Concepts Study

Ratio & Proportion – Concepts, Tricks, Conversions & Exam-Level Applications

2025-12-06T22:58:00.000+05:30

Ratio & Proportion is one of the most important topics for competitive exams. It forms the base for questions of partnership, mixtures, profit & loss, time & work, time–speed–distance, and more.
In this chapter, we will understand basics, shortcut tricks, conversions, and high-level exam questions in the simplest way.

1. Ratio –

What is a Ratio?

A ratio compares two quantities of the same kind (same units).
Example:

5 kg : 2 kg
3 boys : 4 boys
20 minutes : 10 minutes

Ratios show how many times one quantity contains another.

Forms of Ratios

A ratio can be written in three forms:

a : b
a / b
a to b

All three represent the same comparison.

Key Characteristics

(a) Ratios have no units

Because units cancel out.
Example: 10 kg : 5 kg → 10/5 = 2 → ratio = 2 : 1

(b) Ratios can be simplified (like fractions)

You can divide both sides by the GCD or HCF.

Example:
36 : 54
GCD = 18 → 36/18 : 54/18 = 2 : 3

(c) Equivalent Ratios

If you multiply or divide both sides by the same number, the ratio remains the same.

Example:
2 : 5
Multiply both by 3 → 6 : 15
Divide both by 2 → 1 : 2.5

Ratio in Real Life

Speed comparison
Age comparison
Mixture problems (milk : water)
Financial investment (A invests : B invests)
Recipe proportions

Internal & External Ratios

Internal Ratio:

Dividing a quantity internally into parts.
Example: ₹300 divided in ratio 1 : 2 →
Total parts = 3
→ 300 × 1/3 = 100
→ 300 × 2/3 = 200

External Ratio:

Used in geometry (dividing line segments externally).

Conversion Between Ratio & Fraction

Ratio → Fraction

a : b = a/b
Example: 3 : 4 = 3/4

Fraction → Ratio

a/b → a : b
Example: 0.75 = 75/100 → divide by 25 → 3 : 4

Shortcut Ratio Tricks

Trick 1: From Percentage to Ratio

45% : 55%
Remove % → 45 : 55
Divide by 5 → 9 : 11

Trick 2: Fraction to Ratio Quickly

3/8
Ratio = 3 : 8
If two fractions:
(3/4) : (5/6)
LCM of denominators = 12
→ 3/4 = 9/12
→ 5/6 = 10/12
Ratio = 9 : 10

Trick 3: Ratio Scaling (Very Useful in Exams)

Given A : B = 3 : 7
B : C = 14 : 5
Make B equal:
7 × 2 = 14
So A : B = 6 : 14
Hence A : B : C = 6 : 14 : 5

✔ Convert Ratio → Fraction

Ratio a : b
Fraction of a = a / (a + b)

Example:
Ratio = 2 : 3
Fraction of 2 = 2/5
Fraction of 3 = 3/5

✔ Convert Fraction → Ratio

Example:
Fraction = 3/7
Ratio = 3 : (7–3) = 3 : 4

✔ Convert Ratio → Percentage

a : b → a/(a+b) × 100

2. Proportion – Complete, Deep Explanation

What is Proportion?

A proportion states that two ratios are equal.
Example:
2/3 = 4/6 → This is a proportion
2 : 3 :: 4 : 6

Types of Proportion

(A) Direct Proportion

When one quantity increases, the other increases.
Example:
More hours worked → More salary
Mathematically:
a/b = c/d

(B) Inverse Proportion

One quantity increases while the other decreases.
Example:
Speed ↑ → Time ↓
Workers ↓ → Days ↑
Formula:
a × b = c × d

(C) Continued Proportion

If a : b = b : c, then a, b, c are in continued proportion.

(D) Compound Proportion

Combination of multiple proportions.
Example:
If 12 men do a job in 15 days, how many days will 20 men take?

Rules in Proportion

1. Product Rule

In a proportion:
a : b = c : d
So,
a × d = b × c
This is called the Cross-multiplication rule.

2. Mean & Extreme Terms

a : b = c : d
a, d → Extreme terms
b, c → Mean terms
Product of extremes = Product of means

3. Fourth Proportion

a : b = c : x
x = (b × c) / a

4. Third Proportion

a : b = b : x
x = (b²)/a

5. Mean Proportion (Geometric Mean)

a : x = x : b
x² = ab → x = √ab

⭐ Exam-Level Applications (Important for SSC, Banking, CET, Railways, etc.)

📌 1. Ratio-based distribution

Example:
₹1200 is divided between A and B in ratio 3 : 5.

Total parts = 3+5 = 8
A = 1200 × 3/8 = ₹450
B = 1200 × 5/8 = ₹750

📌 2. Changing Ratio Questions

Boys : Girls = 4 : 5
If boys increase by 20, new ratio becomes 6 : 5.
Find girls.

Let original boys = 4k
Original girls = 5k

4k + 20 = 6k
→ 20 = 2k
→ k = 10
→ Girls = 50

📌 3. Exam Trick – Ratios with Difference

Ratio = 5 : 8
Difference = 12

Unit difference = 8 – 5 = 3
Value of 1 unit = 12/3 = 4
Values:
A = 5×4 = 20
B = 8×4 = 32

📌 4. Partnership (Direct Application of Ratios)

A invests ₹4000 for 12 months
B invests ₹6000 for 8 months
Profit ratio = (4000×12) : (6000×8)
= 48000 : 48000
= 1 : 1

📌 5. Mixture & Alligation (Ratio Shortcut)

Ratio = (High value – Mean) : (Mean – Low value)

Example:
Mix milk ₹60/litre with water ₹0/litre to get mixture of ₹40/litre.

Ratio = (60–40) : (40–0) = 20 : 40 = 1 : 2

Deeper Algebra: Solving Proportional Equations

Example:
If $x : y = 4 : 7$ and $x + y = 33$ , find $x, y$ .

Let $x = 4 k$ , $y = 7 k$ . Sum: $4 k + 7 k = 11 k = 33 \Rightarrow k = 3$ . So $x = 12$ , $y = 21$ .

Why this works: Any ratio $a : b$ can be represented as $a \cdot k : b \cdot k$ .

🔷 Advanced Tips & Pitfalls

Always simplify early. Cancelling common factors reduces arithmetic errors.
Watch units. If ratios compare lengths and times, ensure consistent units.
When multiple ratios are chained, find a common term and scale appropriately.
For inverse proportion problems, convert them to direct proportionality by reciprocals.

🔷 Practice Problems (with answers)

Solve these — do them without a calculator to build speed.

Divide ₹600 in the ratio $3 : 7$ .
If $a : b = 5 : 8$ and $b : c = 4 : 9$ , find $a : b : c$ .
If $\frac{x}{12} = \frac{5}{18}$ , find $x$ .
Three people invest ₹2000, ₹3000, ₹5000. Profit ₹6000 distributed in ratio of investments. Find each share.
If $x$ is inversely proportional to $y$ and $x = 6$ when $y = 4$ , find $x$ when $y = 12$ .

Answers (work shown briefly)

Sum = $3 + 7 = 10$ . Parts: $600 \times \frac{3}{10} = 180$ , $600 \times \frac{7}{10} = 420$ .
Make B common: $a : b = 5 : 8$ → multiply by 4 → 20:32. $b : c = 4 : 9$ → multiply by 8 → 32:72. So $a : b : c = 20 : 32 : 72$ . Simplify dividing by 4 → $5 : 8 : 18$ .
$x / 12 = 5 / 18 \Rightarrow x = 12 \times \frac{5}{18} = 12 \times \frac{5}{18} = \frac{60}{18} = \frac{10}{3} \approx 3.333...$ . (Or $x = \frac{10}{3}$ .)
Investment ratio = $2000 : 3000 : 5000 = 2 : 3 : 5$ . Total parts = 10. Profit per part = $6000 / 10 = 600$ . Shares: 2×600=1200, 3×600=1800, 5×600=3000.
$x \propto 1 / y$ . Constant $k = x \times y = 6 \times 4 = 24$ . When $y = 12$ , $x = 24 / 12 = 2$ .

