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 xx and yy. We are given the following information:

  1. x+y=18x + y = 18
  2. 1x+1y=940\frac{1}{x} + \frac{1}{y} = \frac{9}{40}

Step 1: Express yy in terms of xx

From x+y=18x + y = 18, we can write: y=18xy = 18 - x

Step 2: Substitute into the reciprocal equation

Using 1x+1y=940\frac{1}{x} + \frac{1}{y} = \frac{9}{40}, substitute y=18xy = 18 - x: 1x+118x=940\frac{1}{x} + \frac{1}{18 - x} = \frac{9}{40}

Combine the terms under a common denominator: 18x+xx(18x)=940\frac{18 - x + x}{x(18 - x)} = \frac{9}{40}

Simplify the numerator: 18x(18x)=940\frac{18}{x(18 - x)} = \frac{9}{40}

Step 3: Cross-multiply to eliminate fractions

4018=9x(18x)40 \cdot 18 = 9 \cdot x(18 - x)

Simplify both sides: 720=9x(18x)720 = 9x(18 - x)

Distribute on the right-hand side: 720=162x9x2720 = 162x - 9x^2

Rearrange into standard quadratic form: 9x2162x+720=09x^2 - 162x + 720 = 0

Step 4: Simplify the quadratic equation

Divide through by 9: x218x+80=0x^2 - 18x + 80 = 0

Step 5: Solve the quadratic equation

Factorize: x218x+80=(x10)(x8)=0x^2 - 18x + 80 = (x - 10)(x - 8) = 0

Thus, the solutions are: x=10orx=8x = 10 \quad \text{or} \quad x = 8

Step 6: Find the corresponding yy values

If x=10x = 10, then y=1810=8y = 18 - 10 = 8.
If x=8x = 8, then y=188=10y = 18 - 8 = 10.

Final Answer:

The two numbers are 8 and 10.

Would you like more details or explanations?
Here are 5 related questions to consider:

  1. How do you determine if a quadratic equation is factorable?
  2. What are other methods to solve a quadratic equation besides factoring?
  3. How does cross-multiplication work in rational equations?
  4. Why is it important to check both solutions in a reciprocal equation?
  5. 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