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Magazine Chapter 7: Problem 6
A positive integer is 2 more than another. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is \(17 / 35,\) then find the two integers.
Short Answer
Expert verified
The two integers are 5 and 7.
Step by step solution
01
Set Up Variables
Let the smaller integer be \( x \). Then the larger integer is \( x + 2 \).
02
Set Up Reciprocals
The reciprocal of the smaller integer is \( \frac{1}{x} \), and the reciprocal of the larger integer (which is twice) is \( \frac{2}{x+2} \).
03
Write the Equation
According to the problem, \( \frac{1}{x} + \frac{2}{x + 2} = \frac{17}{35} \).
04
Find a Common Denominator
The common denominator for the left side of the equation is \( x(x+2) \). Thus, the equation becomes \( \frac{x+2+2x}{x(x+2)} = \frac{17}{35} \).
05
Simplify the Equation
Simplify the numerator: \( \frac{3x+2}{x(x+2)} = \frac{17}{35} \).
06
Cross-Multiply to Solve for x
Cross-multiply to solve for \( x \): \( 35(3x + 2) = 17x(x+2) \). This becomes \( 105x + 70 = 17x^2 + 34x \).
07
Rearrange into Quadratic Form
Rearrange the equation: \( 17x^2 - 71x - 70 = 0 \).
08
Solve the Quadratic Equation
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 17 \), \( b = -71 \), and \( c = -70 \). Calculate the discriminant and solve for \( x \).
09
Calculate Solutions
The positive integer solutions for \( x \) are found to be \( x = 5 \) and \( x = -\frac{14}{17} \). Since \( x \) must be a positive integer, \( x = 5 \).
10
Determine the Two Integers
The smaller integer is \( x = 5 \). The larger integer is \( x + 2 = 7 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Reciprocals
In algebra, a **reciprocal** is what you get when you switch the numerator and the denominator of a fraction. For example, the reciprocal of a number like 5 is simply \( \frac{1}{5} \). Reciprocals are essential because they show the inverse of operations, particularly in division.
- The reciprocal of \( x \) is \( \frac{1}{x} \).
- For a number to have a reciprocal, it cannot be zero.
Positive Integers
**Positive integers** are numbers greater than zero without any fractions or decimals. They are part of the whole number family. In mathematics, these numbers are often used because they represent countable quantities.
- Examples include 1, 2, 3, and so on.
- In the context of our exercise, we are specifically looking at two such numbers where one is 2 more than the other.
Quadratic Equation
A **quadratic equation** is a polynomial equation of degree 2. It takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable.
- The general shape of its graph is a parabola.
- Solving involves methods like factoring, using the quadratic formula, or completing the square.
Cross-Multiplication
One algebraic technique used in solving equations involving fractions is **cross-multiplication**. This method is useful when we have an equation with two fractions set equal to each other, \( \frac{a}{b} = \frac{c}{d} \). By cross-multiplying, we eliminate the fractions:
- It simplifies the equation to \( a \times d = b \times c \).
- Ensures easier manipulation and solution of the equation.
