Math Problem Statement
the sum of 2 numbers is 18 and the sum of their reciprocals is 9/40. Find the numbers
Solution
Let the two numbers be and . We are given the following information:
Step 1: Express in terms of
From , we can write:
Step 2: Substitute into the reciprocal equation
Using , substitute :
Combine the terms under a common denominator:
Simplify the numerator:
Step 3: Cross-multiply to eliminate fractions
Simplify both sides:
Distribute on the right-hand side:
Rearrange into standard quadratic form:
Step 4: Simplify the quadratic equation
Divide through by 9:
Step 5: Solve the quadratic equation
Factorize:
Thus, the solutions are:
Step 6: Find the corresponding values
If , then .
If , then .
Final Answer:
The two numbers are 8 and 10.
Would you like more details or explanations?
Here are 5 related questions to consider:
- How do you determine if a quadratic equation is factorable?
- What are other methods to solve a quadratic equation besides factoring?
- How does cross-multiplication work in rational equations?
- Why is it important to check both solutions in a reciprocal equation?
- Can this problem be solved graphically, and if so, how?
Tip: Always verify solutions in the original equations to ensure accuracy!
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Equations
Quadratic Equations
Formulas
Reciprocal sum formula: 1/x + 1/y = given value
Quadratic equation: ax^2 + bx + c = 0
Theorems
Factorization of quadratic equations
Cross-multiplication for rational equations
Suitable Grade Level
Grades 9-10