Question

If the sum of the reciprocals of two consecutive odd integers is \( \frac{12}{35} \), then the greater of the two integers is [Official GMAT-2018]

Hint:

When solving for consecutive integers, set up an equation based on their properties and use algebraic techniques to solve.

Answer 1

Step 1: Define the integers. 
Let the two consecutive odd integers be \( x \) and \( x + 2 \). 
Step 2: Set up the equation. 
The sum of the reciprocals is given by: \[ \frac{1}{x} + \frac{1}{x+2} = \frac{12}{35} \] Multiply both sides of the equation by \( (x)(x+2) \) to clear the denominators: \[ (2x+2) \times 35 = 12(x^2 + 2x) \] This simplifies to: \[ (x+1) \times 35 = 6x^2 + 12x \] \[ 6x^2 - 23x - 35 = 0 \] Step 3: Solve the quadratic equation. 
Now solve the quadratic equation: \[ 6x^2 - 30x + 7x - 35 = 0 \quad \Rightarrow \quad (6x + 7)(x - 5) = 0 \] Thus, \( x = 5 \). 
Step 4: Conclusion. 
The greater integer is \( 5 + 2 = 7 \).