Understanding the Mean Proportional Concept

The mean proportional between two numbers, say 'a' and 'b', is defined as the square root of their product. Mathematically, the mean proportional (m) is given by the formula:

\( m = \sqrt{ab} \)

In this problem, we are given a relationship between two mean proportionals and asked to find an unknown number N.

Setting up the Equation from the Problem Statement

The problem states:

1. The mean proportional between 35 and a certain number N.

This can be written as \( \sqrt{35 \times N} \).

2. The mean proportional between 14 and 30.

This can be written as \( \sqrt{14 \times 30} \).

3. The mean proportional between 35 and N is four times the mean proportional between 14 and 30.

Putting this relationship into an equation, we get:

\( \sqrt{35N} = 4 \times \sqrt{14 \times 30} \)

Solving for the Number N

Now, we need to solve this equation to find the value of N. Let's simplify the equation step-by-step:

First, calculate the product inside the second square root:

\( 14 \times 30 = 420 \)

The equation becomes:

\( \sqrt{35N} = 4 \sqrt{420} \)

To eliminate the square roots, we can square both sides of the equation:

\( (\sqrt{35N})^2 = (4 \sqrt{420})^2 \)

\( 35N = 4^2 \times (\sqrt{420})^2 \)

\( 35N = 16 \times 420 \)

Now, we need to isolate N. We can do this by dividing both sides by 35:

\( N = \frac{16 \times 420}{35} \)

We can simplify the fraction by dividing 420 by 35. We know that \( 420 = 42 \times 10 \) and \( 35 = 7 \times 5 \). Also, \( 42 = 6 \times 7 \). So, \( 420 = 6 \times 7 \times 10 \). Dividing 420 by 35:

\( \frac{420}{35} = \frac{6 \times 7 \times 10}{7 \times 5} \)

Cancel out the 7:

\( \frac{6 \times 10}{5} \)

Now, simplify \( \frac{10}{5} = 2 \):

\( 6 \times 2 = 12 \)

So, \( \frac{420}{35} = 12 \).

Substitute this value back into the equation for N:

\( N = 16 \times 12 \)

Calculate the final product:

\( 16 \times 12 = 192 \)

Therefore, the number N is 192.

Verifying the Solution

Let's quickly check if N = 192 satisfies the original condition:

Mean proportional between 35 and 192: \( \sqrt{35 \times 192} \)

\( 35 \times 192 = 35 \times (200 - 8) = 7000 - 280 = 6720 \)

Mean proportional between 14 and 30: \( \sqrt{14 \times 30} = \sqrt{420} \)

Is \( \sqrt{6720} = 4 \times \sqrt{420} \)?

Square both sides:

\( 6720 = 16 \times 420 \)

\( 16 \times 420 = 16 \times (400 + 20) = 6400 + 320 = 6720 \)

The equation holds true. Thus, N = 192 is correct.

Conclusion

Based on our calculation, the number N is 192.

Revision Table: Mean Proportional Concepts

Additional Information: Proportions and Means

Understanding proportions is key to solving problems involving mean proportionals. When three numbers \(a, b, c\) are in continuous proportion, it means that the ratio of the first to the second is equal to the ratio of the second to the third: \( a:b = b:c \). In this sequence, 'b' is called the mean proportional between 'a' and 'c'. This leads to the relationship \( b^2 = ac \).

There are other types of means too:

• Arithmetic Mean: For two numbers a and b, the arithmetic mean is \( \frac{a+b}{2} \).
• Geometric Mean: For two positive numbers a and b, the geometric mean is \( \sqrt{ab} \). The mean proportional is actually the geometric mean of the two numbers.
• Harmonic Mean: For two positive numbers a and b, the harmonic mean is \( \frac{2}{\frac{1}{a} + \frac{1}{b}} = \frac{2ab}{a+b} \).

The problem focused specifically on the geometric mean, referred to as the mean proportional.