⭐ Quick Revision Table

Concept	Meaning	Shortcut
Ratio	Comparison	Simplify with HCF
Proportion	Equality of ratios	Cross multiplication
Ratio → Fraction	a/(a+b)	Direct
Ratio combine	A:B & B:C	Equalize middle term
Exam trick	Ratios with difference	Find 1 unit value

⭐ Key Takeaways

Ratio & Proportion is the foundation for many other chapters.
Always simplify ratios using HCF.
Use cross-multiplication to confirm proportion.
For any distribution, convert ratio to “total parts → unit value”.
Learn to combine ratios – this is the most common exam skill.
Practice speed techniques daily for perfection.

⏭ Next Chapter Preview: – Percentage

In the next chapter we will study:

Basics of percentage
Increase/Decrease shortcuts
Successive percentage tricks
Exam-level percentage questions
Smart shortcuts used in banking & CET exams

Link to:

Fractions & Decimals

2025-12-01T09:00:00.001+05:30

Concepts, Conversions, Tricks & Examples

Welcome back to Concept Studies — Because Concepts Matter!
If you’ve followed the previous chapters on Number System and HCF–LCM, then congratulations! You’re now ready for one of the most scoring and important topics for SSC, Banking, Railways, MPSC, UPSC Prelims and all competitive exams:

⭐ Fractions & Decimals

As a student, I used to get confused between conversions, comparisons and operations. But trust me — once you understand the concept core, everything becomes unbelievably easy.

🔹 What is a Fraction?

A fraction represents a part of a whole and has two parts:

             Numerator (Top)
Fraction =  ----------------
             Denominator (Bottom)

Example: 3/4 means three parts out of four equal parts.

Types of Fractions

Proper Fraction: Numerator < Denominator → (3/5)
Improper Fraction: Numerator > Denominator → (9/4)
Mixed Fraction: 2½, 5⅓
Like Fractions: Same denominator → (5/9, 2/9)
Unlike Fractions: Different denominators → (3/4, 5/6)

🔹 What is a Decimal?

A number with a decimal point, representing tenths, hundredths, thousandths, etc.

Example: 0.5 = 5/10 = 1/2

🔹 Conversion: Fractions ⇆ Decimals (Super Easy!)

Fraction → Decimal

Divide numerator by denominator.

3/4 = 3 ÷ 4 = 0.75
1/8 = 0.125

Decimal → Fraction

Write the decimal over its place value.

0.6 = 6/10 = 3/5
0.125 = 125/1000 = 1/8

🔹 Shortcut Box — Must Remember Fractions

1/2 = 0.5
1/3 = 0.333…
1/4 = 0.25
1/5 = 0.2
1/8 = 0.125
1/16 = 0.0625

🔹 Comparing Fractions – Fastest Exam Trick

Cross Multiplication Method

For fractions a/b and c/d:

If a × d > b × c → a/b is greater

Which is greater: 3/7 or 4/9?  
3 × 9 = 27  
4 × 7 = 28  
28 > 27 → 4/9 is greater

🔹 Converting Mixed Fractions

Mixed → Improper Whole × Denominator + Numerator

Improper → Mixed Divide numerator by denominator.

🔹 Operations on Fractions

✔ Addition / Subtraction (Unlike Fractions)

LCM of denominators → Convert → Solve

3/4 + 5/6  
LCM = 12  
(9/12 + 10/12) = 19/12

✔ Multiplication

2/3 × 4/5 = 8/15

✔ Division

Multiply by reciprocal.

3/5 ÷ 7/9 = 3/5 × 9/7 = 27/35

🔹 Decimal Operations

✔ Addition/Subtraction

Align decimals vertically.

✔ Multiplication

Multiply normally → Count decimal places.

✔ Division

Shift the decimal to make divisor a whole number.

🔹 Shortcut Box – Decimal % to Fraction

25% = 1/4
12.5% = 1/8
37.5% = 3/8
62.5% = 5/8
75% = 3/4

🔹 Real Exam-Type Questions

Convert 7/12 into decimal.
Which is greater: 5/8 or 7/12?
Convert 0.375 into fraction.
Add: 3/5 + 7/10
Divide: 4/9 ÷ 8/27

🔹 Coming Next (Lesson 6)

In the next post, we will cover:

Ratio & Proportion
Shortcut ratio tricks
Conversion between ratio & fraction
Exam-level applications

Stay tuned for the next post — and always remember, Because Concepts Matter!

LCM & HCF – Concepts, Prime Factorization & Shortcuts

2025-11-27T09:36:00.003+05:30

Welcome back to Concept Studies — Because Concepts Matter! As an aspiring competitive exam student, this is one chapter I can confidently say will appear in almost every exam: Factors, Multiples, HCF & LCM. And trust me — if you understand the prime factorization base, this chapter becomes the easiest scoring area in Quantitative Aptitude!

🔹 Why HCF & LCM Are Important?

If you're preparing for SSC, Banking, Railways, MPSC, UPSC Prelims, or any State-level exam, you have already seen these questions:

Find the LCM of 12, 18, 24
Three bells ring together… When will they ring together again?
Find the HCF of 36 and 84 using prime factorization
A man takes steps of 45 cm and 75 cm. What is the minimum distance both will cover together?

All of these can be solved within 10–20 seconds once your concepts are clear.

🔹 Understanding Factors & Multiples

Factor

A factor is a number that divides another number exactly.

Example: Factors of 12 → 1, 2, 3, 4, 6, 12

Multiple

A multiple is the result of multiplying a number.

Example: Multiples of 4 → 4, 8, 12, 16, 20...

🔹 HCF (Highest Common Factor)

The greatest number that divides all the given numbers exactly.

🔹 LCM (Least Common Multiple)

The smallest number that is divisible by all the given numbers.

🔹 Prime Factorization Method (Universal Method)

This is the most accurate and exam-proof method. All we do is break the numbers into prime factors.

Example:
12 = 2 × 2 × 3  
18 = 2 × 3 × 3

How to find HCF?

Take the common prime factors with minimum powers.

Numbers: 12, 18  
Common prime factors: 2 and 3  
Powers: 2¹ and 3¹  
HCF = 2 × 3 = 6

How to find LCM?

Take all prime factors with maximum powers.

2² × 3² = 4 × 9 = 36  
LCM = 36

🔹 Division Method (Fastest for Two Numbers)

Example:

🔹 Shortcut #1 – Relationship Between HCF & LCM

Product of numbers = HCF × LCM

This shortcut is extremely useful in many word problems!

Example:

If HCF = 8 and LCM = 144, product = 8 × 144 = 1152  
If one number = 36  
Other number = 1152 / 36 = 32

🔹 Shortcut #2 – LCM of Fractions

LCM of Fractions = LCM of Numerators / HCF of Denominators

🔹 Shortcut #3 – HCF of Fractions

HCF of Fractions = HCF of Numerators / LCM of Denominators

🔹 Shortcut #4 – LCM of Numbers Close to Each Other

If numbers are consecutive:

LCM of consecutive numbers = Product of numbers

Example: 7, 8, 9 → LCM = 7 × 8 × 9

🔹 Shortcut #5 – HCF of Large Numbers

Subtract the smaller from the larger until numbers become small.

🔹 Common Word Problems

🔸 Type 1: Repetition / Cyclic Events

Example: Three bells ring at 6 sec, 8 sec, 12 sec. When will they ring together?

LCM = 24 sec (Answer)

🔸 Type 2: Minimum Distance / Maximum Length

Example: Find the minimum length of rope that can cut lengths of 20cm, 30cm, 40cm.

HCF = 10cm (Answer)

🔹 Practice Examples (Solve Yourself)

Find the HCF and LCM of 15, 25, 35
Find the LCM of 3/4, 5/6, 7/8
When will 10 min, 15 min and 20 min timers ring together?
Two numbers’ LCM = 180 and HCF = 6. One number is 30. Find the other.
Find the greatest number that divides 75, 90, 135 exactly.

🔹 Coming Next in Lesson 5

Fractions & Decimals
Conversions
Simplification tricks
Decimal-based shortcuts

Stay tuned — and remember, Concepts Matter!

