Understanding Third Proportional

The concept of 'third proportional' relates to ratios and proportions. If we have three numbers, say 'a', 'b', and 'c', and they are in proportion, it means the ratio of the first two numbers ('a' to 'b') is equal to the ratio of the second and third numbers ('b' to 'c').

Mathematically, this is represented as:

a : b :: b : c

Or, in fractional form:

$$ \frac{a}{b} = \frac{b}{c} $$

In this context, 'c' is called the third proportional to 'a' and 'b'.

Calculating the Third Proportional Value of Z

The question states that the third proportional of 36 and Z is 4. Here:

• The first term (a) is 36.
• The second term (b) is Z.
• The third proportional (c) is 4.

Using the definition of the third proportional, we can set up the proportion:

36 : Z :: Z : 4

Now, we convert this proportion into an equation:

$$ \frac{36}{Z} = \frac{Z}{4} $$

To solve for Z, we cross-multiply:

$$ 36 \times 4 = Z \times Z $$

$$ 144 = Z^2 $$

To find the value of Z, we need to take the square root of both sides of the equation:

$$ Z = \sqrt{144} $$

$$ Z = \pm 12 $$

The question specifically asks for the positive value of Z. Therefore, we choose the positive result from the square root.

The positive value of Z is 12.

Matching with Options

Comparing our result with the given options:

• 1. 14
• 2. 11
• 3. 9
• 4. 12
• 5.

The calculated positive value of Z matches option 4.