Mathematics : Arithmetic Concepts

Average:

Average is defined as “The sum of observations divided by the number of observations”.

Important concepts: Here are some important points to remember,

• When a person replaces another in a group of n persons with an average of group A, then
• If the average of the group is increasing, then
  Age of new person = Age of person who left + (n ´ Increase in average)
• If the average of the group decreases, then
  Age of new person = Age of person who left – (n ´ Decrease in average)
• When a person joins the group of n persons with an average of group A, then

• If the average of the group is increasing, then
  Age of new member = Previous average + (n + 1) ´ Increase in average
• When a person joins the group and the average of the group is decreases, then

Age of new member = Previous average – (n + 1) ´ Decrease in average

• When a person left the group of n persons with an average of group A, then

• If the average of the group is increasing, then
  Age of new member = Previous average –  (n + 1)´ Increase in average
• When a person joins the group and the average of the group is decreases, then

Age of new member = Previous average + (n + 1)´ Decrease in average

HCF and LCM:

Highest Common Factor (HCF):

The highest common factor of two or more numbers is the greatest common divisor, which divides each of those numbers an exact number of times. The process to find the HCF is:

1. Express the numbers given as a product of prime numbers separately i.e. find factors of
2. Take the product of prime numbers common to all the given numbers.

Least Common Multiple (LCM):

The least common multiple (LCM) of two or more numbers is the smallest of the numbers, which is exactly divisible by each of them, e.g. consider two numbers 18 and 24.

The multiples of 18 are: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, ....

The multiples of 24 are: 24, 48, 72, 96, 120, 144, 168, 192, 216, ......

The common multiples of both 18 and 24 are 72, 144, 216, ....

The least common multiple is 72.

Here again, try to break the words in reverse order and understand the concept. Firstly, find the multiples of the numbers. Secondly, the common multiples of the numbers and finally the least out of those will be the LCM. The process to find the LCM is:

1. Express the numbers given as a product of prime numbers separately i.e. find factors of numbers
2. Take the product of prime factors of the given numbers after eliminating repetition of the common. 

Ratio & Proportion:

Time, Speed, and Distance:

Speed: Speed is a very basic concept in motion which is all about how fast or slow any object moves.

Speed is defined as speed as distance divided by time.

Distance = Speed × Time

It means that:

(i) Distance is directly proportional to ‘Speed’ when ‘Time’ is constant.

• If the ratio of the distance covered by A and B in same time is a : b, then the ratio of the speed of A and B = a : b

(ii) Distance is directly proportional to ‘Time’ when ‘Speed’ is constant.

• If the ratio of the distance covered by A and B with same speed is a : b, then the ratio of the times taken by then = a : b

(iii) Time is inversely proportional to Speed when ‘Distance’ is constant.

Profit, Loss & Discount:

• Cost Price: The amount, at which an article is purchased or manufactured is known as its cost price.

The cost price is abbreviated as C.P.

• Selling Price: The price at which an article is sold is known as its selling price.

The selling price is abbreviated as S.P.

• Profit = Selling price – Cost price (Profit = S.P. – C.P.)

Note: Profit is generated, when

      Selling price > Cost price

 Loss = Cost price – Selling price (Loss = C.P. – S.P.)

Note: Loss is incurred, when Cost price > Selling price

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