Operations on Numbers

2025-11-23T15:23:00.001+05:30

(The Real Power of Number System)

Because Concepts Matter! | Concept Studies

Hi Champions! 👋 Today’s post is one of my favourites because this is the point where numbers start behaving like magic. If you understand how operations work on different types of numbers, you will solve 40% of Quantitative Aptitude questions instantly.

Let’s begin our journey into the real machinery of Maths — the “Operations on Numbers”. I’m preparing just like you, and sharing everything that has helped me speed up calculations, save time in exams, and avoid silly mistakes.

🔢 1. Basic Operations on Numbers

These four operations are the backbone of all quantitative chapters:

Addition (+)
Subtraction (–)
Multiplication (×)
Division (÷)

Understanding their behaviour with integers, decimals, fractions and negative numbers is essential.

➤ Addition Rules

Adding two positives → positive
Adding two negatives → negative
Adding a positive & negative → subtract and keep bigger number’s sign

Example: 25 + (–18) = 25 – 18 = 7

➤ Subtraction Rules

Subtracting a number is the same as adding its negative.

Example: 12 – (–8) = 12 + 8 = 20

➤ Multiplication Rules

+ × + = +
- × - = +
+ × - = -

Example: (–6) × (–3) = +18

➤ Division Rules

Same sign → positive result, different sign → negative result.

Example: (–42) ÷ 6 = –7

🧮 2. BODMAS – The Heart of Calculations

Without BODMAS, many aptitude questions become confusing. Remember the sequence:

BODMAS = Brackets → Orders → Division → Multiplication → Addition → Subtraction

Example:

Solve: 25 – 5 × 3 + 4
Step 1: 5 × 3 = 15
Step 2: 25 – 15 + 4 = 10 + 4 = 14

➗ 3. Division Algorithm & Remainders

Every number follows this relationship:

Dividend = Divisor × Quotient + Remainder

This forms the basis of remainders, modular arithmetic, cyclicity and unit digit tricks.

Example:

Divide 29 by 4.
4 × 7 = 28 → remainder = 1
Hence, 29 = 4 × 7 + 1

🧩 4. Modulus (|a|) – Absolute Value

Absolute value indicates distance from zero.

|5| = 5
|–5| = 5
|–7 + 2| = |–5| = 5

🔁 5. Cyclicity (Unit Digit Patterns)

Numbers follow patterns when raised to powers. This is the secret weapon for solving power-based questions in seconds.

Example: Unit Digit of 7ⁿ

7¹ → 7
7² → 49 → 9
7³ → 343 → 3
7⁴ → 2401 → 1

The cycle is 7, 9, 3, 1 (repeats every 4 steps).

➕ 6. Digital Sum (Casting Out 9’s)

Digital sum is used for:

Quick checking
Multiplication shortcuts
Error elimination in options

How to Find Digital Sum:

Add digits until a single digit remains.

Example: 583 → 5 + 8 + 3 = 16 → 1 + 6 = 7

🧠 7. Special Properties of Numbers

➤ Even & Odd Rules

Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Even × Even = Even
Even × Odd = Even
Odd × Odd = Odd

➤ Factors-based Behaviour

Multiplying increases factors.
Adding usually does not increase factors.

📘 8. Mixed Operation Examples

Example 1

Evaluate: 18 ÷ 3 + 12 × 2 – 5
Step 1: 18 ÷ 3 = 6
Step 2: 12 × 2 = 24
Step 3: 6 + 24 – 5 = 25

Example 2

Simplify: 45 – (12 ÷ 4) × 3
Step 1: 12 ÷ 4 = 3
Step 2: 3 × 3 = 9
Step 3: 45 – 9 = 36

📝 9. Practice Questions for You

Find the value: 32 – 18 ÷ 3 + 4 × 2
Unit digit of 9⁵⁷
Find digital sum of 8723
Solve: 45 – 6 × 5 + 12 ÷ 3
If 87 = 5 × Q + R, find remainder R.

Answers will be provided in the next follow-up post!

🌟 10. What’s Next?

In the next post, we will cover:

Advanced Order of Numbers
Comparing Fractions
Decimal Operations
Converting between forms

Stay tuned! We are building a rock-solid foundation — concept by concept, lesson by lesson.

Because Concepts Matter!

Number System Basics for Competitive Exams

2025-11-18T22:55:00.001+05:30

A Foundation Lesson for Quantitative Aptitude Aspirants

Hello future achiever! 👋 Welcome to Concept Studies, a place where we learn with clarity, confidence and real-life understanding. Today you are stepping into your very first chapter of Quantitative Aptitude — the Number System.

I still remember the first day I started preparing for competitive exams — banking, SSC, railways, insurance — and almost every topper I met told me the same thing:

“If your Number System is strong, 40% of Quant becomes simple.”

So today, let’s learn this chapter like a real aspirant — not just reading but understanding why every concept matters. Because concepts truly matter! 💙

🔢 1. What Are Numbers?

Numbers are not just digits written on paper. Numbers are the language of mathematics. Every calculation you do — profit, loss, interest, speed, average, Data Interpretation — everything starts here.

Let’s start with the simplest classification:

📌 Types of Numbers

Natural Numbers — 1, 2, 3, 4 …
Whole Numbers — 0, 1, 2, 3 …
Integers — … -3, -2, -1, 0, 1, 2, 3 …
Rational Numbers — fractions p/q
Irrational Numbers — √2, π (non-terminating & non-repeating)
Real Numbers — All rational numbers + irrationals numbers combined

Shortcut Trick: Every fraction p/q (q ≠ 0) is ALWAYS a rational number — even if its decimal looks long and confusing.

🧮 2. Understanding the Number Line (Visual Thinking)

Imagine a straight endless highway. That’s your number line. Left side is negative (cold side ❄), right side is positive (warm side 🔥), and 0 is the center.

This helps you understand ordering, comparison and future topics like inequalities.

✨ 3. Properties of Numbers (Small but Important!)

These apply everywhere — algebra, simplification, HCF-LCM, equations.

✔ Closure Property

If you add or multiply two whole numbers, the result is still a whole number.

✔ Commutative Property

a + b = b + a and a × b = b × a

✔ Associative Property

(a + b) + c = a + (b + c)

💥 4. Divisibility Rules

These rules are LIFESAVERS in exams. They reduce 30 seconds questions into 3 seconds.

📘 Divisibility Rules (Complete Updated List)

Number	Rule
2	Last digit even (0,2,4,6,8)
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Last digit is 0 or 5
6	Number must be divisible by 2 and 3
7	Double last digit & subtract (special rule)
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
10	Last digit is 0
11	(Sum of odd-position digits – even-position digits) divisible by 11

Super Shortcut: If a number is divisible by both 3 and 4 → it is automatically divisible by 12.

🌟 5. Prime & Composite Numbers (Simplified)

A prime number has only two factors — 1 and itself. Composite numbers have more than two factors.

Shortcut: You only need to check divisibility up to √n to test if a number is prime.

🧩 6. Remainders & Modular Thinking

This topic will be fully covered in future chapters, but here’s the basic idea:

Dividend = Divisor × Quotient + Remainder

If you divide 27 by 5 → remainder is 2

📝 7. Practice Questions (Solve & Learn!)

Classify: √7, -12, 0, 4.25, 22/7
Check whether 836 is divisible by 4, 6, 8, and 11.
Find the remainder when 523 is divided by 7.
Identify prime numbers from: 29, 57, 91, 97

🎯 8. What’s Coming Next?

This is just the first brick in your Quant foundation. Next post will be:

👉 Lesson 2: Divisibility Rules — With Advanced Tricks!

Make sure to follow the blog so you never miss the upcoming posts!

Remember: Concepts Matter. You Matter. Keep learning! 🚀

Quantitative Aptitude — The Foundation of All Competitive Exams !

2025-11-13T21:00:00.002+05:30

Quantitative Aptitude — The Foundation of All Competitive Exams!

By Concept Studies — Because Concepts Matter!

🌟 Hey Everyone!

Welcome to my first blog on Concept Studies! I’m an aspiring student just like many of you — preparing for competitive exams and learning something new every day. When I started my journey, Quantitative Aptitude used to scare me. Numbers, formulas, ratios — they all looked complicated. But slowly, I realized something powerful:

Once you understand the concept behind a problem, it becomes easy — even fun!

