Understanding the Proportion Problem

The given question involves a proportion problem presented in the format 405 : y :: y : 125. This notation means that the ratio of 405 to y is equal to the ratio of y to 125. The double colon (::) symbol indicates equality between two ratios.

We are also given the condition that y > 0, which means y must be a positive value.

Setting up the Equation

A proportion can be written as an equation of fractions. The proportion 405 : y :: y : 125 can be written as:

$\frac{405}{y} = \frac{y}{125}$

In this form, y is known as the mean proportional between 405 and 125.

Solving for y using Cross-Multiplication

To solve this equation for y, we can use cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction.

Applying cross-multiplication to $\frac{405}{y} = \frac{y}{125}$:

$405 \times 125 = y \times y$

This simplifies to:

$y^2 = 405 \times 125$

Calculating the Product 405 × 125

Now, we need to calculate the product of 405 and 125.

$405 \times 125 = 50625$

So the equation becomes:

$y^2 = 50625$

Finding the Value of y

To find the value of y, we need to take the square root of both sides of the equation $y^2 = 50625$:

$y = \sqrt{50625}$

We can find the square root by factoring the numbers or using prime factorization. Let's use prime factorization:

• Prime factorization of 405: $405 = 5 \times 81 = 5 \times 9^2 = 5 \times (3^2)^2 = 5 \times 3^4$
• Prime factorization of 125: $125 = 5^3$

So, $405 \times 125 = (5 \times 3^4) \times 5^3 = 5^{1+3} \times 3^4 = 5^4 \times 3^4$

Now, $y^2 = 5^4 \times 3^4 = (5 \times 3)^4 = 15^4$

Taking the square root:

$y = \sqrt{15^4} = 15^{\frac{4}{2}} = 15^2$

Calculating $15^2$:

$15^2 = 15 \times 15 = 225$

So, $y = 225$.

The question also states that y > 0. Our calculated value, y = 225, is positive, so it satisfies this condition.

Conclusion

Based on the proportion 405 : y :: y : 125 and the condition y > 0, the value of y is 225.

Revision Table: Key Concepts

Additional Information: Geometric Mean

The mean proportional y between two numbers a and b is the same as their geometric mean. The geometric mean of two positive numbers a and b is defined as $\sqrt{ab}$. In this problem, y is the geometric mean of 405 and 125. Calculating the geometric mean directly:

Geometric Mean = $\sqrt{405 \times 125} = \sqrt{50625} = 225$.

This confirms the result obtained through solving the proportion equation. The geometric mean is often used in contexts like calculating average growth rates or in geometry related to similar figures.