Numbers Finder by Product and Quotient

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Calculate Two Numbers Using Their Product and Quotient

Result

How to Calculate Two Numbers Using Their Product and Quotient

Given:

\( P \): the product of the two numbers (\( x \cdot y \))
\( Q \): the quotient of the two numbers (\( \frac{x}{y} \))

Steps:

Set up relationships: \( x \cdot y = P, \quad \frac{x}{y} = Q \)
Express \( x \) in terms of \( y \): \( x = Q \cdot y \)
Substitute into the product equation: \( (Q \cdot y) \cdot y = P \)
Solve for \( y \): \( y = \sqrt{\frac{P}{Q}} \)
Find \( x \): \( x = Q \cdot y \)

Examples

Example 1: The product of two numbers is 50, and their quotient is 2. What are the numbers?

Solution:

1. Solve for \( y \):

\( y = \sqrt{\frac{50}{2}} = \sqrt{25} = 5 \)

2. Calculate \( x \):

\( x = 2 \cdot 5 = 10 \)

Result: The numbers are 10 and 5.

Example 2: The product of two numbers is 96, and their quotient is 6. What are the numbers?

Solution:

1. Solve for \( y \):

\( y = \sqrt{\frac{96}{6}} = \sqrt{16} = 4 \)

2. Calculate \( x \):

\( x = 6 \cdot 4 = 24 \)

Result: The numbers are 24 and 4.

Example 3: The product of two numbers is 9375, and their quotient is 15. What are the numbers?

Solution:

1. Solve for \( y \):

\( y = \sqrt{\frac{9375}{15}} = \sqrt{625} = 25 \)

2. Calculate \( x \):

\( x = 15 \cdot 25 = 375 \)

Result: The numbers are 375 and 25.