So, I decided to start this series where I share everything I learn — from core concepts to short tricks, and from basic to advanced levels. Today’s post is the foundation, because Quantitative Aptitude is not just a section — it’s the base of logical thinking for everything else, including Reasoning and English.

🧮 What Exactly Is Quantitative Aptitude?

Quantitative Aptitude (or “Quant”) checks how well you can work with numbers, analyze situations, and solve problems logically. It’s not about being a mathematician — it’s about being smart, quick, and accurate.

It mainly includes:

Number System
Simplification & Approximation
Percentage & Ratio-Proportion
Average, Profit & Loss, Simple & Compound Interest
Time, Speed & Distance, Time & Work
Algebra & Mensuration
Data Interpretation

If you master these, Quant becomes your strongest weapon! 💪

🔢 1. Number System — The Root of All Numbers

Every big question starts with small numbers. Understanding divisibility, factors, LCM, and HCF forms your base.

Tip: Every number can be expressed as a product of its prime factors.

Example:
LCM of 12 and 18
→ 12 = 2² × 3¹
→ 18 = 2¹ × 3²
LCM = 2² × 3² = 36
HCF = 2¹ × 3¹ = 6
Shortcut: LCM × HCF = Product of the Numbers

📉 2. Simplification & Approximation

This is your speed booster section! Questions test how quickly you can handle BODMAS, fractions, and decimals.

Example:
(40% of 250) + (25% of 120)
= (1/2 × 100) + (1/4 × 120)
= 100 + 30 = 130

Shortcut Tip: Replace “%” with “÷100” and cancel zeros smartly!

🎯 3. Percentage — The Core Concept of Arithmetic

Percentages connect everything — Profit & Loss, Discounts, Interest, Data, etc.

Concept: “Percent” means per hundred. So, 20% = 1/5 and 50% = 1/2.

%	Fraction	Trick
10%	1/10	Easy 1 zero cut
20%	1/5	Divide by 5
25%	1/4	Quarter
50%	1/2	Half
75%	3/4	¾ rule
33.33%	1/3	One-third

Example: Find 25% of 640 → 640 × 1/4 = 160

💸 4. Profit, Loss & Discount

Directly connected with percentages and ratio.

Profit% = (Profit / Cost Price) × 100
Loss% = (Loss / Cost Price) × 100

Example: A pen costs ₹100 and is sold at ₹120.
Profit = 20, Profit% = (20/100)×100 = 20%

Shortcut: If Profit% = Loss%, then Overall Loss% = (Common % × Common %)/100

Example: Sold at 20% profit once and 20% loss again → Loss% = (20×20)/100 = 4%

⚙️ 5. Ratio & Proportion

Ratios are comparisons — they make big numbers simple.

Example: If A:B = 2:3 and B:C = 4:5,
Then A:B:C = (2×4):(3×4):(3×5) = 8:12:15

Shortcut: Always multiply the middle term to link both ratios.

🕐 6. Time, Speed & Distance

The golden formula:

Speed = Distance / Time

Example: A car covers 180 km in 3 hours.
Speed = 180 ÷ 3 = 60 km/hr

Shortcut: When the same distance is covered at two speeds,
Average Speed = (2xy)/(x + y)

Example: Speed = 60 km/hr & 90 km/hr → Avg Speed = (2×60×90)/(60+90) = 72 km/hr

💡 7. Time & Work

Work problems are just speed-distance problems in disguise.

Example:
A can do a job in 10 days → 1-day work = 1/10
B can do in 15 days → 1-day work = 1/15
Together = (1/10 + 1/15) = (5/30) = 1/6
So, they complete in 6 days together.

Shortcut: Use the LCM of days to find total work easily.

💰 8. Simple & Compound Interest

Simple Interest (SI) = (P × R × T)/100
Compound Interest (CI) = P(1 + R/100)^T

Example:
Find CI on ₹10,000 at 10% for 2 years:
A = 10000(1+10/100)² = 10000×1.21 = 12,100
CI = ₹2,100

Shortcut: Use “Interest-on-Interest” Method:
1st Year = ₹1000
2nd Year = ₹1000 + ₹100 (on interest) = ₹2100

📊 9. Data Interpretation (DI)

All those charts and graphs are just arithmetic in disguise! You’ll apply everything — percentage, ratio, average — here.

Example: If total expenditure = ₹50,000 and Education = 30%,
Then expense = 50,000 × 3/10 = ₹15,000

✍️ Practice Corner — Try Yourself!

Find the HCF and LCM of 24 and 36.
A product costs ₹400 and is sold at 20% profit. Find the selling price.
A train covers 240 km at 60 km/hr. How long does it take?
A can do a job in 8 days, B in 12 days. How long together?
Find CI on ₹12,000 at 8% p.a. for 2 years.
Out of 600 students, 40% are girls. How many boys are there?

Post your answers in comments — solutions will be shared in the next update!

🧭 Wrapping It Up

That’s all for today’s foundation post on Quantitative Aptitude. I know it’s a lot to take in — but remember:

Concepts are the seeds, and practice is the sunlight. The more you nurture, the stronger you grow.

As you move ahead, everything in Quant will feel connected — Percentages link with Profit & Loss, Ratio with Mixtures, Speed with Time & Work, and everything comes together in Data Interpretation.

I’ll be covering each topic separately in upcoming posts with:

Visual concept images
Shortcut cards
Formula charts
Trick-based problem solving

🧩 Coming Up Next on Concept Studies:

Part 2 – Logical Reasoning Simplified: Master puzzles, coding-decoding, and seating arrangements logically.
Part 3 – English for Exams: Grammar hacks, vocab boosters, and comprehension tricks.
Part 4 – Quant Short Tricks Series: Mind formulas to solve within seconds!

Stay tuned, practice daily, and remember — Because Concepts Matter! 🌱

Word Problems Chapter-1 :- Percentage | Introduction to Percentage

2020-05-22T20:55:00.000+05:30

Percentage (%)

Introduction:-

In mathematics, a percentage is a number or simply a ratio expressed as a fraction of 100.

"Percent" comes from the Latin "Per Centum". The Latin word Centum means 100 When we say "Percent" we are actually saying "per 100 or out of 100".

Percentage is usually denoted by using ' % 'sign. % sign is an easy way to write any number divided by 100.

A percentage is a pure number. (Dimensionless Number)

Ex. :- Instead of saying " I got 70 marks out of 100 in English, one can say " I got 70 % marks in English.

" The percent value is computed by multiplying the numeric value of the ratio by 100."

NOTE :- 1) X % of Y = Y % of X.

Ex. 10 % of 50 is 50 % of 10

Sol. ⇒ 10 % of 50 = 5 as well as 50 % of 10 is 5.

Representation of Percentage :-

Percent can be represented or expressed in the form of decimals and fractions.

Calculations :-

1. The percent value is calculated by multiplying the numeric value ratio by 100.

For E.g. 450 students of class IX & X forms what percentage of total number of students in school which is 1800.

Sol. ⇒ 450/1800 = 1/4

⇒ 1/4 × 100 = 25 %

2. Percentage of percentage is calculated by converting both percentage to fractions of 100

For E.g. 50 % of 60 % = ?
Sol. ⇒ 50/100 × 60/100 = 30/100 = 30 %

3-Ways to ask a problem:-

1. What is 30 % of 200 ?

2. 60 is what percent of 200 ?

3. 60 is 30 % of what ?.

In such questions students get confused as what should exactly be done ?

Hence, to tackle this we can use a simple formula for calculation by which confusion will get removed and calculation become easier.

Formula :-

Now let's try to answer all the 3 questions.

1) What is 30 % of 200 ?

Sol. ⇒ 30/100 × 200 = 30 × 2 = 60

2. 60 is what percent of 200 ?

Sol. ⇒ Here, we will put 60 in place of ' is ' and 200 in place of ' of '

hence we get, 60/200 × 100 = 60/2 = 30 %

3. 60 is 30 % of what ?.

