Solving the Math Problem: Sum and Quotient of Two Numbers
Mathematics often involves solving problems where we need to determine the values of two or more unknowns based on given conditions. In this article, we will solve a specific problem involving the sum and quotient of two numbers. Let's break down the problem step by step to arrive at the solution and understand the underlying mathematical concepts.
Problem Statement
The sum of two numbers is 120, and their quotient is 5. We need to find the two numbers and then determine their difference.
Step-by-Step Solution
Let's denote the two numbers by x and y, where x ≥ y.
Using the Sum Condition
From the problem, we know that the sum of the two numbers is 120:
x y 120
Using the Quotient Condition
Additionally, the quotient of the two numbers is 5:
x / y 5
From the quotient condition, we can express x in terms of y as:
x 5y
Solving the System of Equations
Now we have two equations:
x y 120
x 5y
Substitute the value of x from the second equation into the first equation:
5y y 120
Simplify the equation:
6y 120
Solve for y:
y 20
Now, substitute the value of y back into the equation x 5y to find x:
x 5 * 20 100
Therefore, the two numbers are 100 and 20.
Verifying the Solution
Let's verify our solution:
Sum: 100 20 120 (Condition satisfied)
Quotient: 100 / 20 5 (Condition satisfied)
The solution is correct.
Conclusion
The difference between the two numbers is calculated as:
x - y 100 - 20 80
Therefore, the difference between the two numbers is 80.
This problem demonstrates the importance of translating word problems into algebraic equations and solving systems of equations to find unknown values. By breaking down the problem into manageable steps, we can arrive at a clear and concise solution.