Numbers Finder by Sum and Quotient

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Calculate Two Numbers Using Their Sum and Quotient

Result

How to Calculate Two Numbers Using Their Sum and Quotient

Given:

\( S \): the sum of the two numbers (\( x + y \))
\( Q \): the quotient of the two numbers (\( \frac{x}{y} \))

Steps:

Set up relationships: \( x + y = S, \quad \frac{x}{y} = Q \)
Express \( x \) in terms of \( y \): \( x = Q \cdot y \)
Substitute into the sum equation: \( Q \cdot y + y = S \)
Factorize \( y \): \( (Q + 1) \cdot y = S \)
Solve for \( y \): \( y = \frac{S}{Q + 1} \)
Find \( x \): \( x = S - y \)

Examples

Example 1: The sum of two numbers is 84, and their quotient is 20. What are the numbers?

Solution:

1. Solve for \( y \):

\( y = \frac{84}{20 + 1} = \frac{84}{21} = 4 \)

2. Calculate \( x \):

\( x = 84 - 4 = 80 \)

Result: The numbers are 80 and 4.

Example 2: The sum of two numbers is 240, and their quotient is 7. What are the numbers?

Solution:

1. Solve for \( y \):

\( y = \frac{240}{7 + 1} = \frac{240}{8} = 30 \)

2. Calculate \( x \):

\( x = 240 - 30 = 210 \)

Result: The numbers are 210 and 30.