Sol. ⇒ Here, we will put 60 in place of ' is ' and an unknown variable 'x' in place of ' of '

hence we get, 60/x × 100 = 30 %

6000/x = 30 %

x = 6000/30 = 200

Conversions :-

1) Percent to Decimal :-

⇒ Divide by 100 & remove % sign

(or you can simply shift the decimal point to 2 places to the left)

2) Decimal to Percent :-

⇒ Multiply by 100 & add % sign

(or you can simply shift decimal point to 2 places to the right)

3) Fraction to Decimal :-

⇒ Simply divide numerator by denominator.

4) Decimal to Fraction :-

⇒ Remove the decimal point by dividing it by power of 10 equals to decimal place and then simplify the fraction.

5) Fraction to Percentage :-

⇒ Simply multiply the fraction by 100.

6) Percentage to Fraction :-

⇒ Simply remove % sign (i.e. divide by 100) and simplify..

Percentage increase & decrease / Percent Change :-

When value of a quantity is increases or decreases, we compare the increased or decreased value to the initial (original) value of the quantity.

The difference thus obtained is further divided by the initial (original) value and the fraction thus obtained is multiplied by 100 to get value of Percentage increase/decrease in the value.

Percentage Difference :-

" Percentage difference is calculated as the difference between two values divided by the average of two values. "

Percent Error :-

" Percent error is the difference between False/ approximate / Estimated value and the True/Actual value in comparison to the actual value and is expressed as a percentage. "

Ex. Instead of writing his age 16, Rohit wrote it as 14 in application form. Find percentage Error.

Sol.⇒ | 16 - 14 |

___________________ × 100 = 1/8 × 100 = 25 %

16 %

Percentage Points :-

News :- SBI increases Home Loan interest rates from 10 % to 12 %.

Whether it is a 2 % rise or 20 % rise ?

⇒ Let's calculate,

12 % - 10 %

____________________ × 100 = 20 % rise.

10 %

" We can also say that it was a rise of 2 percentage points."

When we subtract one percentage from another you get "Percentage Point"

1 Percentage Point = 1 % as a simple difference.

NOTE :- It does not mean a relative change it is difference between two percentage.

In the next article we will discuss word problems based on Percetage.

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Word Problems Chapter-1 :- Percentage | Basics Of Percentage | Important Multiples of Fraction Values

2020-05-19T15:29:00.001+05:30

In earlier post, we have studied about important fraction value and their conversion in percentage forms. Now, in this post we will be studying some important multiples of fractional values.

Learning these multiples will be beneficial for every aspirant for increasing their calculations speed. They are easy to remember and follow some patterns.

Let us study each one of them one by one.

You must have noticed that each of these sets follow certain pattern and hence are easy to remember.

If you practice basics daily you will be able to remember each of these values correctly.

So practice daily and believe in yourself.

Soon we will be posting questions for practice and quizes based on the tricks we lernt in aptitude

which will increase your calculation speed and develop your approach.

A seperate answer key will also be posted on the same day to make comarisions.

So, go now and discover all the posts in the Aptitude Section and learn all the tricks and try to write it by remembering.

Solve 5 questions based on each trick which will enhance your preparation.

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Word Problems Chapter-1 :- Percentage | Basics Of Percentage | Important Fraction Values

2020-05-16T21:41:00.002+05:30

In order to get command on this chapter, it becomes essential to learn the basics of this chapter which is Conversion of percentages into fractional values and vice versa.

In this, we will remember some common fractional value's conversion into percentage value.

Learning this will boost up the basic calculations related to percentage whether it is a Word Problem or a Simplification.

Here you go, Let's learn them !

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Basics of Simplification | Multiplication of Two Digits By Two Digits

2020-05-10T16:21:00.000+05:30

To find product of two 2-digit numbers quickly, take a look at the following steps-

Step 1 :- Multiply both the unit place digits with each other.

Step 2 :- Cross Multiply and add both products.

Step 3 :- Multiply both the Ten's place digits with each other.

Note - You must make sure take carry if the result in each part exceeds 9 from right to left.

Let's look at some examples.

For Ex. 31 x 21

Step 1 :- Multiply both the unit place digits with each other (i.e. 1 x 1 = 1)

Step 2 :- Cross Multiply and add both products [i.e. (3 x 1) + (1 x 2) = 5].

Step 3 :- Multiply both the Ten's place digits with each other (i.e. 3 x 2 = 6) and add the '1' carried which results 7.

Final Result :- 651.

For Ex. 43 x 12

Step 1 :- Multiply both the unit place digits with each other (i.e. 3 x 2 = 6)

Step 2 :- Cross Multiply and add both products [i.e. (4 x 2) + (3 x 1) = 11]. As this results exceeds 9, 1 will be carried.

Step 3 :- Multiply both the Ten's place digits with each other (i.e. 4 x 1 = 4) and add the '1' carried which results 5.

Final Result :- 516.

For Ex. 53 x 27

Step 1 :- Multiply both the unit place digits with each other (i.e. 3 x 7 = 21) As this results exceeds 9, 2 will be carried.

Step 2 :- Cross Multiply and add both products [i.e. (5 x 7) + (3 x 2) = 41]. Add the '2' carried which results 43

As this results exceeds 9, 4 will be carried.

Step 3 :- Multiply both the Ten's place digits with each other (i.e. 5 x 2 = 10) and add the '4' carried which results 14.

Final Result :- 1431.

For Ex. 64 x 39

Step 1 :- Multiply both the unit place digits with each other (i.e. 4 x 9 = 36) As this results exceeds 9, 3 will be carried.

Step 2 :- Cross Multiply and add both products [i.e. (6 x 9) + (4 x 3) = 66]. Add the '3' carried which results 69

As this results exceeds 9, 6 will be carried.

Step 3 :- Multiply both the Ten's place digits with each other (i.e. 6 x 3 = 18) and add the '6' carried which results 24.

Final Result :- 2496.

For Ex. 78 x 83

Step 1 :- Multiply both the unit place digits with each other (i.e. 8 x 3 = 24) As this results exceeds 9, 2 will be carried.

Step 2 :- Cross Multiply and add both products [i.e. (7 x 3) + (8 x 8) = 85]. Add the '2' carried which results 87.

As this results exceeds 9, 8 will be carried.

Step 3 :- Multiply both the Ten's place digits with each other (i.e. 7 x 8 = 56) and add the '8' carried which results 64.

Final Result :- 6474.

Problems For Practice

1) 54 x 63

2) 32 x 49

3) 87 x 36

4) 16 x 91

5) 77 x 46

6) 28 x 65

7) 86 x 44

8) 33 x 53

9) 69 x 74

10) 78 x 83

Basics Of Simplification | List of Cubes from 1 to 50

2020-05-06T13:48:00.001+05:30

Here is the table of cubes from 1 to 50.It is recommended to learn Cubes up to 30 (Min.)

Number	Cube
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1,000
11	1,331
12	1,728
13	2,197
14	2,744
15	3,375
16	4,096
17	4,913
18	5,832
19	6,859
20	8,000
21	9,261
22	10,648
23	12,167
24	13,824
25	15,625
26	17,576
27	19,683
28	21,952
29	24,389
30	27,000
31	29,791
32	32,768
33	35,967
34	39,304
35	42,875
36	46,656
37	50,653
38	54,872
39	59,319
40	64,000
41	68,921
42	74,088
43	79,507
44	85,184
45	91,125
46	97,336
47	1,03,823
48	1,10,592
49	1,17,649
50	1,25,000

Basics of Simplification | Squares | Learn Squares form 1 - 100

2020-05-06T13:33:00.000+05:30

Following is the table of squares from 1 to 100.To increase the calculation speed it is essential for an aspirant to learn Multiplication Tables up to 30, Squares up to 60 (min.), Cubes up to 30.

Number	Square
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900
31	961
32	1,024
33	1,089
34	1,156
35	1,225
36	1,296
37	1,369
38	1,444
39	1,521
40	1,600
41	1,681
42	1,764
43	1,849
44	1,936
45	2,025
46	2,116
47	2,209
48	2,304
49	2,401
50	2,500
51	2,601
52	2,704
53	2,809
54	2,916
55	3,025
56	3,136
57	3,249
58	3,364
59	3,481
60	3,600
61	3,721
62	3,844
63	3,969
64	4,096
65	4,225
66	4,356
67	4,489
68	4,624
69	4,761
70	4,900
71	5,041
72	5,184
73	5,329
74	5,476
75	5,625
76	5,776
77	5,929
78	6,084
79	6,241
80	6,400
81	6,561
82	6,724
83	6,889
84	7,056
85	7,225
86	7,396
87	7,569
88	7,744
89	7,921
90	8,100
91	8,281
92	8,464
93	8,649
94	8,836
95	9,025
96	9,216
97	9,309
98	9,604
99	9,801
100	10,000

How to Find Square of a Number ? | Squaring Tricks | Square of a number whose all digits are 1

2020-04-29T16:12:00.001+05:30

Everyone knows squares of 1 and 11 are 1 and 121 respectively. But did you try to find squares of 111, 1111, 11111 etc.

