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

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