Method 2: Division Method (Euclidean Algorithm)

1. Divide the larger number by the smaller number.

2. Take the remainder.

3. Divide the smaller number by the remainder.

4. Repeat until the remainder becomes 0.
The last divisor is the HCF.

Example:
Find HCF of 36 and 60:

60/36=1 remainder 24
36/24=1 remainder 12
24/12=2 remainder 0

HCF = 12

Another way to find HCF when Two numbers are there and LCM is Given :

Formula: HCF × LCM = N1 × N2

e.g: N1 = 15, N2 = 20, LCM = 60, HCF= ?

HCF = 15×20/60
= 300/60
HCF = 5


Note: This formula is valid for only two numbers.

Face value And Place value
1. Face Value
The face value of a digit is the actual value of the digit itself, irrespective of its position in the number.

#Example
For the number 4725:
Face value of 4 = 4
Face value of 7 = 7
Face value of 2 = 2
Face value of 5 = 5
✅ Face value = digit itself

2. Place Value
It is the place at which Digit located like 10th position, 100th position etc.

#Example:
For the number 4725:
Place value of 4 = 4000
Place value of 7 = 700
Place value of 2 = 20
Place value of 5 = 5

Quick Trick to Remember
Face value → What you see 👀
Place value → Where it is 📍

Types of Fractions

1. Proper Fraction:
• A Fraction where the Numerator < denominator.
• Value of fraction is less than 1.
• Example: 3/5, 6/7, 9/11.

2. Improper Fraction:
• A Fraction where the Numerator > denominator.
• Value of fraction is greater than 1.
•Example: 8/5, 7/3, 11/7.

3. Mixed Fraction:
    • A fraction that is combination of a whole number and a proper fraction.
    • Every mixed fraction can be converted into an improper fraction, and vice versa
•Example: 2(1/3) = 7/3,
5(2/4) = 22/4.

Fractions Decimals Percentage
1/2 0.5 50%
1/4 0.25 25%
1/5 0.20 20%
1/10 0.10 10%
1/20 0.05 5%
1/8 0.125 12.5%
3/4 0.75 75%
4/5 0.80 80%
1/3 0.3333 33.33%
2/3 0.6666 66.66%
1/6 0.1666 16.66%
5/6 0.8333 83.33%
6/5 1.2 120%
5/4 1.25 125%
3/2 1.5 150%
1/7 0.1428 14.28%
1/9 0.1111 11.11%
1/11 0.0909 9.09%
1/12 0.0833 8.33%
1/15 0.0666 6.66%
1/16 0.0625 6.25%

Comparing Fractions

1. Same Denominator
If fractions have the Same Denominator, Just compare numerators.
Example: 5/9 < 7/9

2. Same Numerator
If fractions have the same numerator, the one with the smaller denominator is larger.
Example: 3/5 > 3/7

3. Different Numerator and Denominator
There are two methods:

i. Cross Multiplication
a/b and c/d is there,
check a×d and c×b,
If a×d > c×b then a/b > c/d,
else c/d > a/b
Example:
3/5 and 4/8, check 3×8 and 4×5, i.e. 24 > 20, 3/5 > 4/8.

ii. Convert to same Denominator
Example:
5/7 and 3/4, 5/7 × 4/4 and 3/4 × 7/7, i.e 20/28 and 21/28
Therefore, we can say that,
5/7 < 3/5.

Percentage Increase & Decrease:

1. Percentage increase:
When the value goes up, use,
Percentage increase= ((New value - Old value) ÷ Old value) × 100

Example:
Old price = 100
New price = 120
Increase = 120 − 100 = 20
Percentage increase = (20 / 100) × 100 = 20% Increase

2. Percentage decrease:
          When the value goes down, use,
Percentage decrease (Old value - New value) ÷ Old value) × 100

Example:
Old price = 100
New price = 70
Decrease = 100 − 70 = 20
Percentage decrease = (30 / 100) × 100 = 30% Decrease

Shortcut Trick:
-Increase → New − Old
-Decrease → Old − New
-Always divide by Old value & multiply by 100.

1. 20 is 25% of 80. So, 80 getting 100 means 25% increase in 80.
Don't be in a hurry, Solve with formulas.
2. 100-80=20 increase
(20÷(80 old value))×100 = 25%

We will be consistent starting from today. Any query, You can DM me @indefatigable07 .

Exam Trick (Important):

Net % change = a+b+(ab÷100)
(where a and b are % changes, use negative for decrease)

Or simply: Consider 100 is the value, 20% increase is 120, 10% decrease 120-12=108, that means 8% increase in 100.