Distributing the 15 Formula

Oct 13, 2025 by magicalceylontours.com 42 views

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Introduction

In mathematics, proportions are a fundamental concept that helps us understand the relationship between different quantities. A proportion is a statement that two ratios are equal. In this article, we will focus on solving a proportion, specifically the equation $\frac{x+2}{5}=\frac{6}{15}$ . We will break down the solution into step-by-step instructions, making it easy to understand and follow.

Understanding the Problem

The given equation is a proportion, where the ratio of $x+2$ to $5$ is equal to the ratio of $6$ to $15$ . Our goal is to solve for the value of $x$ .

What is a Proportion?

A proportion is a statement that two ratios are equal. It can be written in the form $\frac{a}{b}=\frac{c}{d}$ , where $a$ , $b$ , $c$ , and $d$ are numbers. In this case, the proportion is $\frac{x+2}{5}=\frac{6}{15}$ .

Why is Solving Proportions Important?

Solving proportions is an essential skill in mathematics, as it helps us understand the relationship between different quantities. It is used in various real-world applications, such as finance, science, and engineering.

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.

Cross-Multiplication Formula

The cross-multiplication formula is:

a \times d = b \times c

In this case, the formula becomes:

(x+2) \times 15 = 5 \times 6

Simplifying the Equation

Now, we will simplify the equation by multiplying the numbers.

(x+2) \times 15 = 30

Step 2: Distribute the 15

Next, we will distribute the $15$ to the terms inside the parentheses.

Distributing the 15 Formula

The distributing the 15 formula is:

a \times (b+c) = a \times b + a \times c

In this case, the formula becomes:

15(x+2) = 15x + 30

Simplifying the Equation

Now, we will simplify the equation by combining like terms.

15x + 30 = 30

Step 3: Subtract 30 from Both Sides

Next, we will subtract $30$ from both sides of the equation to isolate the term with the variable.

Subtracting 30 Formula

The subtracting 30 formula is:

a - b = a - (b+c)

In this case, the formula becomes:

15x = 0

Simplifying the Equation

Now, we will simplify the equation by combining like terms.

15x = 0

Step 4: Divide Both Sides by 15

Finally, we will divide both sides of the equation by $15$ to solve for the value of $x$ .

Dividing Both Sides Formula

The dividing both sides formula is:

\frac{a}{b} = \frac{c}{d}

In this case, the formula becomes:

x = \frac{0}{15}

Simplifying the Equation

Now, we will simplify the equation by combining like terms.

x = 0

Conclusion

In this article, we solved the proportion $\frac{x+2}{5}=\frac{6}{15}$ using the method of cross-multiplication. We broke down the solution into step-by-step instructions, making it easy to understand and follow. We also discussed the importance of solving proportions in mathematics and its real-world applications.

Final Answer

The final answer is $\boxed{0}$ .

References

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Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It can be written in the form $\frac{a}{b}=\frac{c}{d}$ , where $a$ , $b$ , $c$ , and $d$ are numbers.

Q: Why is solving proportions important?

A: Solving proportions is an essential skill in mathematics, as it helps us understand the relationship between different quantities. It is used in various real-world applications, such as finance, science, and engineering.

Q: How do I solve a proportion?

A: To solve a proportion, you can 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.

Q: What is cross-multiplication?

A: Cross-multiplication is a method of solving proportions by multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.

Q: How do I use cross-multiplication to solve a proportion?

A: To use cross-multiplication to solve a proportion, follow these steps:

Write the proportion in the form $\frac{a}{b}=\frac{c}{d}$ .
Multiply the numerator of the first ratio by the denominator of the second ratio.
Multiply the numerator of the second ratio by the denominator of the first ratio.
Set the two products equal to each other.
Solve for the variable.

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

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

Not using cross-multiplication correctly
Not simplifying the equation
Not isolating the variable
Not checking the solution

Q: How do I check my solution to a proportion?

