The Concept of Equivalent Ratios

Oct 13, 2025 by miodeya.com 128 views

Understanding the Problem

When solving for an unknown value in a proportion, we need to use the concept of equivalent ratios. A proportion is a statement that two ratios are equal. In this case, we are given the proportion $\frac{4}{5} = \frac{N}{15}$ , and we need to solve for the value of $N$ .

The Concept of Equivalent Ratios

Equivalent ratios are ratios that have the same value. In other words, if we multiply or divide both the numerator and the denominator of a ratio by the same non-zero number, we get an equivalent ratio. This concept is crucial in solving proportions.

Solving the Proportion

To solve the proportion $\frac{4}{5} = \frac{N}{15}$ , we need to find the value of $N$ that makes the two ratios equivalent. We can do this by cross-multiplying, which means multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.

Cross-Multiplying

Cross-multiplying the proportion $\frac{4}{5} = \frac{N}{15}$ gives us:

4 \times 15 = 5 \times N

Simplifying the Equation

Simplifying the equation $4 \times 15 = 5 \times N$ gives us:

60 = 5N

Solving for N

To solve for $N$ , we need to isolate $N$ on one side of the equation. We can do this by dividing both sides of the equation by 5:

\frac{60}{5} = N

Calculating the Value of N

Calculating the value of $N$ gives us:

N = 12

Conclusion

In conclusion, we have solved the proportion $\frac{4}{5} = \frac{N}{15}$ and found the value of $N$ to be 12. This is the correct answer, and we can verify it by plugging it back into the original proportion.

Answer

The correct answer is C) 12.

Why is this Important?

Solving proportions is an important skill in mathematics, as it allows us to solve problems involving equivalent ratios. This skill is used in a variety of real-world applications, such as finance, science, and engineering.

Real-World Applications

Solving proportions has many real-world applications, including:

Finance: Solving proportions is used in finance to calculate interest rates, investment returns, and other financial metrics.
Science: Solving proportions is used in science to calculate the concentration of a solution, the volume of a gas, and other scientific metrics.
Engineering: Solving proportions is used in engineering to calculate the stress on a material, the flow rate of a fluid, and other engineering metrics.

Tips and Tricks

Here are some tips and tricks for solving proportions:

Use cross-multiplication: Cross-multiplication is a powerful tool for solving proportions.
Simplify the equation: Simplifying the equation can make it easier to solve.
Isolate the variable: Isolating the variable on one side of the equation can make it easier to solve.
Check your answer: Checking your answer by plugging it back into the original proportion can help you verify that it is correct.

Common Mistakes

Here are some common mistakes to avoid when solving proportions:

Not using cross-multiplication: Not using cross-multiplication can make it difficult to solve the proportion.
Not simplifying the equation: Not simplifying the equation can make it difficult to solve.
Not isolating the variable: Not isolating the variable on one side of the equation can make it difficult to solve.
Not checking your answer: Not checking your answer by plugging it back into the original proportion can lead to incorrect solutions.

Conclusion

In conclusion, solving proportions is an important skill in mathematics that has many real-world applications. By using cross-multiplication, simplifying the equation, isolating the variable, and checking your answer, you can solve proportions with ease. Remember to avoid common mistakes, such as not using cross-multiplication, not simplifying the equation, not isolating the variable, and not checking your answer. With practice and patience, you can become proficient in solving proportions and apply this skill to a variety of real-world problems.

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It is a way of expressing the relationship between two quantities.

Q: How do I solve a proportion?

A: To solve a proportion, you need to find the value of the unknown quantity that makes the two ratios equivalent. You can do this by cross-multiplying, which means 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 technique used to solve proportions. It involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.

Q: Why do I need to simplify the equation?

A: Simplifying the equation makes it easier to solve. It helps to eliminate any unnecessary terms and makes the equation more manageable.

Q: How do I isolate the variable?

A: To isolate the variable, you need to get it on one side of the equation by itself. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: Why do I need to check my answer?

A: Checking your answer is an important step in solving proportions. It helps to verify that your solution is correct and ensures that you have not made any mistakes.

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
Not simplifying the equation
Not isolating the variable
Not checking your answer

Q: How do I apply proportions in real-world situations?

A: Proportions have many real-world applications, including finance, science, and engineering. You can use proportions to calculate interest rates, investment returns, concentration of a solution, volume of a gas, and other scientific and engineering metrics.

Q: What are some examples of proportions in real-world situations?

A: Some examples of proportions in real-world situations include:

Calculating the interest rate on a loan
Determining the concentration of a solution
Calculating the volume of a gas
Determining the stress on a material
Calculating the flow rate of a fluid

Q: How do I practice solving proportions?

A: You can practice solving proportions by working through examples and exercises. You can also use online resources, such as calculators and worksheets, to help you practice.

Q: What are some tips for solving proportions?

A: Some tips for solving proportions include:

Use cross-multiplication
Simplify the equation
Isolate the variable
Check your answer
Practice regularly

Q: What are some common misconceptions about proportions?

A: Some common misconceptions about proportions include:

Thinking that proportions are only used in mathematics
Believing that proportions are only used in simple calculations
Assuming that proportions are only used in specific fields, such as finance or science

Q: How do I overcome common misconceptions about proportions?

A: To overcome common misconceptions about proportions, you need to understand the concept of proportions and its applications. You can do this by learning about proportions, practicing solving proportions, and applying proportions in real-world situations.

Conclusion

In conclusion, solving proportions is an important skill that has many real-world applications. By understanding the concept of proportions, practicing solving proportions, and applying proportions in real-world situations, you can become proficient in solving proportions and apply this skill to a variety of real-world problems.