Solving for the Unknown in Product and Multiples: A Step-by-Step Guide

Mathematics often involves solving for unknowns through equations that represent relationships between numbers. This article provides a detailed explanation on solving a specific type of problem: finding the value of unknowns when given the product and a multiple relationship. We will go through the steps to solve the given example, xy 900 and 3x 45, to find the values of x and y.

Problem Statement

Given:

xy 900 3x 45

Our goal is to find the values of x and y given the two equations above.

Step-by-Step Solution

Solving for x in the Second Equation

The second equation, 3x 45, can be solved directly.

Solution

Divide both sides of the equation by 3 to find x:

3x 45

x 45 / 3 15

Solving for y in the First Equation

Now that we know x 15, we can substitute this value into the first equation, xy 900, to solve for y.

Solution

Substitute x 15 into the first equation:

xy 900

15y 900

Divide both sides by 15 to solve for y:

y 900 / 15 60

Conclusion

The solutions to the given equations are x 15 and y 60. Therefore, the other number is 60.

Verification

To verify our solution, let's check if the values satisfy both original equations:

xy 900 3x 45

Substituting x 15 and y 60:

15 * 60 900 (True) 3 * 15 45 (True)

Additional Examples

Example 1

Let the two numbers be N and n.

Equations:

3Nn 900 … eqn 1 3N 45 … eqn 2

Solution:

From eqn 2, solve for N:

3N 45

N 45 / 3 15

Substitute N 15 into eqn 1:

3Nn 900

15n 900

Solve for n:

n 900 / 15 60

The numbers are 15 and 60.

Example 2

Let the two numbers be x and y.

Solution:

From the second equation, solve for x:

3x 45

x 45 / 3 15

Substitute x 15 into the first equation:

xy 900

15y 900

Solve for y:

y 900 / 15 60

The numbers are 15 and 60.

Both verification methods illustrate that 15 and 60 satisfy the original equations, confirming our solution is correct.