Solve The Proportion:7.) { \frac{8}{12} = \frac{a}{21}$}$

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Introduction

Proportions are a fundamental concept in mathematics, and solving them is a crucial skill to master. In this article, we will focus on solving a specific proportion problem: 812=a21\frac{8}{12} = \frac{a}{21}. We will break down the solution step by step, using clear and concise language to ensure that you understand the process.

What is a Proportion?

A proportion is a statement that two ratios are equal. It is often written in the form ab=cd\frac{a}{b} = \frac{c}{d}, where aa, bb, cc, and dd are numbers. Proportions can be used to solve a wide range of problems, from simple equations to complex real-world applications.

The Problem

The problem we will be solving is 812=a21\frac{8}{12} = \frac{a}{21}. Our goal is to find the value of aa.

Step 1: Cross-Multiply

To solve the proportion, we will use the method of cross-multiplication. This involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.

812=a21\frac{8}{12} = \frac{a}{21}

Cross-multiplying gives us:

8×21=12×a8 \times 21 = 12 \times a

Step 2: Simplify the Equation

Now that we have cross-multiplied, we can simplify the equation by multiplying the numbers.

168=12a168 = 12a

Step 3: Divide Both Sides by 12

To isolate aa, we need to divide both sides of the equation by 12.

16812=a\frac{168}{12} = a

Step 4: Simplify the Fraction

Now that we have divided both sides by 12, we can simplify the fraction.

14=a14 = a

Conclusion

We have successfully solved the proportion 812=a21\frac{8}{12} = \frac{a}{21}. The value of aa is 14.

Why is Solving Proportions Important?

Solving proportions is an essential skill in mathematics, and it has many real-world applications. Here are a few examples:

  • Cooking: When you are cooking, you may need to scale up or down a recipe. Solving proportions can help you do this accurately.
  • Building: When you are building a structure, you may need to calculate the proportions of different components. Solving proportions can help you do this accurately.
  • Science: When you are conducting scientific experiments, you may need to calculate the proportions of different substances. Solving proportions can help you do this accurately.

Tips for Solving Proportions

Here are a few tips for solving proportions:

  • Read the problem carefully: Before you start solving the proportion, make sure you understand what the problem is asking.
  • Use cross-multiplication: Cross-multiplication is a powerful tool for solving proportions.
  • Simplify the equation: Simplifying the equation can make it easier to solve.
  • Check your answer: Once you have solved the proportion, make sure to check your answer to ensure that it is correct.

Common Mistakes to Avoid

Here are a few common mistakes to avoid when solving proportions:

  • Not reading the problem carefully: Failing to read the problem carefully can lead to mistakes.
  • Not using cross-multiplication: Failing to use cross-multiplication can make it harder to solve the proportion.
  • Not simplifying the equation: Failing to simplify the equation can make it harder to solve.
  • Not checking your answer: Failing to check your answer can lead to mistakes.

Conclusion

Solving proportions is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can solve proportions accurately and confidently. Remember to read the problem carefully, use cross-multiplication, simplify the equation, and check your answer. With practice and patience, you can become proficient in solving proportions.

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Introduction

In our previous article, we covered the basics of solving proportions and walked through a step-by-step solution to a specific problem. In this article, we will answer some common questions that people have about solving proportions.

Q: What is a proportion?

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

Q: How do I know if a problem is a proportion?

A: To determine if a problem is a proportion, look for the following characteristics:

  • The problem involves two ratios.
  • The problem states that the two ratios are equal.
  • The problem is written in the form ab=cd\frac{a}{b} = \frac{c}{d}.

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 solve proportions because it allows us to eliminate the fractions and work with whole numbers.

Q: How do I simplify the equation after cross-multiplication?

A: After cross-multiplication, simplify the equation by multiplying the numbers. This will help you isolate the variable.

Q: What if I get a fraction as my answer?

A: If you get a fraction as your answer, you can simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q: Can I use a calculator to solve proportions?

A: Yes, you can use a calculator to solve proportions. However, it's always a good idea to check your answer by hand to ensure that it's correct.

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

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

  • Not reading the problem carefully.
  • Not using cross-multiplication.
  • Not simplifying the equation.
  • Not checking your answer.

Q: How can I practice solving proportions?

A: There are many ways to practice solving proportions, including:

  • Working through practice problems in a textbook or online resource.
  • Using a calculator to generate random proportions and solving them by hand.
  • Creating your own proportions and solving them.

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

A: Solving proportions has many real-world applications, including:

  • Cooking: When you are cooking, you may need to scale up or down a recipe. Solving proportions can help you do this accurately.
  • Building: When you are building a structure, you may need to calculate the proportions of different components. Solving proportions can help you do this accurately.
  • Science: When you are conducting scientific experiments, you may need to calculate the proportions of different substances. Solving proportions can help you do this accurately.

Conclusion

Solving proportions is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article and practicing regularly, you can become proficient in solving proportions. Remember to read the problem carefully, use cross-multiplication, simplify the equation, and check your answer. With practice and patience, you can become a master of solving proportions.