Subtracting 104 from Both Sides

Oct 13, 2025 by miodeya.com 124 views

Understanding the Problem

In this problem, we are given a proportion in the form of a ratio, where the first ratio is equivalent to the second ratio. The goal is to find the value of the variable x in the second ratio. To do this, we need to use the concept of equivalent ratios and solve for x.

What are Equivalent Ratios?

Equivalent ratios are ratios that have the same value, but are expressed in different forms. For example, the ratios 1:2 and 2:4 are equivalent because they both represent the same relationship between the two quantities. In this problem, we are given the ratio 5:8, which is equivalent to the ratio (2x + 13):56.

Setting Up the Equation

To solve for x, we need to set up an equation based on the given proportion. We can do this by cross-multiplying the two ratios, which means multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa. This gives us the equation:

5 × 56 = 8 × (2x + 13)

Simplifying the Equation

Now that we have the equation, we can simplify it by multiplying the numbers and combining like terms. This gives us:

280 = 16x + 104

Subtracting 104 from Both Sides

To isolate the term with x, we need to subtract 104 from both sides of the equation. This gives us:

176 = 16x

Dividing Both Sides by 16

Finally, we can solve for x by dividing both sides of the equation by 16. This gives us:

x = 11

Conclusion

In this problem, we used the concept of equivalent ratios to solve for the value of x in a proportion. We set up an equation based on the given proportion, simplified it, and solved for x. The final answer is x = 11.

Why is this Problem Important?

This problem is important because it illustrates the concept of equivalent ratios and how to solve for a variable in a proportion. This is a fundamental concept in mathematics, and it has many real-world applications, such as finance, science, and engineering.

Real-World Applications

The concept of equivalent ratios has many real-world applications, such as:

Finance: When comparing the prices of two different products, we need to use equivalent ratios to ensure that we are comparing apples to apples.
Science: When measuring the concentration of a substance, we need to use equivalent ratios to ensure that we are measuring the correct amount.
Engineering: When designing a system, we need to use equivalent ratios to ensure that the system is functioning correctly.

Tips and Tricks

Here are some tips and tricks to help you solve problems like this:

Read the problem carefully: Make sure you understand what the problem is asking for.
Use equivalent ratios: When comparing two ratios, use equivalent ratios to ensure that you are comparing apples to apples.
Simplify the equation: Simplify the equation by combining like terms and multiplying numbers.
Isolate the variable: Isolate the variable by subtracting or adding numbers to both sides of the equation.

Common Mistakes

Here are some common mistakes to avoid when solving problems like this:

Not reading the problem carefully: Make sure you understand what the problem is asking for.
Not using equivalent ratios: When comparing two ratios, use equivalent ratios to ensure that you are comparing apples to apples.
Not simplifying the equation: Simplify the equation by combining like terms and multiplying numbers.
Not isolating the variable: Isolate the variable by subtracting or adding numbers to both sides of the equation.

Conclusion

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It is often written in the form of a fraction, where the first ratio is equal to the second ratio.

Q: How do I know if two ratios are equivalent?

A: Two ratios are equivalent if they have the same value, but are expressed in different forms. For example, the ratios 1:2 and 2:4 are equivalent because they both represent the same relationship between the two quantities.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to solve proportions by multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.

Q: How do I simplify an equation?

A: To simplify an equation, you need to combine like terms and multiply numbers. For example, if you have the equation 2x + 3 = 5x - 2, you can simplify it by combining the like terms and multiplying the numbers.

Q: How do I isolate a variable?

A: To isolate a variable, you need to get the variable by itself on one side of the equation. This can be done by adding or subtracting numbers to both sides of the equation.

Q: What is the difference between a proportion and an equation?

A: A proportion is a statement that two ratios are equal, while an equation is a statement that two expressions are equal. While both proportions and equations can be used to solve for a variable, they are used in different contexts.

Q: How do I use proportions in real-life situations?

A: Proportions are used in many real-life situations, such as finance, science, and engineering. For example, when comparing the prices of two different products, you can use proportions to ensure that you are comparing apples to apples.

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

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

Not reading the problem carefully
Not using equivalent ratios
Not simplifying the equation
Not isolating the variable

Q: How do I check my answer when solving a proportion?

A: To check your answer when solving a proportion, you need to plug the value of the variable back into the original equation and see if it is true. If it is true, then your answer is correct.

Q: What are some tips for solving proportions?

A: Some tips for solving proportions include:

Read the problem carefully
Use equivalent ratios
Simplify the equation
Isolate the variable
Check your answer

Q: Can I use proportions to solve for more than one variable?

A: Yes, you can use proportions to solve for more than one variable. However, you need to make sure that the proportion is consistent and that the variables are related in a logical way.

Q: How do I use proportions to solve for a variable in a word problem?

A: To use proportions to solve for a variable in a word problem, you need to:

Read the problem carefully and identify the variables
Write a proportion based on the information given in the problem
Solve the proportion using the techniques described above
Check your answer to make sure it is correct

Conclusion

In conclusion, proportions are an important concept in mathematics that has many real-world applications. By understanding how to solve proportions, you can use them to solve for variables in a variety of contexts. Remember to read the problem carefully, use equivalent ratios, simplify the equation, and isolate the variable to avoid common mistakes.