Basic numeracy (numbers and their relations, orders of magnitude, etc.) (Class X level)

What is meant by basic numeracy? How to prepare numbers and their relations, orders of magnitude etc?

The Civil Services Aptitude Test (CSAT) is a crucial part of the UPSC Preliminary Examination. One of the key areas that aspirants often find challenging is the Basic Numeracy section.

This section tests the mathematical skills of candidates at the Class X level. However, candidates may also expect questions of a little bit higher standard than Class X level like Permutations and Combination, Probability etc in connection with problem solving.

In this post, we will delve into the topic of Basic Numeracy and quantitative aptitude, focusing on numbers, their relations, orders of magnitude, and more. We will also provide model questions with detailed solutions to help you prepare effectively.

Understanding Basic Numeracy

At the heart of basic numeracy lies the understanding of numbers and their relationships. This includes:

• Recognizing different types of numbers (integers, fractions, decimals, etc.)
• Understanding the relations between numbers (greater than, less than, equal to)
• Grasping the concept of orders of magnitude and how numbers can be compared based on their size.

Numbers and their relations

Numbers are fundamental entities in mathematics and everyday life. They help us count, measure, and represent quantities. Beyond mere counting, numbers have intricate relationships and properties that form the basis of arithmetic, algebra, and advanced mathematical studies.

Types of Numbers:

1. Natural Numbers: These are the set of positive integers starting from 1. (e.g., 1, 2, 3, 4, …)
2. Whole Numbers: Natural numbers including zero. (e.g., 0, 1, 2, 3, …)
3. Integers: This set includes all positive, negative numbers, and zero. (e.g., …, -3, -2, -1, 0, 1, 2, 3, …)
4. Rational Numbers: Numbers that can be expressed as a fraction of two integers, where the denominator is not zero. (e.g., 1/2, 7/3)
5. Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal representation is non-recurring and non-terminating. (e.g., √2, π)
6. Real Numbers: This set encompasses both rational and irrational numbers.
7. Complex Numbers: Numbers that have both a real and an imaginary part. (e.g., 3 + 2i)

Relations Between Numbers:

1. Equality: Two numbers are said to be equal if they represent the same value. (e.g., 2 + 3 = 5)
2. Inequality: Numbers can be greater than, less than, greater than or equal to, or less than or equal to other numbers. (e.g., 5 > 3, 2 ≤ 4)
3. Divisibility: One number is divisible by another if, upon division, the remainder is zero. (e.g., 8 is divisible by 2)
4. Factors: If ‘a’ is divisible by ‘b’, then ‘b’ is a factor of ‘a’. (e.g., 1, 2, 4 are factors of 8)
5. Multiples: The result of multiplying a number by an integer. (e.g., 10, 20, 30 are multiples of 10)

Properties of Numbers:

1. Commutative Property: The order of numbers does not affect the result (applies to addition and multiplication).
   a + b = b + a
   a × b = b × a
2. Associative Property: Grouping of numbers does not affect the result (applies to addition and multiplication).
   (a + b) + c = a + (b + c)
   (a × b) × c = a × (b × c)
3. Distributive Property: Multiplying a number by a group of numbers added together is the same as doing each multiplication separately.
   a × (b + c) = (a × b) + (a × c)
4. Identity Property: The sum of any number and zero is the number itself (additive identity). The product of any number and one is the number itself (multiplicative identity).

Orders of magnitude

Orders of magnitude provide a broad way to understand and compare the relative sizes or extents of quantities. It’s a concept that helps us grasp the scale of numbers, especially when they range from very large to very small, by representing numbers as powers of ten.

Understanding Orders of Magnitude:
When we say that two quantities differ by an order of magnitude, we essentially mean that one is roughly ten times larger than the other. For instance, 100 is one order of magnitude larger than 10, and 1,000 is two orders of magnitude larger than 10.

Model Questions with Detailed Solutions

1. Question:
If the sum of two consecutive even numbers is 130, what are the numbers?

Solution:
Let the two consecutive even numbers be x and x + 2.
Given, x + (x + 2) = 130
2x + 2 = 130
2x = 128
x = 64
So, the numbers are 64 and 66.

2. Question:
If the diameter of a typical bacterial cell is approximately 1×10−61×10−6 meters and the diameter of a typical human hair is approximately 1×10−41×10−4 meters, by how many orders of magnitude is the human hair thicker?

A) 1
B) 2
C) 10
D) 100

Solution:
Difference in orders of magnitude = $(-4)-(-6)=2$.
Answer: B) 2

3. Question:
If a number is increased by 20% and then decreased by 20%, what is the net percentage change?

Solution:
Let the original number be 100.
After a 20% increase, the number becomes 120.
After a 20% decrease on 120, the decrease is 24 (20% of 120).
So, the new number is 96.
Net change = 100 – 96 = 4% decrease.

4. Question:
How many ways can 5 books be arranged on a shelf?

Solution:
The number of ways to arrange 5 books is given by 5! (5 factorial)
5! = 5 × 4 × 3 × 2 × 1 = 120
So, there are 120 ways to arrange the 5 books.

5. Question:
In how many ways can a committee of 3 members be selected from a group of 7 people?

Solution:
The number of ways to select 3 members from 7 is given by 7C3 (7 choose 3)
7C3 = 7! / (3! × (7-3)!)
= 7! / (3! × 4!)
= (7 × 6 × 5) / (3 × 2 × 1)
= 35
So, there are 35 ways to select the committee.

6. Question:
A bag contains 4 red balls and 6 blue balls. What is the probability of drawing a red ball?

Solution:
Total balls = 4 + 6 = 10
Probability of drawing a red ball = Number of red balls / Total balls
= 4/10
= 2/5
So, the probability is 2/5.

7. Question:
A die is rolled. What is the probability of getting an even number?

Solution:
There are 3 even numbers on a die (2, 4, 6).
Total possible outcomes = 6 (since a die has 6 faces)
Probability of getting an even number = Number of even numbers / Total possible outcomes
= 3/6
= 1/2
So, the probability is 1/2.

8. Question:
From a standard deck of 52 cards, one card is drawn. What is the probability that the card is a queen?

Solution:
There are 4 queens in a standard deck of 52 cards.
Probability of drawing a queen = Number of queens / Total cards
= 4/52
= 1/13
So, the probability is 1/13.

How to study Basic Numeracy for CSAT?

Students may note that this article on Basic numeracy is just an overview of the topic. There is a lot more to learn about basic numeracy and quantitative aptitude in the CSAT paper.

We recommend the below sources to learn the subject.

1. Join the ClearIAS CSAT Course.
2. Join ClearIAS Prelims Test Series.
3. Join ClearIAS Prelims cum Mains Course.
4. Go through ClearIAS YouTube Classes on CSAT.
5. Read books on CSAT.

Tips for Tackling Basic Numeracy Questions in CSAT

1. Practice Regularly: The more you practice, the more comfortable you will become with different types of questions.
2. Understand the Concepts: Instead of rote learning, focus on understanding the underlying concepts.
3. Time Management: Basic numeracy questions can be time-consuming. Practice solving questions within a set timeframe to improve your speed.

Also read: CSAT Course: UPSC Prelims Paper 2 Program

Conclusion

Basic numeracy is an essential part of the CSAT paper, and with regular practice and a clear understanding of concepts, you can ace this section.

We hope the model questions and solutions provided above will aid in your preparation.

Best of luck with your CSAT preparation!