $ \frac{a}{b} = \frac{c}{d} }$, where aa, bb, cc, and dd are numbers. In the given problem, we have ${ \frac{3{8} = \frac{\square}{48} }$. Our goal is to find the value of the missing number, denoted by â–¡\square.

Step 1: Identify the Given Information

The given proportion is: ${ \frac{3}{8} = \frac{\square}{48} }$. We can see that the first ratio is 38\frac{3}{8}, and the second ratio is â–¡48\frac{\square}{48}. We need to find the value of â–¡\square.

Step 2: Cross-Multiply

To solve the proportion, we can cross-multiply, which means multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa. This gives us: ${ 3 \times 48 = 8 \times \square }$. Simplifying the equation, we get: ${ 144 = 8 \times \square }$.

Step 3: Solve for â–¡\square

Now that we have the equation 144=8×□144 = 8 \times \square, we can solve for □\square. To do this, we need to isolate □\square on one side of the equation. We can do this by dividing both sides of the equation by 8: ${ \square = \frac{144}{8} }$. Simplifying the fraction, we get: ${ \square = 18 }$.

Conclusion

In this article, we solved the proportion: ${ \frac{3}{8} = \frac{\square}{48} }$. We broke down the solution into three steps: identifying the given information, cross-multiplying, and solving for â–¡\square. By following these steps, we were able to find the value of â–¡\square, which is 18.

Real-World Applications

Proportions have many real-world applications, including:

  • Finance: Proportions are used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Proportions are used to describe the relationships between different physical quantities, such as the ratio of a substance's mass to its volume.
  • Engineering: Proportions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Tips and Tricks

Here are some tips and tricks to help you solve proportions:

  • Use cross-multiplication: Cross-multiplication is a powerful tool for solving proportions. It allows you to eliminate the fractions and work with whole numbers.
  • Simplify the equation: Simplify the equation by combining like terms and canceling out common factors.
  • Check your answer: Always check your answer by plugging it back into the original proportion.

Common Mistakes

Here are some common mistakes to avoid when solving proportions:

  • Forgetting to cross-multiply: Failing to cross-multiply can lead to incorrect solutions.
  • Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
  • Not checking the answer: Failing to check the answer can lead to incorrect solutions.

Conclusion

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It can be written in the form: ${ \frac{a}{b} = \frac{c}{d} }$, where aa, bb, cc, and dd are numbers.

Q: How do I solve a proportion?

A: To solve a proportion, you need to follow these steps:

  1. Identify the given information: Write down the proportion and identify the given information.
  2. Cross-multiply: Multiply the numerator of the first ratio by the denominator of the second ratio, and vice versa.
  3. Solve for the unknown value: Use the resulting equation to solve for the unknown value.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to solve proportions. It involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.

Q: Why do I need to cross-multiply?

A: Cross-multiplication is necessary to eliminate the fractions and work with whole numbers. It allows you to solve for the unknown value more easily.

Q: How do I simplify the equation?

A: To simplify the equation, you need to combine like terms and cancel out common factors.

Q: What are some common mistakes to avoid when solving proportions?

A: Some common mistakes to avoid when solving proportions include:

  • Forgetting to cross-multiply: Failing to cross-multiply can lead to incorrect solutions.
  • Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
  • Not checking the answer: Failing to check the answer can lead to incorrect solutions.

Q: How do I check my answer?

A: To check your answer, you need to plug it back into the original proportion and verify that it is true.

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

A: Proportions have many real-world applications, including:

  • Finance: Proportions are used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Proportions are used to describe the relationships between different physical quantities, such as the ratio of a substance's mass to its volume.
  • Engineering: Proportions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: Can I use proportions to solve other types of problems?

A: Yes, you can use proportions to solve other types of problems, such as:

  • Percentages: Proportions can be used to calculate percentages and solve problems involving percentages.
  • Ratios: Proportions can be used to solve problems involving ratios and proportions.
  • Fractions: Proportions can be used to solve problems involving fractions and decimals.

Q: How can I practice solving proportions?

A: You can practice solving proportions by:

  • Working through examples: Practice solving proportions by working through examples and exercises.
  • Using online resources: Use online resources, such as video tutorials and practice problems, to help you practice solving proportions.
  • Seeking help: Seek help from a teacher, tutor, or classmate if you need additional support.

Conclusion

In conclusion, solving proportions is a straightforward process that involves identifying the given information, cross-multiplying, and solving for the unknown value. By following these steps and using the tips and tricks outlined in this article, you can become proficient in solving proportions and apply them to real-world problems.