A: To check your solution to a proportion, follow these steps:

Plug the solution back into the original proportion.
Simplify the equation.
Check if the solution is true.

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

A: Some real-world applications of solving proportions include:

Finance: Solving proportions is used to calculate interest rates, investment returns, and other financial metrics.
Science: Solving proportions is used to calculate rates of change, such as the rate of decay of a radioactive substance.
Engineering: Solving proportions is used to calculate the dimensions of a structure, such as the height of a building.

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 solution by hand to ensure that it's correct.

Q: How do I practice solving proportions?

A: To practice solving proportions, try the following:

Work through practice problems in a textbook or online resource.
Use a calculator to check your solutions.
Try solving proportions with different variables and coefficients.
Practice solving proportions with different types of ratios, such as equivalent ratios and non-equivalent ratios.

Q: What are some resources for learning more about solving proportions?

A: Some resources for learning more about solving proportions include:

Textbooks: "Algebra" by Michael Artin, "Mathematics for the Nonmathematician" by Morris Kline
Online resources: Khan Academy, Math Open Reference, Wolfram Alpha
Video tutorials: 3Blue1Brown, Crash Course, Vi Hart
Practice problems: IXL, Mathway, Symbolab

Q: Can I use solving proportions to solve other types of equations?

A: Yes, you can use solving proportions to solve other types of equations, such as linear equations and quadratic equations. However, you may need to use different methods and techniques to solve these types of equations.

Q: How do I know if I'm solving proportions correctly?

A: To know if you're solving proportions correctly, follow these steps:

Check your work by plugging the solution back into the original proportion.
Simplify the equation to ensure that it's true.
Check if the solution is reasonable and makes sense in the context of the problem.

Q: Can I use solving proportions to solve word problems?

A: Yes, you can use solving proportions to solve word problems. However, you may need to translate the word problem into a mathematical equation and then use solving proportions to solve the equation.

Q: How do I use solving proportions to solve word problems?

A: To use solving proportions to solve word problems, follow these steps:

Read the word problem carefully and identify the key information.
Translate the word problem into a mathematical equation.
Use solving proportions to solve the equation.
Check your solution by plugging it back into the original equation.

Q: What are some common word problems that involve solving proportions?

A: Some common word problems that involve solving proportions include:

Calculating the cost of a product based on its price and quantity.
Determining the rate of change of a quantity over time.
Finding the dimensions of a structure based on its volume and surface area.

Q: Can I use solving proportions to solve problems in other subjects?

A: Yes, you can use solving proportions to solve problems in other subjects, such as science and engineering. However, you may need to use different methods and techniques to solve these types of problems.

Q: How do I know if I'm using solving proportions correctly in other subjects?

A: To know if you're using solving proportions correctly in other subjects, follow these steps:

Check your work by plugging the solution back into the original equation.
Simplify the equation to ensure that it's true.
Check if the solution is reasonable and makes sense in the context of the problem.

Q: Can I use solving proportions to solve problems in real-world applications?

A: Yes, you can use solving proportions to solve problems in real-world applications, such as finance, science, and engineering. However, you may need to use different methods and techniques to solve these types of problems.

Q: How do I know if I'm using solving proportions correctly in real-world applications?

A: To know if you're using solving proportions correctly in real-world applications, follow these steps:

Check your work by plugging the solution back into the original equation.
Simplify the equation to ensure that it's true.
Check if the solution is reasonable and makes sense in the context of the problem.

Q: Can I use solving proportions to solve problems in other areas of mathematics?

A: Yes, you can use solving proportions to solve problems in other areas of mathematics, such as algebra and geometry. However, you may need to use different methods and techniques to solve these types of problems.

Q: How do I know if I'm using solving proportions correctly in other areas of mathematics?

A: To know if you're using solving proportions correctly in other areas of mathematics, follow these steps:

Check your work by plugging the solution back into the original equation.
Simplify the equation to ensure that it's true.
Check if the solution is reasonable and makes sense in the context of the problem.