Here is the way to calculate the squares of such types of numbers without doing any lengthy writing work or without stressing or recalling any basics.

Want to know how ????

Follow these steps and I am damn sure that you will be able to calculate squares of any long digit numbers whose all digits are 1.

Here you go !

➤ Calculate the number of digits in the given number.

➤ Start writing from 1 to the highest number (i.e. Value = No. of Digits in the given number) and back again to 1.

For Ex. (111)²

Step 1 :- Number of digits in the given number = 3.

Step 2 :- Now we will write from 1 to 3 and then back to 1 (i.e. 12321)

Final Result :- 12321.

For Ex. (1111)²

Step 1 :- Number of digits in the given number = 4.

Step 2 :- Now we will write from 1 to 4 and then back to 1 (i.e. 1234321)

Final Result :- 123421.

For Ex. (11111)²

Step 1 :- Number of digits in the given number = 5.

Step 2 :- Now we will write from 1 to 3 and then back to 1 (i.e. 123454321)

Final Result :- 123454321.

Note :- (This is not a trick or a method it is just a pattern of numbers observed.)

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How to find Square of a number ? | Squaring Tricks | 100 Base Method

2020-04-28T19:38:00.002+05:30

We have already learnt 25 Base Method which can easily calculate squares of two digit numbers quickly. Here we will study, how to find squares using 100 as base.

Using 25 Base Method and a daily consistent practice of it we can orally calculate squares from 25 to 75.

For Faster calculation speed, it is very essential to learn Multiplication Tables up to 30 (must), Squares up to 60 & Cubes up to 30.

Here, in this article we will learn how to find a square of a number using 100 as base.

➤ Divide the result portion in two parts.

Firstly, we will compare the given number with 100.

if the given number is less than 100, (let's assume by 'n' units) we will simply subtract it from the given number.

We will write it in the first part.

➤ We will Calculate Square of 'n' and write it in the second part.

(Note that if the second part contains 3 digit number we will simply carry it and add it in the first part.)

So let's understand this method by some examples.

For Ex. (106)²

Step 1 :- As 106 is greater than 100 by 6 (hence, here n= 6) we will add it to the given number (i.e. 106) which results 112.

Step 2 :- Square of 6 which is 36.
Final Result :- 11236.

For Ex. (92)²

Step 1 :- As 92 is smaller than 100 by 8 (hence, here n= 8) we will subtract it from the given number (i.e. 92) which results 84 .

Step 2 :- Square of 8 which is 64.
Final Result :- 8464.

For Ex. (112)²

Step 1 :- As 112 is greater than 100 by 12 (hence, here n= 12) we will add it to the given number (i.e. 112) which results 124.

Step 2 :- Square of 12 which is 144. (Here we got 3 digit number hence 1 will get carried and we will add it in 124)

Final Result :- 12544.

For Ex. (86)²

Step 1 :- As 86 is smaller than 100 by 14 (hence, here n= 14) we will subtract it from the given number (i.e. 86) which results 72

Step 2 :- Square of 14 which is 196 (Here we got 3 digit number hence 2 will be carried and we will add it in 72)
Final Result :- 7496.

Problems For Practice

1) (82)²

2) (109)²

3) (113)²

4) (91)²

5) (96)²

6) (103)²

7) (84)²

8) (88)²

9) (114)²

10) (119)²

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How to square a number ? | Squaring Tricks | 25 Base Method

2020-04-28T12:00:00.000+05:30

It is not possible to remember all the squares orally. But with some prefixed bases it became easy to calculate such Squares

For Faster calculation speed, it is very essential to learn Multiplication Tables up to 30 (must), Squares up to 60 & Cubes up to 30.

In some critical conditions (such as during live examination) when the timer is continuously running and when you can't remember the basics these methods work.

Here, in this article we will learn how to find a square of a number using 25 as base.

➤ Divide the result portion in two parts.

Firstly, we will compare the given number with 50.

if the given number is less than 50, (let's assume by 'n' units) we will simply subtract it from the base (i.e. 25)

We will write it in the first part.

➤ We will Calculate Square of 'n' and write it in the second part.

(Note that if the second part contains 3 digit number we will simply carry it and add it in the first part.)

So let's understand this method by some examples.

For Ex. (56)²

Step 1 :- As 56 is greater than 50 by 6 (hence, here n= 6) we will add it to the base 25 which results 31.

Step 2 :- Square of 6 which is 36.
Final Result :- 3136.

For Ex. (43)²

Step 1 :- As 43 is smaller than 50 by 7 (hence, here n= 7) we will subtract it from the base 25 which results 18.

Step 2 :- Square of 7 which is 49.
Final Result :- 1849.

For Ex. (63)²

Step 1 :- As 63 is greater than 50 by 13 (hence, here n= 13) we will add it to the base 25 which results 38

Step 2 :- Square of 13 which is 169. (Here we got 3 digit number hence 1 will get carried and we will add it in 38)

Final Result :- 3969.

For Ex. (34)²

Step 1 :- As 34 is smaller than 50 by 16 (hence, here n= 16) we will subtract it from the base 25 which results 9

Step 2 :- Square of 14 which is 256 (Here we got 3 digit number hence 2 will be carried and we will add it in 11)
Final Result :- 1156.

Problems For Practice

1) (37)²

2) (53)²

3) (44)²

4) (39)²

5) (62)²

6) (47)²

7) (58)²

8) (41)²

9) (64)²

10) (36)²

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Basics of Multiplication | Multiplication Tricks | Multiplication of two consecutive Even/Odd Numbers

2020-04-27T13:44:00.000+05:30

Two Consecutive Even or Odd number can be multiplied in a fraction of second.
Want to know how ?

For this you should know squares of numbers.

Let us look at a simple Example.

Ex. 27 × 29

Ex. 44 × 46

Ex. 55 × 53

Ex. 34 × 32

Well if you calculate this by traditional method it will take little bit long duration of time. But with some logic and a knowledge of squares you will be able to calculate it's answers as 783, 2024, 2915 & 1088 respectively.

Isn't it interesting ?

so let's start with the steps you should follow to calculate these types of products.

Step 1 :- Calculate the square of the middle number. ( If you can remember it will increase your calculation speed)

Step 2 :- Subtract 1 from the obtained result. This will be your answer.

Ex. 27 × 29

Step 1 :- We will calculate square of 28 which is 784.

Step 2 :- Subtract 1 from 784.

Final Result :- 783.

Ex. 44 × 46

Step 1 :- We will calculate square of middle no. (i.e 45) which is 2025.

Step 2 :- Subtract 1 from 2025.

Final Result :- 2024.

Ex. 55 × 53

Step 1 :- We will calculate square of middle no. (i.e 54) which is 2916.

Step 2 :- Subtract 1 from 2916.

Final Result :- 2915.

Ex. 34 × 32

Step 1 :- We will calculate square of middle no. (i.e 33) which is 1089.

Step 2 :- Subtract 1 from 1089.

Final Result :- 1088.

Logic :- This is based on the most popular squaring formula we have learnt in schooling.

(a + b)² × (a – b)² = a² – b²

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How to Square a Number ? | Squaring Tricks | Square of a Two Digit Number

2020-04-22T22:54:00.001+05:30

In this Method we will learn how to find a square of any two digit number in a very short span of time.

In this we will use our basic formula but in a different way.

Take a look at the fomula for finding a square.

(a + b)²= a² + 2ab + b²

➤ Divide the result portion into 3 parts. Consider Tens place digit as ‘a’ and Unit place digit as ‘b’ as shown.

