2. Insert Three Rational Numbers Between: (i) 4 And 11 (ii) 9 And 16

Feb 28, 2025 by ADMIN 70 views

**Inserting Rational Numbers between Given Intervals**

Introduction

In mathematics, rational numbers are a type of real number that can be expressed as the ratio of two integers. They play a crucial role in various mathematical operations and are used extensively in algebra, geometry, and other branches of mathematics. One of the essential concepts in mathematics is inserting rational numbers between given intervals. This concept is used to create a sequence of rational numbers that are evenly spaced and can be used to approximate the value of a given number.

Inserting Rational Numbers between 4 and 11

To insert three rational numbers between 4 and 11, we need to find the average of the two numbers and then add or subtract a fraction to create a sequence of rational numbers. The average of 4 and 11 is (4 + 11) / 2 = 7.5. To create a sequence of rational numbers, we can add or subtract a fraction from the average.

Let's start by adding a fraction to the average. We can add 1/2, 1/4, or 1/8 to the average to create a sequence of rational numbers. Adding 1/2 to the average gives us 7.5 + 1/2 = 8. Adding 1/4 to the average gives us 7.5 + 1/4 = 7.75. Adding 1/8 to the average gives us 7.5 + 1/8 = 7.625.

Now, let's subtract a fraction from the average. We can subtract 1/2, 1/4, or 1/8 from the average to create a sequence of rational numbers. Subtracting 1/2 from the average gives us 7.5 - 1/2 = 7. Subtracting 1/4 from the average gives us 7.5 - 1/4 = 7.25. Subtracting 1/8 from the average gives us 7.5 - 1/8 = 7.375.

Inserting Rational Numbers between 9 and 16

To insert three rational numbers between 9 and 16, we need to find the average of the two numbers and then add or subtract a fraction to create a sequence of rational numbers. The average of 9 and 16 is (9 + 16) / 2 = 12.5. To create a sequence of rational numbers, we can add or subtract a fraction from the average.

Let's start by adding a fraction to the average. We can add 1/2, 1/4, or 1/8 to the average to create a sequence of rational numbers. Adding 1/2 to the average gives us 12.5 + 1/2 = 13. Adding 1/4 to the average gives us 12.5 + 1/4 = 12.75. Adding 1/8 to the average gives us 12.5 + 1/8 = 12.625.

Now, let's subtract a fraction from the average. We can subtract 1/2, 1/4, or 1/8 from the average to create a sequence of rational numbers. Subtracting 1/2 from the average gives us 12.5 - 1/2 = 12. Subtracting 1/4 from the average gives us 12.5 - 1/4 = 12.25. Subtracting 1/8 from the average gives us 12.5 - 1/8 = 12.375.

Conclusion

Inserting rational numbers between given intervals is an essential concept in mathematics. It helps to create a sequence of rational numbers that are evenly spaced and can be used to approximate the value of a given number. By finding the average of the two numbers and adding or subtracting a fraction, we can create a sequence of rational numbers that are evenly spaced. This concept is used extensively in algebra, geometry, and other branches of mathematics.

Example Problems

Insert three rational numbers between 2 and 7.
Insert three rational numbers between 5 and 12.
Insert three rational numbers between 8 and 15.

Solutions

To insert three rational numbers between 2 and 7, we need to find the average of the two numbers and then add or subtract a fraction to create a sequence of rational numbers. The average of 2 and 7 is (2 + 7) / 2 = 4.5. Adding 1/2, 1/4, or 1/8 to the average gives us 5, 4.75, or 4.625, respectively. Subtracting 1/2, 1/4, or 1/8 from the average gives us 4, 4.25, or 4.375, respectively.
To insert three rational numbers between 5 and 12, we need to find the average of the two numbers and then add or subtract a fraction to create a sequence of rational numbers. The average of 5 and 12 is (5 + 12) / 2 = 8.5. Adding 1/2, 1/4, or 1/8 to the average gives us 9, 8.75, or 8.625, respectively. Subtracting 1/2, 1/4, or 1/8 from the average gives us 8, 8.25, or 8.375, respectively.
To insert three rational numbers between 8 and 15, we need to find the average of the two numbers and then add or subtract a fraction to create a sequence of rational numbers. The average of 8 and 15 is (8 + 15) / 2 = 11.5. Adding 1/2, 1/4, or 1/8 to the average gives us 12, 11.75, or 11.625, respectively. Subtracting 1/2, 1/4, or 1/8 from the average gives us 11, 11.25, or 11.375, respectively.

Tips and Tricks

To insert rational numbers between given intervals, find the average of the two numbers and then add or subtract a fraction to create a sequence of rational numbers.
Use fractions such as 1/2, 1/4, or 1/8 to add or subtract from the average.
Make sure to check your work by plugging in the values into the original equation.
Practice inserting rational numbers between given intervals to become more comfortable with the concept.

Common Mistakes

Failing to find the average of the two numbers.
Adding or subtracting the wrong fraction from the average.
Not checking the work by plugging in the values into the original equation.

Conclusion

Q: What is the purpose of inserting rational numbers between given intervals?

A: The purpose of inserting rational numbers between given intervals is to create a sequence of rational numbers that are evenly spaced and can be used to approximate the value of a given number.

Q: How do I find the average of two numbers?

A: To find the average of two numbers, you add the two numbers together and then divide by 2. For example, the average of 4 and 11 is (4 + 11) / 2 = 7.5.

Q: What are some common fractions that I can add or subtract from the average?

A: Some common fractions that you can add or subtract from the average include 1/2, 1/4, and 1/8. You can also use other fractions such as 1/3, 1/6, or 1/10.

Q: How do I know which fraction to add or subtract from the average?

A: The choice of fraction to add or subtract from the average depends on the specific problem you are working on. You may need to experiment with different fractions to find the one that works best for your problem.

Q: Can I use decimals instead of fractions?

A: Yes, you can use decimals instead of fractions. For example, instead of adding 1/2 to the average, you can add 0.5. However, be careful when working with decimals, as they can be more difficult to work with than fractions.

Q: How do I check my work when inserting rational numbers between given intervals?

A: To check your work, you can plug in the values into the original equation and see if they are true. You can also use a calculator to check your work.

Q: What are some common mistakes to avoid when inserting rational numbers between given intervals?

A: Some common mistakes to avoid when inserting rational numbers between given intervals include:

Failing to find the average of the two numbers
Adding or subtracting the wrong fraction from the average
Not checking the work by plugging in the values into the original equation

Q: Can I use this concept to solve other types of problems?

A: Yes, you can use this concept to solve other types of problems. For example, you can use it to find the midpoint of a line segment, or to solve problems involving proportions.

Q: How can I practice inserting rational numbers between given intervals?

A: You can practice inserting rational numbers between given intervals by working on problems in your textbook or online. You can also try creating your own problems and solving them.

Q: What are some real-world applications of inserting rational numbers between given intervals?

A: Some real-world applications of inserting rational numbers between given intervals include:

Finding the midpoint of a line segment
Solving problems involving proportions
Creating a sequence of rational numbers that are evenly spaced
Approximating the value of a given number