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 xx and yy:

  1. The sum of the numbers is 18:
    x+y=18x + y = 18

  2. The sum of their reciprocals is 940\frac{9}{40}:
    1x+1y=940\frac{1}{x} + \frac{1}{y} = \frac{9}{40}

Step 1: Use the equation for the sum of the numbers

From the first equation, we know that: x+y=18x + y = 18 We can express yy in terms of xx: y=18xy = 18 - x

Step 2: Use the equation for the sum of the reciprocals

From the second equation: 1x+1y=940\frac{1}{x} + \frac{1}{y} = \frac{9}{40} Substitute y=18xy = 18 - x into this equation: 1x+118x=940\frac{1}{x} + \frac{1}{18 - x} = \frac{9}{40} Now, to solve for xx, find a common denominator on the left-hand side: (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} Now cross-multiply to solve for xx: 18×40=9×x(18x)18 \times 40 = 9 \times x(18 - x) 720=9x(18x)720 = 9x(18 - x) Now expand the right-hand side: 720=9x(18)9x2720 = 9x(18) - 9x^2 720=162x9x2720 = 162x - 9x^2 Rearrange the equation into a standard quadratic form: 9x2162x+720=09x^2 - 162x + 720 = 0 Now, divide the entire equation by 9: x218x+80=0x^2 - 18x + 80 = 0

Step 3: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula: x=(18)±(18)24(1)(80)2(1)x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(1)(80)}}{2(1)} x=18±3243202x = \frac{18 \pm \sqrt{324 - 320}}{2} x=18±42x = \frac{18 \pm \sqrt{4}}{2} x=18±22x = \frac{18 \pm 2}{2} So, the two possible values for xx are: x=18+22=10orx=1822=8x = \frac{18 + 2}{2} = 10 \quad \text{or} \quad x = \frac{18 - 2}{2} = 8

Step 4: Find the corresponding values for yy

Using x+y=18x + y = 18:

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

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:

  1. How does the quadratic formula work in solving quadratic equations?
  2. What are other methods of solving systems of equations besides substitution?
  3. How can we check our solution by verifying the given conditions?
  4. Can we solve problems with reciprocal sums in other ways, such as using algebraic identities?
  5. 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