➤ In leftmost part we will calculate a² , in the rightmost part we will calculate b² and in the middle part we will calculate product of 2ab.

➤ If any of the parts whether it is middle or rightmost or both contain two digit number we will simply take carry their tens digits.

For Ex. (41)²

Step 1 :- We will consider '4' as 'a' and '1' as 'b'

and we will calculate

a²= (4)² =16

2ab = 2 * 4 * 1 = 8

b²= (1)²=1

Step 2 :- As we can see middle part and rightmost part do not contain two digit number they will be taken as it is in the Final result.

Final Result :- 1681

For Ex. (67)²

Step 1 :- We will consider '6' as 'a' and '7' as 'b'

and we will calculate

a²= (6)² =36

2ab = 2 * 6 * 7 = 84

b²= (7)²=49

Step 2 :- As we can see middle part and rightmost part both contain two digit numbers they will be taken carry as shown.

Final Result :- 4489.

For Ex. (72)²

Step 1 :- We will consider '7' as 'a' and '2' as 'b'

and we will calculate

a²= (7)² =49

2ab = 2 * 7 * 2 = 28

b²= (2)²=4

Step 2 :- As we can see only middle part contains two digit number 2 will be taken carry as shown.

Final Result :- 5184.

Problems For Practice

1) (34)²

2) (57)²

3) (49)²

4) (63)²

5) (14)²

6) (24)²

7) (77)²

8) (58)²

9) (94)²

10) (87)²

How To Square a number ? | Squaring Tricks | Square of a Number Ending with -5

2020-04-21T13:08:00.002+05:30

There are plenty of methods to find squares of numbers. Here, we are going to discuss How to find square of a number ending with 5 ?

➤ Divide the result space into 2 parts as shown.

➤ In 2^nd part write square of 5 i.e. 25.

➤ In first part write the product of remaining number with it's successor.

For Ex. (65)²

^{Step 1 :-}As the given number ends with -5 we will write 25 in the 2^nd part.

^{Step 2 :-}In first part we will calculate product of remaining number (i.e. 6) with it’s successor (i.e. 7) which is 42

Final Result :- 4225.

For Ex. (135)²

Step 1 :- As the given number ends with -5 we will write 25 in the 2^nd part.

^{Step 2 :-}In first part we will calculate product of remaining number (i.e. 13) with it’s successor (i.e. 14) which is 182.

(We have learnt this in Basics of Multiplications | Multiplication Tricks | Multiplication of Two Consecutive Numbers )

Final Result :- 18225.

Problems For Practice.

1) (85)²

2) (55)²

3) (35)²

4) (75)²

5) (95)²

6) (165)²

7) (225)²

8) (345)²

9) (415)²

10) (195)²

Basics Of Multiplication | Multiplication Tricks | Multiplication by 11 (Part 2)

2020-04-20T18:17:00.003+05:30

Earlier, we have studied how to multiply a two digit number by 11. Here, we will study how to multiply any number by 11.

If you have not studied Multiplication of 11 ( Part 1) you can visit and learn it From the link given below.

Basics Of Multiplications | Multiplication Tricks | Multiply two digit number by 11 (Part 1)

So, let's begin the Part 2 of the Multiplication by 11.

➤ Step 1 :- Multiply the given number by 10

➤ Step 2 :- Add the obtained result in the given number.

For Ex. - 143 × 11

Step 1 :- 143 × 10 = 1430

.Step 2 :- 1430 + 143 = 1573

Final Result :- 1573.

For Ex. - 228 × 11

Step 1 :- 228 × 10 = 2280

Step 2 :- 2280 + 228 = 2508

Final Result :- 2508.

Problems for Practice

1) 136 × 11

2) 221× 11

3) 364 × 11

4) 453 × 11

5) 389 × 11

Basics of Multiplication | Multiplication Tricks | Multiplication by 125

2020-04-19T12:13:00.002+05:30

Here is the Trick to quickly multiply any number by 125.

➤ Add three zeros at the end of the given number which we are going to multiply by 125.

➤ Divide by 8 to the obtained result.

Logic-

125 can be written as 1000/8 (1000 by 8), hence first we put three zeros at the end of the given number and then divide it by 8.

For Ex. - 56 × 125

Step 1 :- We will add two zeros at the end which results 56000.

Step 2 :- Then we will divide it by 8 (i.e. 56000/8 = 7000).

Final Result :- 7000.

Problems For Practice

1) 84 × 125

2) 144 × 125

3) 68 × 125

4) 76 × 125

5) 88 × 125

6) 29 × 125

7) 49 × 125

8) 79 × 125

9) 38 × 125

10) 97 × 125 .

Basics of Multiplication | Multiplication Tricks | Multiplication by 25

2020-04-18T15:34:00.001+05:30

Multiplication by 25 seems little beat lengthy if we directly multiply it.
However if we do it with logic it will be easy to find solution of it in a very short time.

Here is the Trick to quickly multiply any number by 25.

➤ Add two zeros at the end of the given number which we are going to multiply by 25.

➤ Divide by 4 to the obtained result.

Logic-

25 can be written as 100/4 (100 by 4), hence first we put two zeros at the end of the given number and then divide it by 4.

For Ex. - 36 × 25

Step 1 :- We will add two zeros at the end which results 3600.

Step 2 :- Then we will divide it by 4 (i.e. 3600/4 = 900).

Final Result :- 900.

Problems For Practice

1) 62 × 25

2) 44 × 25

3) 76 × 25

4) 37 × 25

5) 69 × 25

6) 32 × 25

7) 86 × 25

8) 48 × 25

9) 93 × 25

10) 49 × 25 .

Basics Of Multiplication | Multiplication Tricks | Multiplication of Two Digit Number Having Same Tens Digit and Sum of Unitary Digits is 10

2020-04-17T10:38:00.000+05:30

Let us take a look at some pairs of numbers given below.

1) 33 & 37 2) 52 & 58 3) 41 & 49 4) 34 & 36 5) 23 & 27

If we observe each pair carefully, we will notice that in each pair the digit at tens place are equal & Sum of the digits at ones place is 10.

Some Pairs of numbers have a different Property.

Here, we are going to discuss "How to find product of two digit numbers where Tens Digits are Equal and Sum of Unitary Digits is Equals to 10 ? "

For Ex. - 41 × 49

Step 1 :- Divide the Result Portion in in 2 parts.

In left most part we will multiply tens digit by it's successor. (i.e. 4 × 5 = 20)

Step 2 :- In rightmost part multiply both the digits of ones place. (i.e. 1 × 9 = 9) and if the result is in single digit Put a '0' in front of it. (i.e. 09)

Final Result :- 2009.

For Ex. - 33 × 37

Step 1 :- Divide the Result Portion in in 2 parts.

In left most part we will multiply tens digit by it's successor. (i.e. 3 × 4 = 12)

Step 2 :- In rightmost part multiply both the digits of ones place. (i.e. 3 × 7 = 21) and if the result is in single digit Put a '0' in front of it.

(As you can see the result is in Two digit Hence we don't need to put '0' in front of it.)

Final Result :- 1221.

Problems For Practice

1) 91 × 99

2) 84 × 86

3) 67 × 63

4) 59 × 51

5) 47 × 43

Basics Of Multiplication | Multiplication Tricks | Multiplication by 5

2020-04-16T12:34:00.001+05:30

Multiplication by 5 seems easy as all of us know the table of 5. But when we multiply it by a Long Number then it becomes a little bit long.

Here is the Trick to quickly multiply any number by 5.

➤ Half the given number.

If it is an Even number we can simply take half of the given number But if it is an Odd number then we will subtract 1 from it and then take the half of the obtained result

➤ Check the given number which we are going to multiply is even number or odd number ?

If it is an Odd number the unit digit of result will be '5' and if it is an Even number then the unit digit of the result will be '0' .

Logic-

5 can be written as 10/2 (10 by 2), hence first we take half of the given number and then multiply it by 10 or simply put '0' at it's end.

For Ex. - 46 × 5

Step 1 :- As 46 is an even number we will simply take half of it which is 23

Step 2 :- As we know it is an even number we will put a '0' in the end.

Final Result :- 230.

