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.

Discount & Offers :
Discount is reduction on Marked Price.

Discount = Marked Price(MP) - Selling Price(SP)

Discount% = (Discount ÷ MP) × 100

SP = MP - Discount

Example:
MP = 1000, Discount = 20%, SP = ?

SP = 1000-(20% Of 1000)
= 1000-200
SP = 800

By using Formula:
SP = MP × (100-discount%)/100
= 1000 × (100-20)/100
= 1000 × 80/100
= 10 × 80
= 800

Two discounts are NOT added directly:
If discount given is a% and b% then:
Total discount = a+b-(ab/100)
Example: a = 20%, b = 30% then
Total disc = 20+30-(20×30)/100
= 50 - 600/100
= 50 - 6 = 44%

Find Profit after Discount:
Profit = (SP - CP)
Profit = ((SP - CP)/CP)×100

[CP = Cost Price]

Profit% = (Profit/CP)×100

When shopkeeper gives discount still gains profit, then

MP/CP = (100+profit%)/(100-discount%)

Ratios, Proportion & Averages ⚖️
1.Ratio & its applications
2.Proportion & inverse proportion
3.Averages - Mean, Median, Mode
4.Weighted averages

1. Ratio & it's Applications
A ratio compares two quantities in a single environment.

i.e. if A = 10, B = 20,
Then A:B is 10:20 i.e 1:2

Multipy or divide both terms ratio remains same.

[Applications]
1.Sharing quantity
Divide 300 into 2 quantities such that ratio remains 2:3.
i.e (2/5)×300=120 & (3/5)×300=180

2. Mixing Problem
If milk:water=5:3, total=8 then milk=5/8 & water=3/8

3.Salary or Age ratio
If A:B=2:3,
let A=2x & B=3x

2. Proportion
Equality of two ratios
a:b = c:d (a/b = c/d)

Cross multiplication rules
If A/B=C/D, then A×D=B×C.

1.Direct Proportion:
When one quantity increases, the other also increases.
Examples
More workers → More work
More distance → More fuel
More items → More cost

Example: If 5 pens cost ₹50, what is cost of 8 pens?
Set proportion:
5 : 50 = 8 : x
Cross multiply:
5x = 400
x = ₹80

2.Inverse Proportion:
When one quantity increases, the other decreases.
Examples
More workers → Less time
Higher speed → Less travel time

Example: 5 workers complete work in 12 days.
10 workers will complete it?
5 × 12 = 10 × x
x = 6 days

•Continued Proportion:
If a:b=b:c, then b²=ac.
a,b,c are in continued proportion.

•Fourth Proportional:
Find d?
If a:b=c:d, then d is called the Fourth Proportional

3. Averages-Mean, Median and Mode
These are the three main measures of average used in mathematics and aptitude exams.

1. Mean (Arithmetic Mean)
The mean is the average value of all numbers.

Formula: Sum of all observations /Number of all observations

Example:
Find the mean of: 2, 4, 6, 8, 10.
Sum = 2 + 4 + 6 + 8 + 10 = 30
Total numbers = 5
Mean = 30 ÷ 5 = 6

2. Median
The median is the middle value after arranging numbers in ascending or descending order.

Rules
1.If total numbers are odd → middle number is median.
2.If total numbers are even → average of two middle numbers.

Example 1 (Odd Numbers)
Numbers: 3, 5, 7, 9, 11.
Middle number = 7
Median = 7
Example 2 (Even Numbers)
Numbers: 2, 4, 6, 8.
Middle two numbers = 4 and 6
Median = 10 ÷ 2 = 5

3. Mode
The mode is the number that appears most frequently.
Example
Numbers: 2, 3, 3, 5, 7, 3, 8.
Number repeated most = 3
Mode = 3