Today, I want to talk about a type of GMAT/GRE/CAT quant question that might look intimidating but becomes quite manageable once you grasp a core concept: Maxima-Minima. This isn't a standalone chapter, but it's a way of thinking that applies to many GMAT/GRE/CAT questions. It's all about thinking in terms of extremes.
Let's dive into a problem that perfectly illustrates this principle.
If S is the sum of the reciprocals of the 10 consecutive integers from 21 to 30, then S is between which of the following two fractions?
- 1/3 and 1/2
- 1/4 and 1/3
- 1/5 and 1/4
- 1/6 and 1/5
If you try to calculate `S = 1/21 + 1/22 + ... + 1/30` precisely, you'll be there all day! This is a classic GMAT/GRE/CAT move – giving you something that seems computationally intensive but actually tests your conceptual understanding and approximation skills.
My Approach: Thinking in Extremes
The key here, as I always tell my students, is to find the range for 'S' by calculating its maximum and minimum possible values. This simplifies the entire problem.
1. Finding the Upper Bound (Maxima)
To make the sum as large as possible, we need to make each fraction as large as possible. This means using the smallest possible denominator (21 or approx 20).
2. Finding the Lower Bound (Minima)
To make the sum as small as possible, we need to make each fraction as small as possible. This means using the largest possible denominator (30).
The Solution
By applying this "maxima-minima" approach, we've quickly determined that the actual sum 'S' must be greater than 1/3 and less than 1/2.
This question, which initially might seem like a nightmare of calculations, transforms into a simple exercise in estimation and understanding how fractions behave when you think about extremes. This is the kind of strategic thinking that helps you ace the GMAT Quant section. Keep practicing this mindset, and you'll find even the toughest problems yield to a systematic approach!