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.

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