Question

The average rainfall in the months of January and February is \(6 \mathrm{~cm}\) and in the months of March to June is \(5 \mathrm{~cm}\) and July to October is \(10 \mathrm{~cm}\) and in the November and December, it is \(6 \mathrm{~cm}\). The average rainfall for the whole year is: (a) 7 (b) \(5.5\) (c) \(7.5\) (d) none of these

Short answer

Answer: The average rainfall for the whole year is 7 cm.

Step-by-step solution

Divide the year into different sections

We already have the information for: 1. January and February, average rainfall = \(6\mathrm{~ cm}\) 2. March to June, average rainfall = \(5\mathrm{~ cm}\) 3. July to October, average rainfall = \(10\mathrm{~ cm}\) 4. November and December, average rainfall = \(6\mathrm{~ cm}\)

Calculate the weighted average

First, we need to find the total rainfall for each section. 1. For January and February (2 months): \(2 \times 6 = 12\mathrm{~ cm}\) 2. For March to June (4 months): \(4 \times 5 = 20\mathrm{~ cm}\) 3. For July to October (4 months): \(4 \times 10= 40\mathrm{~ cm}\) 4. For November and December (2 months): \(2 \times 6 = 12\mathrm{~ cm}\) Then, calculate the total rainfall for the year by adding up the total rainfall for each section: \(12 + 20 + 40 + 12 = 84\mathrm{~ cm}\)

Divide the total rainfall by the total number of months (12) to find the average rainfall for the whole year

\(\frac{84\mathrm{~ cm}}{12\mathrm{~ months}} = 7\mathrm{~ cm}\) So the average rainfall for the whole year is 7 cm which corresponds to option (a).

Key concepts

Weighted Average

Understanding the concept of a 'weighted average' is crucial in accurately evaluating data sets where different elements have varying levels of significance. Simply put, a weighted average is used when we want to calculate the average of a set of values, with each value having a different level of importance or 'weight'.

In the context of average rainfall calculation, the 'weight' corresponds to the duration of the time period each rainfall value spans. For instance, the rainfall average for January and February cannot be treated the same as the average for March to June, as the time periods are different.

Here's how to approach the weighted average calculation:

Mathematical Problem-solving

Tackling mathematical problems like calculating averages involves a structured approach. It typically consists of several steps that build upon each other to reach a solution. Here's a breakdown of this approach:

First, you identify the relevant information given in the exercise. For rainfall averages, this includes recognizing the specific months and their corresponding average rainfall values. Next, you break down the year into manageable segments where these values apply, ensuring you're clearly aware of the duration each average stands for.

The problem-solving process then requires you to apply appropriate mathematical operations - in this case, a series of multiplications and additions that will lead to the weighted average. Lastly, to find the average for the year, you distribute this total evenly across all months – that's where division comes into play. This methodical process ensures that all elements of the problem are accounted for and the solution is both comprehensive and accurate.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numerical and mathematical data accurately and efficiently. It's a skill that is vital not only in academic pursuits but in everyday life as well.

Having a strong quantitative aptitude means that you can quickly interpret numerical information, apply mathematical operations like addition, subtraction, multiplication, and division, and solve complex problems with confidence. In the case of calculating average rainfall, it requires an appreciation of the context (driven by the significance of months and seasonal variations) and the capability to translate it into a quantitative framework. The exercise given illustrates this aptitude perfectly by combining the mathematical concept of weighted averages with the practical application of calculating rainfall over different times of the year.

By exercising and improving quantitative aptitude, students become better equipped to deal with a range of mathematical challenges, ensuring they can approach tasks efficiently and with a greater degree of accuracy.