Which Numbers Are The Extremes Of The Proportion Shown Below?$\frac{3}{4}=\frac{6}{8}$A. 3 And 8 B. 3 And 6 C. 4 And 6 D. 4 And 8

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What are Proportions?

A proportion is a mathematical statement that two ratios are equal. It is often expressed as a fraction, where the two ratios are set equal to each other. For example, the proportion 34=68\frac{3}{4}=\frac{6}{8} states that the ratio of 3 to 4 is equal to the ratio of 6 to 8.

Identifying Extreme Numbers in a Proportion

When working with proportions, it is essential to identify the extreme numbers, which are the numbers that are not in the middle of the proportion. In the given proportion 34=68\frac{3}{4}=\frac{6}{8}, we need to determine which numbers are the extremes.

Step 1: Find the Greatest Common Divisor (GCD)

To identify the extreme numbers, we need to find the greatest common divisor (GCD) of the two ratios. The GCD is the largest number that divides both numbers without leaving a remainder. In this case, the GCD of 3 and 6 is 3, and the GCD of 4 and 8 is 4.

Step 2: Divide Both Ratios by the GCD

Once we have found the GCD, we need to divide both ratios by the GCD. This will give us the simplified ratios. In this case, we divide both ratios by 3:

34=68\frac{3}{4}=\frac{6}{8}

143=283\frac{1}{\frac{4}{3}}=\frac{2}{\frac{8}{3}}

34=68\frac{3}{4}=\frac{6}{8}

Step 3: Identify the Extreme Numbers

Now that we have simplified the ratios, we can identify the extreme numbers. The extreme numbers are the numbers that are not in the middle of the proportion. In this case, the extreme numbers are 3 and 8.

Conclusion

In conclusion, the extreme numbers of the proportion 34=68\frac{3}{4}=\frac{6}{8} are 3 and 8. This is because the GCD of the two ratios is 3, and when we divide both ratios by 3, we get the simplified ratios 143=283\frac{1}{\frac{4}{3}}=\frac{2}{\frac{8}{3}}. The extreme numbers are the numbers that are not in the middle of the proportion, which are 3 and 8.

Answer

The correct answer is A. 3 and 8.

Why is this Important?

Understanding proportions and identifying extreme numbers is crucial in mathematics, particularly in algebra and geometry. It helps us to simplify complex equations and solve problems more efficiently. In addition, it is essential in real-world applications, such as finance, engineering, and science.

Real-World Applications

Proportions and extreme numbers are used in various real-world applications, such as:

  • Finance: Understanding proportions and extreme numbers is essential in finance, particularly in investment and risk management.
  • Engineering: Proportions and extreme numbers are used in engineering to design and optimize systems, such as bridges and buildings.
  • Science: Proportions and extreme numbers are used in science to understand and describe natural phenomena, such as the behavior of particles and the movement of celestial bodies.

Conclusion

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 ratios are set equal to each other.

Q: How do I identify the extreme numbers in a proportion?

A: To identify the extreme numbers in a proportion, you need to find the greatest common divisor (GCD) of the two ratios. Then, divide both ratios by the GCD to simplify them. The extreme numbers are the numbers that are not in the middle of the proportion.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 3 and 6 is 3, and the GCD of 4 and 8 is 4.

Q: How do I simplify a proportion?

A: To simplify a proportion, you need to find the GCD of the two ratios and divide both ratios by the GCD. This will give you the simplified ratios.

Q: What are the extreme numbers in the proportion 34=68\frac{3}{4}=\frac{6}{8}?

A: The extreme numbers in the proportion 34=68\frac{3}{4}=\frac{6}{8} are 3 and 8.

Q: Why is it important to understand proportions and extreme numbers?

A: Understanding proportions and extreme numbers is crucial in mathematics, particularly in algebra and geometry. It helps us to simplify complex equations and solve problems more efficiently. In addition, it is essential in real-world applications, such as finance, engineering, and science.

Q: Can you give me an example of a real-world application of proportions and extreme numbers?

A: Yes, here's an example: Imagine you are a civil engineer designing a bridge. You need to ensure that the bridge can hold a certain amount of weight. You can use proportions and extreme numbers to calculate the weight that the bridge can hold.

Q: How do I calculate the weight that a bridge can hold using proportions and extreme numbers?

A: To calculate the weight that a bridge can hold, you need to use the proportion weightlength=loaddistance\frac{weight}{length}=\frac{load}{distance}. You can then simplify the proportion by finding the GCD of the two ratios and dividing both ratios by the GCD. This will give you the simplified ratio, which you can use to calculate the weight that the bridge can hold.

Q: What are some common mistakes to avoid when working with proportions and extreme numbers?

A: Some common mistakes to avoid when working with proportions and extreme numbers include:

  • Not finding the GCD of the two ratios before simplifying the proportion
  • Not dividing both ratios by the GCD
  • Not checking if the simplified ratio is equal to the original ratio
  • Not using the correct units when working with proportions and extreme numbers

Q: How can I practice working with proportions and extreme numbers?

A: You can practice working with proportions and extreme numbers by:

  • Solving problems that involve proportions and extreme numbers
  • Using online resources, such as calculators and worksheets
  • Working with a tutor or teacher who can provide guidance and support
  • Practicing with real-world applications, such as finance, engineering, and science.