What Is The Value Of $x$ In This Proportion? 4 11 = − 3 X + 5 \frac{4}{11} = \frac{-3}{x+5} 11 4 ​ = X + 5 − 3 ​ A. X = − 13 1 4 X = -13 \frac{1}{4} X = − 13 4 1 ​ B. X = − 9 1 2 X = -9 \frac{1}{2} X = − 9 2 1 ​ C. X = − 7 X = -7 X = − 7 D. X = − 3 1 4 X = -3 \frac{1}{4} X = − 3 4 1 ​

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Understanding the Concept of Proportions

A proportion is a mathematical statement that two ratios are equal. It is often expressed as a fraction, where the two fractions have the same value. In this article, we will explore how to solve a proportion and find the value of $x$ in the given equation.

The Given Equation

The given equation is 411=−3x+5\frac{4}{11} = \frac{-3}{x+5}. This equation represents a proportion, where the two ratios are equal. Our goal is to find the value of $x$ that makes this proportion true.

Solving the Proportion

To solve the proportion, we can use the following steps:

  1. Cross-multiply: Multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the numerator of the second fraction by the denominator of the first fraction.
  2. Simplify: Simplify the resulting equation by combining like terms.
  3. Solve for $x$: Solve for $x$ by isolating it on one side of the equation.

Cross-multiplying

Let's start by cross-multiplying the given equation:

4(x+5)=−3(11)4(x+5) = -3(11)

Simplifying

Now, let's simplify the equation by distributing the numbers to the terms inside the parentheses:

4x+20=−334x + 20 = -33

Solving for $x$

Next, let's solve for $x$ by isolating it on one side of the equation. We can do this by subtracting 20 from both sides of the equation:

4x=−534x = -53

Then, we can divide both sides of the equation by 4 to solve for $x$:

x=−534x = -\frac{53}{4}

Converting the Fraction to a Mixed Number

To convert the fraction to a mixed number, we can divide the numerator by the denominator:

x=−1314x = -13 \frac{1}{4}

Conclusion

In this article, we have learned how to solve a proportion and find the value of $x$ in the given equation. We have used the steps of cross-multiplying, simplifying, and solving for $x$ to find the value of $x$.

Comparison with the Given Options

Now, let's compare our answer with the given options:

A. x=−1314x = -13 \frac{1}{4} B. x=−912x = -9 \frac{1}{2} C. x=−7x = -7 D. x=−314x = -3 \frac{1}{4}

Our answer matches option A, which is x=−1314x = -13 \frac{1}{4}.

Final Thoughts

Solving proportions is an important concept in mathematics, and it has many real-world applications. In this article, we have learned how to solve a proportion and find the value of $x$ in the given equation. We have used the steps of cross-multiplying, simplifying, and solving for $x$ to find the value of $x$.

Q: What is a proportion?

A: A proportion is a mathematical statement that two ratios are equal. It is often expressed as a fraction, where the two fractions have the same value.

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

A: A problem is a proportion if it can be expressed as a statement that two ratios are equal. For example, the problem 411=−3x+5\frac{4}{11} = \frac{-3}{x+5} is a proportion because it states that the ratio of 4 to 11 is equal to the ratio of -3 to x+5x+5.

Q: What are the steps to solve a proportion?

A: The steps to solve a proportion are:

  1. Cross-multiply: Multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the numerator of the second fraction by the denominator of the first fraction.
  2. Simplify: Simplify the resulting equation by combining like terms.
  3. Solve for $x$: Solve for $x$ by isolating it on one side of the equation.

Q: What is cross-multiplication?

A: Cross-multiplication is the process of multiplying the numerator of the first fraction by the denominator of the second fraction, and multiplying the numerator of the second fraction by the denominator of the first fraction.

Q: Why do I need to cross-multiply?

A: Cross-multiplication is necessary to eliminate the fractions and make it easier to solve for $x$.

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

A: To simplify the equation, combine like terms and eliminate any fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

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

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.

Q: How do I solve for $x$?

A: To solve for $x$, isolate it on one side of the equation by performing the necessary operations to get $x$ by itself.

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 simplifying the equation
  • Not solving for $x$ correctly
  • Not checking the answer to make sure it is reasonable

Q: How do I check my answer to make sure it is reasonable?

A: To check your answer, plug it back into the original equation and make sure it is true. If the answer is not reasonable, recheck your work and try again.

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 and investment returns.
  • Science: Proportions are used to calculate the concentration of solutions and the amount of a substance in a mixture.
  • Engineering: Proportions are used to design and build structures, such as bridges and buildings.

Q: Can proportions be used to solve problems that involve variables?

A: Yes, proportions can be used to solve problems that involve variables. In these cases, the proportion is used to set up an equation that can be solved for the variable.

Q: Can proportions be used to solve problems that involve inequalities?

A: Yes, proportions can be used to solve problems that involve inequalities. In these cases, the proportion is used to set up an inequality that can be solved for the variable.

Q: What are some common types of proportions?

A: Some common types of proportions include:

  • Simple proportions: These are proportions that involve two fractions with the same value.
  • Complex proportions: These are proportions that involve two fractions with different values.
  • Inverse proportions: These are proportions that involve two fractions that are inversely related.

Q: How do I know which type of proportion to use?

A: The type of proportion to use depends on the problem. If the problem involves two fractions with the same value, use a simple proportion. If the problem involves two fractions with different values, use a complex proportion. If the problem involves two fractions that are inversely related, use an inverse proportion.