54 is to be divided into two parts such that the sum of 10 times the first and 22 times the second is 780. The bigger part is :

Method 1 to solve the question.
Let us assume the numerator be x.
Then the denominator will be 3x + 2.
According the question,
if the numerator as well as denominator is increased by one , the fraction becomes 1/3,
(x + 1)/(3x + 2 + 1) = 1/3
? (x + 1)3 = 1(3x + 2 + 1)
? 3x + 3 = 3x + 3
Since Left hand side equation is same to right hand side equation. So we cannot solve the equation and cannot determine the answer.

Method 2 to solve the question.
Let us assume the numerator be x and denominator be y.
According to question,
The denominator of a fraction is 2 more than thrice its numerator.
y = 3x + 2 .....................(1)
Again According to question,
if the numerator as well as denominator is increased by one , the fraction becomes 1/3.
(x + 1)/(y + 1) = 1/3
? 3(x + 1) = y + 1
? 3x + 3 = y + 1
? y = 3x + 2............(2)
Since Equation (1) and (2) is same. So we cannot determine the number.

Method 3 to solve the question.
Hit and Trail method.
Solve all option one by one as per given question.
First took Option (A) 4/13
Numerator 4 in given fraction in Option (A)
The denominator of a fraction is 2 more than thrice its numerator.
So denominator = 2 + 4 x 3 = 2 + 12 = 14
So fraction will be 4/14 So this is wrong answer.
Now took Option (B) 3/11
Numerator 3 in given fraction in Option (B)
The denominator of a fraction is 2 more than thrice its numerator.
So denominator = 2 + 3 x 3 = 2 + 9 = 11
So fraction will be 3/11 So this is right answer.

Let us assume the ten's place digit be x.
Then, the number = 10x + 3
and sum of digits = x + 3
According to question,
So, ( x + 3 ) = 1/7( 10x + 3 )
?7x + 21 =10x +3
? 3x = 18
? x = 6
The number = 10x + 3 = 10 x 6 + 3 = 60 + 3 = 63

Let us assume the number be N.
According to question,
A number when subtracted by 1/7 of itself gives the same value as the sum of all the angles of a triangle,
N - N x 1/7 = 180 ( As we know sum of all angels of a triangle is 180 .)
? N - N/7 = 180
? (7N - N)/7 = 180
? (7N - N) = 180 x 7
? 6N = 180 x 7
? N = 180 x 7/6
? N = 30 x 7
? N = 210

Suppose the number of cows = a therefore , the number of herdsmen = a
The total number of legs = Legs of cows + Legs of herdsmen ( Cow has 4 legs and herdsmen has 2 legs )
The total number of legs = a x 4 + a x 2 = 4a + 2a = 6a
The total number of heads = Heads of cows + Heads of herdsmen ( Cow has 1 head and herdsmen has also 1 head )
The total number of heads = a + a = 2a
According to question,
The total number of legs was 28 less than four times the number of heads,
6a = 4 x 2a - 28
? 8a - 28 = 6a
? 8a - 6a = 28
? a = 28/2 = 14

Let us assume the ten's digit of the number is n. Then the unit's digit will be 3n.
According to question,
The sum of the digits is equal to 8.
sum of the digit = 8
n + 3n = 8
? 4n = 8
? n = 2
So, ten's digit is 2 and unit's digit is 6.
So the number = 10n + 3n = 10 x 2 + 3 x 2 = 20 + 6 = 26.

Let number of notes of each denomination be N.
According to question,
A man has ?480 in the denominations of 1 rupee notes , 5 rupee notes and 10 rupee notes.
Total value of notes = 480
1 x N + 5 x N + 10 x N = 480
? N + 5N + 10N = 480
? 16N = 480
? N= 30.
Hence, total number of notes = 3N = 3 x 30 = 90

Let us assume the one adult fare be a and one child fare be c.
According to question,
Bus fare of one Adult is six times the fare of one child.
a = 6c ........................(1)
If an adult's bus fare is 114.
a = 114
put the value of a in equation (1). We will get
6c = 114
? c = 114/6
? c = 19
Now we have to calculate amount paid by 4 adults and 5 children for Bus fare
Bus fare of 4 adults and 5 child = 4a + 5c
Put the value of a and c in above equation. We will get
? Bus fare of 4 adults and 5 child = 4 x 114 + 5 x 19
? Bus fare of 4 adults and 5 child = 456 + 95
? Bus fare of 4 adults and 5 child =?/- 551

The present ages of two persons are 36 and 50 years respectively
After n years the age of both persons will be 36 + n and 50 + n respectively.
According to question,
After n years the ratio of their ages will be 3:4,
(36 + n)/(50 + n) = 3/4
? (36 +n) x 4 = 3 x (50 + n)
? 36 x 4 + 4 x n = 50 x 3 + 3 x n
? 144 + 4n = 150 + 3n
? 4n - 3n =150 -144
? n = 6

Let us assume the number of brown socks = B
Let us assume the price of brown socks = ? P per pair
Then price of black socks = ? 2P per pair
Before interchanging the shocks the amount = amount of black shocks + amount of brown shocks
? Before interchanging the shocks the total amount = 4 x 2P + B x P
? Before interchanging the shocks the total amount = 8P + BP
After interchanging the shocks the total amount = amount of black shocks + amount of brown shocks
? After interchanging the shocks the total amount = 4 x P + B x 2P
? After interchanging the shocks the total amount = 4P + 2BP
According to question,
After interchanging the shocks the total amount = Before interchanging the shocks the amount + Before interchanging the shocks the amount x 50 %
4P + 2BP = 8P + BP + (8P + BP ) x 50%
? 4P + 2BP = 8P + BP + (8P + BP ) x 50/100
? 4P + 2BP = 8P + BP + (8P + BP ) x 1/2
? 4P + 2BP = ( 16P + 2BP + 8P + BP ) x 1/2
? (4P + 2BP) x 2 = ( 16P + 2BP + 8P + BP )
? 8P + 4BP = 24P + 3BP
? 8P + 4BP = 24P + 3BP
? 4BP - 3BP = 24P - 8P
? BP = 16P
? B = 16
The ratio of the number of black and brown pairs of socks = Number of block shocks / Number of Brown Shocks
? The ratio of the number of black and brown pairs of socks in the original order = 4/16 = 1/4 = 1: 4

Let total bill would be ? P
Each one will to pay = ? P/11
10 friends could pay 10 x 60 = ? 600
According to question
600 + P/11 + 50 = P
? 650 = P - P/11
? 650 = (11P - P)/11
? 650 x 11 = (11P - P)
? 650 x 11 = 10P
? P = 11 x 65
? P = 715
Amount paid by 11^th friend = 715/11 + 50 = ? 115