Solve The Equation For \[$ S \$\]:$\[ \frac{10}{s} = \frac{4}{19} \\]

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Introduction

In this article, we will delve into solving a simple yet crucial equation in mathematics. The equation in question is 10s=419\frac{10}{s} = \frac{4}{19}, where we need to find the value of ss. This equation is a fundamental concept in algebra and is used extensively in various mathematical operations. In this guide, we will break down the solution step by step, making it easy to understand and follow.

Understanding the Equation

The given equation is 10s=419\frac{10}{s} = \frac{4}{19}. To solve for ss, we need to isolate the variable ss on one side of the equation. The equation is a simple proportion, where the ratio of two quantities is equal to the ratio of two other quantities.

What is a Proportion?

A proportion is a statement that two ratios are equal. In this case, the ratio of 10 to ss is equal to the ratio of 4 to 19. Proportions are used to compare the relationships between different quantities.

Why is Solving Proportions Important?

Solving proportions is essential in various mathematical operations, such as solving equations, graphing functions, and working with ratios. It is also used in real-world applications, such as finance, engineering, and science.

Solving the Equation

To solve the equation 10s=419\frac{10}{s} = \frac{4}{19}, we can use the following steps:

Step 1: Cross-Multiply

Cross-multiplying is a technique used to eliminate the fractions in an equation. To cross-multiply, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.

10s=419\frac{10}{s} = \frac{4}{19}

10×19=4×s10 \times 19 = 4 \times s

190=4s190 = 4s

Step 2: Divide Both Sides by 4

To isolate the variable ss, we need to divide both sides of the equation by 4.

1904=4s4\frac{190}{4} = \frac{4s}{4}

47.5=s47.5 = s

Step 3: Check the Solution

To ensure that our solution is correct, we can plug it back into the original equation and check if it is true.

10s=419\frac{10}{s} = \frac{4}{19}

1047.5=419\frac{10}{47.5} = \frac{4}{19}

0.2105≈0.21050.2105 \approx 0.2105

Since the two sides of the equation are approximately equal, we can conclude that our solution is correct.

Conclusion

In this article, we solved the equation 10s=419\frac{10}{s} = \frac{4}{19} for the variable ss. We used the technique of cross-multiplication to eliminate the fractions and then divided both sides of the equation by 4 to isolate the variable. We also checked our solution by plugging it back into the original equation. The final answer is s=47.5s = 47.5.

Applications of Solving Proportions

Solving proportions has numerous applications in various fields, including:

Finance

Solving proportions is used in finance to calculate interest rates, investment returns, and loan payments.

Engineering

Solving proportions is used in engineering to design and optimize systems, such as bridges, buildings, and electronic circuits.

Science

Solving proportions is used in science to analyze data, make predictions, and understand complex phenomena.

Tips and Tricks

When solving proportions, it is essential to follow these tips and tricks:

Use Cross-Multiplication

Cross-multiplication is a powerful technique used to eliminate fractions in an equation.

Check Your Solution

To ensure that your solution is correct, plug it back into the original equation and check if it is true.

Practice, Practice, Practice

Solving proportions requires practice to become proficient. Start with simple equations and gradually move on to more complex ones.

Final Thoughts

Solving proportions is a fundamental concept in mathematics that has numerous applications in various fields. By following the steps outlined in this article, you can solve equations like 10s=419\frac{10}{s} = \frac{4}{19} with ease. Remember to use cross-multiplication, check your solution, and practice regularly to become proficient in solving proportions.

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Introduction

In our previous article, we solved the equation 10s=419\frac{10}{s} = \frac{4}{19} for the variable ss. In this article, we will provide a Q&A guide to help you understand the concept of solving proportions and address any questions you may have.

Q&A

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. In this case, the ratio of 10 to ss is equal to the ratio of 4 to 19.

Q: Why is solving proportions important?

A: Solving proportions is essential in various mathematical operations, such as solving equations, graphing functions, and working with ratios. It is also used in real-world applications, such as finance, engineering, and science.

Q: How do I solve a proportion?

A: To solve a proportion, you can use the following steps:

  1. Cross-multiply: Multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.
  2. Divide both sides by the denominator: Divide both sides of the equation by the denominator to isolate the variable.
  3. Check your solution: Plug your solution back into the original equation to ensure that it is true.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to eliminate fractions in an equation. It involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa.

Q: Why do I need to check my solution?

A: Checking your solution ensures that your answer is correct. If you plug your solution back into the original equation and it is true, then you know that your answer is correct.

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

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

  • Not cross-multiplying
  • Not dividing both sides by the denominator
  • Not checking your solution
  • Not following the order of operations

Q: How can I practice solving proportions?

A: You can practice solving proportions by working on simple equations and gradually moving on to more complex ones. You can also use online resources, such as math websites and apps, to practice solving proportions.

Tips and Tricks

Use Cross-Multiplication

Cross-multiplication is a powerful technique used to eliminate fractions in an equation.

Check Your Solution

To ensure that your solution is correct, plug it back into the original equation and check if it is true.

Practice, Practice, Practice

Solving proportions requires practice to become proficient. Start with simple equations and gradually move on to more complex ones.

Common Misconceptions

Misconception 1: Solving proportions is only for math problems.

A: Solving proportions is not only for math problems. It is used in various fields, such as finance, engineering, and science.

Misconception 2: Solving proportions is only for simple equations.

A: Solving proportions is not only for simple equations. It can be used to solve complex equations and real-world problems.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of solving proportions and address any questions you may have. We also provided tips and tricks to help you practice solving proportions and avoid common mistakes. Remember to use cross-multiplication, check your solution, and practice regularly to become proficient in solving proportions.

Final Thoughts

Solving proportions is a fundamental concept in mathematics that has numerous applications in various fields. By following the steps outlined in this article and practicing regularly, you can become proficient in solving proportions and apply it to real-world problems.