Math Problem Statement
The sum of two numbers is 18 and the sum of their reciprocals is 9/40. Find the numbers.
Solution
We are given two pieces of information about two numbers, let's call them and :
-
The sum of the numbers is 18:
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The sum of their reciprocals is :
Step 1: Use the equation for the sum of the numbers
From the first equation, we know that: We can express in terms of :
Step 2: Use the equation for the sum of the reciprocals
From the second equation: Substitute into this equation: Now, to solve for , find a common denominator on the left-hand side: Simplify the numerator: Now cross-multiply to solve for : Now expand the right-hand side: Rearrange the equation into a standard quadratic form: Now, divide the entire equation by 9:
Step 3: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula: So, the two possible values for are:
Step 4: Find the corresponding values for
Using :
- If , then .
- If , then .
Thus, the two numbers are 10 and 8.
Final Answer:
The numbers are 10 and 8.
Would you like a detailed breakdown or have any questions about this solution? Here are some related questions to explore further:
- How does the quadratic formula work in solving quadratic equations?
- What are other methods of solving systems of equations besides substitution?
- How can we check our solution by verifying the given conditions?
- Can we solve problems with reciprocal sums in other ways, such as using algebraic identities?
- What would change if the sum of reciprocals was a different fraction?
Tip: When solving reciprocal problems, it's often helpful to simplify the equation as early as possible to avoid complex algebraic steps.
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Math Problem Analysis
Mathematical Concepts
Algebra
Reciprocals
Quadratic Equations
Formulas
x + y = 18
1/x + 1/y = 9/40
Quadratic equation formula: ax^2 + bx + c = 0
Theorems
Quadratic formula
Suitable Grade Level
Grades 7-9
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