Solving for Arithmetic Means with Given Ratios: A Comprehensive Guide

Arithmetic means are a fundamental concept in arithmetic and algebra, often appearing in problems that require the application of algebraic techniques. In this article, we will explore a specific problem where we need to find the value of n (the number of arithmetic means) between two numbers, 5 and 32, given the ratio of the 3rd and 7th means. We will break down the problem step-by-step, showcasing the application of algebra and the importance of checking the validity of the solution.

Problem Explanation

There are n arithmetic means between 5 and 32 such that the ratio of the 3rd and 7th means is 7:13. We need to find the value of n.

Step-by-Step Solution

Let's start by establishing the general formula for the arithmetic means. We denote the first term by 5 and the last term by 32. The n arithmetic means can be expressed as:

5 5 d 5 2d... 32

Step 1: Determine the Common Difference d

The common difference d can be expressed in terms of n as follows:

5 nd 32

Solving for d:

d (32 - 5) / n 27 / n

Step 2: Express the 3rd and 7th Means

The 3rd mean is:

5 2d 5 2(27 / n) 5 54 / n

The 7th mean is:

5 6d 5 6(27 / n) 5 162 / n

Step 3: Apply the Given Ratio

The problem states that the ratio of the 3rd to the 7th means is 7:13:

a3 / a7 7 / 13

Substituting the expressions for a3 and a7:

(5 54 / n) / (5 162 / n) 7 / 13

Cross-multiplying to eliminate the fraction:

13(5 54 / n) 7(5 162 / n)

Expanding both sides:

65 654 / n 35 1134 / n

Rearranging the equation:

65 - 35 1134 / n - 654 / n

20 480 / n

Solving for n:

n 480 / 20 13.4

Since n must be an integer, we need to verify the solution with integer values.

Step 4: Verify the Solution

We consider n 13:

d 27 / 14 27 / 14

a3 5 2(27 / 14) 5 54 / 14 5 27 / 7

a7 5 6(27 / 14) 5 162 / 14 5 81 / 7

Calculating the ratio:

a3 / a7 (5 27 / 7) / (5 81 / 7) (35 27) / (35 81) 62 / 116 7 / 13

This confirms that our value of n 13 is correct.

Conclusion

The value of n is: 13.

Key Takeaways:

Understanding the arithmetic mean formula is crucial for solving such problems. Checking the solution with integer values is essential to ensure accuracy. Applying algebraic techniques such as cross-multiplication and solving equations effectively.

By mastering these steps, you can tackle similar arithmetic mean and ratio problems with confidence.