Solve For { X $}$ In The Equation:${ \frac{7}{8} = \frac{x}{24} }$

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Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes the equation true. In this article, we will focus on solving a simple equation involving fractions. The equation we will be solving is 78=x24\frac{7}{8} = \frac{x}{24}, where xx is the variable we need to solve for.

Understanding the Equation

Before we start solving the equation, let's understand what it means. The equation 78=x24\frac{7}{8} = \frac{x}{24} states that the ratio of 7 to 8 is equal to the ratio of xx to 24. In other words, if we have a quantity that is 7 parts out of 8, it is equivalent to a quantity that is xx parts out of 24.

Solving the Equation

To solve the equation, we need to isolate the variable xx. We can do this by multiplying both sides of the equation by the reciprocal of the fraction on the right-hand side. The reciprocal of 241\frac{24}{1} is 124\frac{1}{24}, so we multiply both sides by 124\frac{1}{24}.

78=x24\frac{7}{8} = \frac{x}{24}

78⋅124=x24⋅124\frac{7}{8} \cdot \frac{1}{24} = \frac{x}{24} \cdot \frac{1}{24}

7192=x576\frac{7}{192} = \frac{x}{576}

Simplifying the Equation

Now that we have the equation in the form 7192=x576\frac{7}{192} = \frac{x}{576}, we can simplify it by multiplying both sides by the least common multiple (LCM) of the denominators. The LCM of 192 and 576 is 576, so we multiply both sides by 576.

7192=x576\frac{7}{192} = \frac{x}{576}

7192⋅576=x576⋅576\frac{7}{192} \cdot 576 = \frac{x}{576} \cdot 576

7⋅576192=x⋅576576\frac{7 \cdot 576}{192} = \frac{x \cdot 576}{576}

4032192=x⋅576576\frac{4032}{192} = \frac{x \cdot 576}{576}

Solving for xx

Now that we have the equation in the form 4032192=x⋅576576\frac{4032}{192} = \frac{x \cdot 576}{576}, we can solve for xx by multiplying both sides by the reciprocal of the fraction on the right-hand side. The reciprocal of 5761\frac{576}{1} is 1576\frac{1}{576}, so we multiply both sides by 1576\frac{1}{576}.

4032192=x⋅576576\frac{4032}{192} = \frac{x \cdot 576}{576}

4032192⋅1576=x⋅576576⋅1576\frac{4032}{192} \cdot \frac{1}{576} = \frac{x \cdot 576}{576} \cdot \frac{1}{576}

4032110592=x1\frac{4032}{110592} = \frac{x}{1}

Simplifying the Result

Now that we have the equation in the form 4032110592=x1\frac{4032}{110592} = \frac{x}{1}, we can simplify it by multiplying both sides by 1.

4032110592=x1\frac{4032}{110592} = \frac{x}{1}

4032110592⋅1=x1⋅1\frac{4032}{110592} \cdot 1 = \frac{x}{1} \cdot 1

4032110592=x\frac{4032}{110592} = x

Conclusion

In this article, we solved the equation 78=x24\frac{7}{8} = \frac{x}{24} by multiplying both sides by the reciprocal of the fraction on the right-hand side and simplifying the result. We found that the value of xx is 4032110592\frac{4032}{110592}, which can be simplified to 21576\frac{21}{576}.

Final Answer

The final answer is 21576\boxed{\frac{21}{576}}.

Related Topics

  • Solving equations with fractions
  • Multiplying and dividing fractions
  • Simplifying fractions

References

Introduction

In our previous article, we solved the equation 78=x24\frac{7}{8} = \frac{x}{24} by multiplying both sides by the reciprocal of the fraction on the right-hand side and simplifying the result. In this article, we will answer some common questions that students may have when solving this type of equation.

Q: What is the value of xx in the equation 78=x24\frac{7}{8} = \frac{x}{24}?

A: The value of xx is 4032110592\frac{4032}{110592}, which can be simplified to 21576\frac{21}{576}.

Q: How do I multiply fractions?

A: To multiply fractions, you multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom). For example, to multiply 34\frac{3}{4} and 56\frac{5}{6}, you would multiply the numerators (3 and 5) and multiply the denominators (4 and 6).

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 34\frac{3}{4} is 43\frac{4}{3}.

Q: How do I simplify a fraction?

A: To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify 1218\frac{12}{18}, you would divide both the numerator and the denominator by 6, resulting in 23\frac{2}{3}.

Q: What is the least common multiple (LCM) of two numbers?

A: The LCM of two numbers is the smallest number that both numbers can divide into evenly. For example, the LCM of 4 and 6 is 12.

Q: How do I solve an equation with fractions?

A: To solve an equation with fractions, you can multiply both sides of the equation by the reciprocal of the fraction on the right-hand side. This will eliminate the fraction and allow you to solve for the variable.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change.

Q: How do I check my answer to an equation?

A: To check your answer to an equation, you can plug the value you found into the original equation and see if it is true. If it is true, then your answer is correct.

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

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

  • Not following the order of operations (PEMDAS)
  • Not simplifying fractions
  • Not checking your answer
  • Not using the correct method to solve the equation

Conclusion

Solving equations with fractions can be challenging, but with practice and patience, you can become proficient in solving these types of equations. Remember to follow the order of operations, simplify fractions, and check your answer to ensure that you are getting the correct solution.

Final Answer

The final answer is 21576\boxed{\frac{21}{576}}.

Related Topics

  • Solving equations with fractions
  • Multiplying and dividing fractions
  • Simplifying fractions
  • Least common multiple (LCM)
  • Greatest common divisor (GCD)

References