Solve For $q$ In The Proportion:$\[ \frac{q}{18} = \frac{7}{15} \\]$q =$

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Introduction

In mathematics, proportions are used to describe the relationship between two or more quantities. A proportion is a statement that two ratios are equal. In this article, we will focus on solving for q in the proportion: q18=715\frac{q}{18} = \frac{7}{15}. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Proportion

A proportion is a statement that two ratios are equal. In this case, the proportion is q18=715\frac{q}{18} = \frac{7}{15}. This means that the ratio of q to 18 is equal to the ratio of 7 to 15.

Step 1: Cross-Multiply

To solve for q, we need to cross-multiply the two ratios. This means that we multiply the numerator of the first ratio by the denominator of the second ratio, and vice versa. In this case, we multiply q by 15 and 18 by 7.

q18=715\frac{q}{18} = \frac{7}{15}

q×15=18×7q \times 15 = 18 \times 7

Step 2: Simplify the Equation

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

q×15=126q \times 15 = 126

Step 3: Divide Both Sides by 15

To solve for q, we need to isolate q on one side of the equation. We can do this by dividing both sides of the equation by 15.

q×1515=12615\frac{q \times 15}{15} = \frac{126}{15}

q=12615q = \frac{126}{15}

Step 4: Simplify the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 126 and 15 is 3.

q=126÷315÷3q = \frac{126 \div 3}{15 \div 3}

q=425q = \frac{42}{5}

Conclusion

In this article, we solved for q in the proportion q18=715\frac{q}{18} = \frac{7}{15}. We broke down the solution step by step and provided a clear explanation of each step. We cross-multiplied the two ratios, simplified the equation, divided both sides by 15, and simplified the fraction. The final answer is q=425q = \frac{42}{5}.

Real-World Applications

Solving proportions is an important skill in mathematics and has many real-world applications. For example, in finance, proportions are used to calculate interest rates and investment returns. In engineering, proportions are used to design and build structures such as bridges and buildings. In science, proportions are used to describe the relationships between different variables and to make predictions about future events.

Tips and Tricks

When solving proportions, it's essential to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps and tips, you can become proficient in solving proportions and apply this skill to a wide range of real-world problems.

Common Mistakes to Avoid

When solving proportions, there are several common mistakes to avoid:

  1. Not cross-multiplying: Failing to cross-multiply the two ratios can lead to incorrect solutions.
  2. Not simplifying the equation: Failing to simplify the equation can make it difficult to solve for q.
  3. Not dividing both sides by the same value: Failing to divide both sides of the equation by the same value can lead to incorrect solutions.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Practice Problems

To practice solving proportions, try the following problems:

  1. x12=916\frac{x}{12} = \frac{9}{16}
  2. y20=724\frac{y}{20} = \frac{7}{24}
  3. z18=1127\frac{z}{18} = \frac{11}{27}

Introduction

In our previous article, we discussed how to solve proportions and provided a step-by-step guide on how to solve for q in the proportion q18=715\frac{q}{18} = \frac{7}{15}. In this article, we will answer some of the most frequently asked questions about solving proportions.

Q: What is a proportion?

A proportion is a statement that two ratios are equal. It is a way of describing the relationship between two or more quantities.

Q: How do I know if a proportion is true or false?

To determine if a proportion is true or false, you can use the following steps:

  1. Write the proportion as an equation.
  2. Cross-multiply the two ratios.
  3. Simplify the equation.
  4. Check if the resulting equation is true or false.

Q: What is cross-multiplication?

Cross-multiplication is the process of multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa. This is done to eliminate the fractions and make it easier to solve for the unknown variable.

Q: Why do I need to simplify the equation?

Simplifying the equation is necessary to make it easier to solve for the unknown variable. By simplifying the equation, you can eliminate any unnecessary terms and make it easier to isolate the variable.

Q: Can I use proportions to solve equations with variables on both sides?

Yes, you can use proportions to solve equations with variables on both sides. However, you will need to use a different method, such as substitution or elimination, to solve for the variable.

Q: How do I know if a proportion is a direct or inverse proportion?

A direct proportion is a proportion where the two ratios are equal, and the variables are directly related. An inverse proportion is a proportion where the two ratios are equal, but the variables are inversely related.

Q: Can I use proportions to solve problems with percents?

Yes, you can use proportions to solve problems with percents. However, you will need to convert the percent to a decimal or fraction before using the proportion.

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

Proportions have many real-world applications, including:

  • Finance: Proportions are used to calculate interest rates and investment returns.
  • Engineering: Proportions are used to design and build structures such as bridges and buildings.
  • Science: Proportions are used to describe the relationships between different variables and to make predictions about future events.

Q: How can I practice solving proportions?

You can practice solving proportions by using online resources, such as Khan Academy or Mathway, or by working on practice problems. You can also try solving proportions with different variables and ratios to see how the solution changes.

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

Some common mistakes to avoid when solving proportions include:

  • Not cross-multiplying the two ratios.
  • Not simplifying the equation.
  • Not dividing both sides of the equation by the same value.
  • Not checking if the resulting equation is true or false.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Conclusion

Solving proportions is an important skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can become proficient in solving proportions and apply this skill to a wide range of problems. Remember to practice regularly and avoid common mistakes to ensure that your solutions are accurate and reliable.