Problems For Practice

1) 72 × 5

1) 88 × 5

1) 90 × 5

1) 54 × 5

1) 28 × 5

For Ex. - 53 × 5

Step 1 :- As 53 is an even number we will subtract 1 from it which results 52 and then take half of it which is 26

Step 2 :- As we know it is an odd number we will put a '5' in the end.

Final Result :- 265.

Problems For Practice

1) 23 × 5

1) 85 × 5

1) 57 × 5

1) 93 × 5

1) 39 × 5 .

Bsics of English | Increase Your Vocabulary | Homonyms | Some Common Words With Similar Pronunciation (List 1)

2020-04-15T22:52:00.002+05:30

Here is the List of Some Common Homonyms we use in our day to day life.

Discover all of them and learn the basic difference between them.

There तेथे Their त्यांचा
Here येथे Hear ऐकणे
Were होते Wear परिधान करणे
Week आठवडा Weak अशक्त
Fare भाडे Fair यात्रा
Cell विजेरिचा सेल Sell विकणे
Sit बसणे Seat आसन
Once एकदा One's एखाद्याचा
Wet ओला Wait वाट पहाणे
Test चाचणी Taste चव
Road रस्ता Rode आरुड झाला
Meet भेटणे Meat मास
Leave सोडणे Live रहाणे
Piece तुकडा Peace शांतता
Rain पाऊस Rein लगाम
Wander भटकणे Wonder आश्चर्य
Reach पोहोचणे Rich श्रीमंत
Besides जवळ Besides शिवाय/आणखी
Feel वाटणे Fill भरणे
Heat तापवणे Hit दाबा
Waste रद्दी West पश्चिम दिशा
Of चा Off दूर
Beat मारणे Bit चावणे
Weather हवा Whether जर का
Floor जमीन Flour पीठ
Fell पडला Fail अयशस्वी
Principal प्राचार्य Principle तत्व
In आत Inn खानावळ
Thrown फेकलेला Throne सिंहासन
Sheep मेंढी Ship जहाज
Sleep झोपणे Slip घसरने
Lone एकटा Loan कर्ज
Whole संपूर्ण Hole बीळ
Rite विधी Write लिहिने
Led मार्गदर्शन केले Laid ठेवला
Lane गल्ली Lain ठेवलेला
Hour तास Our आमचा
Lives रहातो Leaves पाने/सोडून जातो
Deep खोल Dip बुडविने
Prey भक्ष्य Pray प्रार्थना

Basics of Multiplications | Multiplication Tricks | Multiplication of Two Consecutive Numbers

2020-04-15T19:26:00.001+05:30

Want to speed up in Multiplying two consecutive numbers ?

Before learning the Short Cut Method to multiply two consecutive numbers, let us take a look at the Traditional Method to calculate the results.

For Ex. - 21 × 22

Step 1 :- We multiply 21 by 2 which is 42

Step 2 :- Then we write unitary digit i.e 2 at result and we then take carry Tens digit of the obtained result i.e. 4

Step 3 :- Again we multiply 21 by 2 and which results 42 again add the Tens digit we have carried which can be calculated as 42 + 4 = 46

Final Result - We get 462.

Now we will do the same problem by Short Cut Method.

This can be done in two ways.

Way 1 - Add 21 to the square of 21. (21 + 441 = 462)

Way 2 - Subtract 22 from square of 22. (484 - 22 = 462)

Let us understand the logic behind this method.

Logic -

The product m × n can be written as 'n' times addition of 'm' .

Let us try the same example with above logic.

For Way 1

Step 1 :- First we will multiply 21 × 21 which is 441.

Step 2 :- Secondly, we will multiply 21 × 1 which results 21

Step 3 :- Add both the results. 441 + 21 = 462.

Final Result - We get 462.

For Way 2

Step 1 :- First we will multiply 22 × 22 which is 484.

Step 2 :- Secondly, we will multiply 22 × 1 which results 22

Step 3 :- Subtract 22 from 484. (484 - 22 = 462.)

Final Result - We get 462.

Problems for Practice

1) 18 × 19

1) 33 × 34

1) 45 × 46

1) 28 × 29

1) 36 × 37

Concepts Study

Ratio & Proportion – Concepts, Tricks, Conversions & Exam-Level Applications

1. Ratio –

What is a Ratio?

Forms of Ratios

Key Characteristics

(a) Ratios have no units

(b) Ratios can be simplified (like fractions)

(c) Equivalent Ratios

Ratio in Real Life

Internal & External Ratios

Internal Ratio:

External Ratio:

Conversion Between Ratio & Fraction

Ratio → Fraction

Fraction → Ratio

Shortcut Ratio Tricks

Trick 1: From Percentage to Ratio

Trick 2: Fraction to Ratio Quickly

Trick 3: Ratio Scaling (Very Useful in Exams)

✔ Convert Ratio → Fraction

✔ Convert Fraction → Ratio

✔ Convert Ratio → Percentage

2. Proportion – Complete, Deep Explanation

What is Proportion?

A proportion states that two ratios are equal.Example:2/3 = 4/6 → This is a proportion2 : 3 :: 4 : 6

Types of Proportion

(A) Direct Proportion

When one quantity increases, the other increases.Example:More hours worked → More salaryMathematically:a/b = c/d

(B) Inverse Proportion

One quantity increases while the other decreases.Example:Speed ↑ → Time ↓Workers ↓ → Days ↑Formula:a × b = c × d

(C) Continued Proportion

If a : b = b : c, then a, b, c are in continued proportion.

(D) Compound Proportion

Combination of multiple proportions.Example:If 12 men do a job in 15 days, how many days will 20 men take?

Rules in Proportion

1. Product Rule

In a proportion:a : b = c : dSo,a × d = b × cThis is called the Cross-multiplication rule.

2. Mean & Extreme Terms

a : b = c : da, d → Extreme termsb, c → Mean termsProduct of extremes = Product of means

3. Fourth Proportion

a : b = c : xx = (b × c) / a

4. Third Proportion

a : b = b : xx = (b²)/a

5. Mean Proportion (Geometric Mean)

a : x = x : bx² = ab → x = √ab

⭐ Exam-Level Applications (Important for SSC, Banking, CET, Railways, etc.)

📌 1. Ratio-based distribution

📌 2. Changing Ratio Questions

📌 3. Exam Trick – Ratios with Difference

📌 4. Partnership (Direct Application of Ratios)

📌 5. Mixture & Alligation (Ratio Shortcut)

Deeper Algebra: Solving Proportional Equations

🔷 Advanced Tips & Pitfalls

🔷 Practice Problems (with answers)

Answers (work shown briefly)

⭐ Quick Revision Table

⭐ Key Takeaways

⏭ Next Chapter Preview: – Percentage

Fractions & Decimals

Concepts, Conversions, Tricks & Examples

⭐ Fractions & Decimals

🔹 What is a Fraction?

Types of Fractions

🔹 What is a Decimal?

🔹 Conversion: Fractions ⇆ Decimals (Super Easy!)

Fraction → Decimal

Decimal → Fraction

🔹 Shortcut Box — Must Remember Fractions

🔹 Comparing Fractions – Fastest Exam Trick

Cross Multiplication Method

🔹 Converting Mixed Fractions

🔹 Operations on Fractions

✔ Addition / Subtraction (Unlike Fractions)

✔ Multiplication

✔ Division

🔹 Decimal Operations

✔ Addition/Subtraction

✔ Multiplication

✔ Division

A proportion states that two ratios are equal.
Example:
2/3 = 4/6 → This is a proportion
2 : 3 :: 4 : 6

When one quantity increases, the other increases.
Example:
More hours worked → More salary
Mathematically:
a/b = c/d

One quantity increases while the other decreases.
Example:
Speed ↑ → Time ↓
Workers ↓ → Days ↑
Formula:
a × b = c × d

Combination of multiple proportions.
Example:
If 12 men do a job in 15 days, how many days will 20 men take?

In a proportion:
a : b = c : d
So,
a × d = b × c
This is called the Cross-multiplication rule.

a : b = c : d
a, d → Extreme terms
b, c → Mean terms
Product of extremes = Product of means

a : b = c : x
x = (b × c) / a

a : b = b : x
x = (b²)/a

a : x = x : b
x² = ab → x = √ab