Quantitative Aptitude - 1
Numbers
Classification of Numbers
Complex number = a + ib
Value of i = √-1
Rational numbers form = p/q, q is not equal to 0
Integer range:- -infinity to + infinity
Whole number range :- 0 to + infinity
Real numbers range:- 1 to + infinity
Divisibility Rules
2 :- The last number should be divisible by 2
4 :- The last 2 numbers should be divisible by 4
8 :- The last 3 numbers should be divisible by 8
3 :- The sum of the numbers should be divisible by 3
9 :- The sum of the numbers should be divisible by 9
5 :- The last number should be 0 or 5
10 :- The last number should be 0
6 :- Any number which is divisible by 2 & 3 is divisible by 6 as well
12 :- Any number which is divisible by 3 & 4 is divisible by 12 as well
11:- The difference between the sum of odd-placed numbers and even-placed numbers should be equal to 0 or 11
multiple.
7:- The last number should be doubled and subtracted from the remaining numbers. If the resultant is divisible by 7.
Then the whole number is divisible by 7
Power cycles
2:- {2,4,8,6}
3:- {3,9,7,1}
4:- {4,6}
5:- {5}
6:- {6}
7:- {7,9,3,1}
8:- {8,4,2,6}
9:- {9,1}
Remainder cycles
For example:
The remainder cycle of 6n / 7 = {6,1}
LCM & HCF
Factors
Let prime factorization N = ap x bq x cr where, a,b,c are prime numbers.
Number of factors for N = (p + 1) x (q +1) x (r +1)
Sum of factors = (ap+1-1)/(a-1) x (bq+1-1)/(b-1) x (cr+1-1)/(c-1)
Product of factors of N = N(Total No. of Factors)/2
P is said to be a factor of Q if P divides Q without leaving the remainder.
Factors of a number are always less than or equal to that number
HCF of two numbers is the highest factor which is common to both numbers.
Multiples
P is said to be multiple of Q if Q divides P without leaving remainder.
Multiples of a number are always greater than or equal to that number.
LCM of two numbers is the smallest multiple which is common to both the numbers
Properties
Product of two Numbers = LCM x HCF
If the ratio of numbers is a:b, and the HCF of numbers is H then, LCM of the numbers = H x a x b = HCF x Product of
ratios
LCM of fractions =(LCM of Numerators)/(HCF of denominators)
HCF of fractions =(HCF of Numerators)/(LCM of denominators)
Percentages
Basic Calculations
a percentage of b = (a/b)*100
1/7 = 14.2857%
2/7 = 28.5714%
3/7 = 42.8571%
4/7 = 57.1428%
5/7 = 71.4285%
6/7 = 85.7142%
Percentage Increase/Decrease
Successive Increase/Decrease
Net % Change or Overall Percentage Change = x + y + (xy/100)
Here, x and y are the percentage increase or decrease.
Use (+)ve sign for increase and (-)ve sign for decrease.
Profit and Loss
Terminology and Relations
Selling price = Cost Price +profit
Selling price = Cost price - Loss
Profit Percent(P%) =[(SP - CP)/CP]*100
Loss Percent(L%) = [(CP - SP)/CP)]*100
CP - price at which seller buys
SP - price at which seller sells
Profit - amount gain by selling above CP
Loss - amount loss by selling below CP
Discount
Discount = Marked Price(MP) - Selling Price(SP)
Discount Percent(D%) = (Discount/Marked Price) x 100 = [(MP - SP)/MP] x 100
Effective Discount on successive discount% = [- X - Y + (XY))/100]
Where,
X = First discount
Y = Second discount
-ve sign represents reduction in price.
Marked Price - Price on the product at which it is supposed to be sell.
Discount - Amount by which marked price is reduced
Discount percentage - Percentage by which the price of an item is reduced
Discount types:
Up to - May vary from 0 to given amount
Flat - Exact amount is given
Successive - Discounts are given in successive order
Simple Interest & Compound Interest
Definitions
Compound Interest(CI) is the interest that is calculated both on the principal and the previously earned interest.
Simple Interest (SI) is the interest that is calculated only on the principal.
For the same principle, Rate, and Time period CI > SI
Formulas
Amount(A) = Principle(P) + Simple/Compund Interest(SI or CI)
SI = (P x R x T)/100
Where,
P → Principle
R → Rate of Interest
T → Time (in years)
To calculate the Compound Interest(CI) we have to calculate the total amount.
CI = A-P
Where,
A → Amount
P → Principle
t → Time period
r → Rate of Interest
n → number of times interest applied per time period
Ratios and Proportions
Ratios
Ratio is a comparison of two or more quantities of the same unit.
It is unitless quantity means it has no unit.
ak:bk = a:b
a/k:b/k = a:b
if a:b = c:d then b:a = d:c
a:b = c:d ⇒ ad = bc
Proportions
If we have the ratio a:b and it's equal to the ratio c:d,
a:b=c:d, then we can say that a:b is proportional to c:d
If two ratios a:b and c:d are equal then they are represented as,
a:b :: c:d
Where,
a and d → extreme term
b and c → mean term
For two ratios in proportion.
a/b = c/d, can be written as a/c = b/d.
i.e a x d = b x c
Mixtures and Allegations
When we are mixing or combining two or more different substances then the resultant solution is known as mixture
Allegation helps us to find the ratio in which two or more ingredients at the given price must be mixed to produce a
mixture of desired price
Partnerships
Profit = investment × time
Tip: Always try to find ratio of profits of all the partners and distribute total profit among them in the same ratio.
Ages
Age is the amount of time that has passed since someone or something was born or created. It's usually measured in
years.
If someone is 'a' years old now, in 'n' years they will be (a + n) years old.
If we go back 'n' years, they were (a - n) years old.
If we compare two people's ages as a ratio, like p:q, then one person's age is 'pa' and the other's is 'qa', where 'a'
will be a constant.
If someone is 'a' years old now, after 'n' times that age, they will be a*n years old.
If someone is 'a' years old now, then one part out of 'n' parts of their age is a/n.
Averages
Average = Sum of Observation(Sn) / Number of Observation(N) = Sn/N
Addition or subtraction of constant k to each item in the list will result in increase or decrease of average by k.
Multiplication or division of all number by a constant k which result the average to be multiplied or divided by k
respectively.
If the given list of numbers are in AP then average will be:
(middle element) if number of elements are odd,
(average of middle two element) if number of elements are even.
