A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is:

Method 1
Let us assume the number be 3p , 3(p+1) and 3(p+2)
According to question,
3p + 3(p+1) + 3(p+2) = 72
? 3p + 3p + 3 +3p + 6 = 72
? 9p +9 = 72
? 9p = 72 - 9
? 9p = 63
? p = 63/9 = 7
? Largest number = 3(p + 2)
Put the value of p in above equation.
? Largest number = 3 x ( 7 + 2 )
? Largest number = 3 x 9
? Largest number = 27

Method 2
Let us assume the numbers P , P + 3 ,P+ 6
According to Question,
sum of three numbers = 72
P + P + 3 + P + 6 = 72
? 3P + 9 = 72
? 3P = 72 - 9
? 3P = 72 - 9
? P = 63/3
? P = 21
So largest Number = P + 6 = 21 + 6 = 27

Let us assume 1 pen cost be ? x and 1 pencil cost be ? y.
According to question,
The Cost of 36 pens and 42 pencils is 460.
36x + 42y = 460 ...................(1)
then the cost of 18 pens and 21 pencils = 18x + 21y.
Now divide the Equation (1) by 2. We will get,
18x + 21y = 230
So the Cost of 18 pens and 21 pencils = 18x + 21y = 230

3x + 7y = 75 ..(i)
5x - 5y = 25
? x - y = 5 ...(ii)
On multiplying Eq. (ii) by 7 and adding to Eq. (i), we get
3x + 7y = 75
7x - 7y = 35
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10x = 110
x = 11
? Putting the value of x in Eq. (ii), we get
11 - y = 5
? y = 6
? x + y = 6 + 11 = 17

Given, 3^{x + y} = 81
? 3^{x + y} = 3⁴
? x + y = 4 ...(i)
and 81^{x - y} = 3 or (3⁴)^{x - y} = 3
? x - y = 1/4
On solving the Eqs. (i) and (ii), we get
2x = 17/4 ? x = 17/8

Let the number be x and y.
Then, according to the question,
x + y = 15 ...(1)
x - y = 3 ...(ii)
on adding Eqs. (i) and (ii), we get
2x = 18
? x = 9
On putting the value of in Eq. (i), we get
y = 6
? Product = xy = 54

Let us assume the digits of the original number are unit's digit a and ten's digit b.
The Original Number will be 10a + b.
After interchanging the digits the new number will be 10b + a.
According to question,
The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54.
New Number = Original Number - 54
10b + a = 10a + b - 54
? 10b + a - 10a - b = -54
? 9b - 9a = -54
? a - b = 6....................................(1)
Again according to question,
Sum of the digits of original number = 10
a + b = 10..................................................(2)
Add the equation (1) and (2), we will get
a - b + a + b = 10 + 6
2a = 16
a = 8
Put the value of a in Equation (2) , we will get
8 + b = 10
b = 10 - 8
b = 2
Put the value of a and b for original number, we will get
10a + b = 10 x 8 + 2 = 80 + 2 = 82

Suppose that the Ravi present age is A years.
According to the given question,
A/4 + A/5 + A/3 + 13 = A
? (15A + 12A + 20A)/60 = A - 13
? 47A = 60A - 780
?60A - 47A = 780
?13A = 780
? A = 780/13 = 60 years

Method 1
Let us assume the present age of mother be M and daughter be D.
According to question,
Ratio of present age of mother and daughter = 7 : 1
M / D = 7 : 1
? M / D = 7 / 1
? M / D = 7
? M = 7D ..........................(1)
Four years ago the age of mother = M - 4
Four years ago the age of daughter = D - 4
Again according to question,
Four years ago ratio of mother age and daughter age = 19 : 1
? (M - 4 ) / (D - 4) = 19 : 1
? (M - 4 ) / (D - 4) = 19
? (M - 4 ) = 19(D - 4)
? M - 4 = 19D - 76 .....................(2)
Put the value of M from equation (1) in above equation (2), we will get
7D - 4 = 19D - 76
? 76 - 4 = 19D - 7D
? 72 = 12D
? 12D = 72
? D = 72/12
? D = 6
Put the vale of D in equation (1), we will get the present age of mother.
Present age of Mother M = 7D = 7 x 6
Present age of Mother M = 42
? mother's age after 4 yrs = 42 + 4 = 46 yrs

Method 2
Let us assume the ratio factor is x.
According to question,
Ratio of present age of mother and daughter = 7 : 1
Then present age of mother = 7x and present age of daughter = x
Four years ago the age of mother = 7x - 4
Four years ago the age of daughter = x - 4
Again according to question,
Four years ago ratio of mother age and daughter age = 19 : 1
? (7x - 4 ) / (x - 4) = 19 : 1
? (7x - 4 ) / (x - 4) = 19
? (7x - 4 ) = 19(x - 4)
? 7x - 4 = 19x - 76
? 76 - 4 = 19x - 7x
? 72 = 12x
? 12x = 72
? x = 6
Present age of Mother = 7x = 7 x 6 = 42
? mother's age after 4 yrs = 42 + 4 = 46 yrs

Let the numbers be x and y.
According to question,
The ratio of two numbers is 4:7.
? x/y = 4/7
? 7x = 4y
? 7x - 4y = 0 ........................(1)
Again According to question,
If each of those numbers increased by 30, their ratio will become 5:8
(x + 30) / (y + 30) = 5/8
? 8(x + 30) = 5(y + 30)
? 8x + 240 = 5y + 150
? 8x - 5y = -90 ..........................(2)
Multiply 5 with equation (1) , we will get.
35x - 20y = 0 ..................(3)
Multiply 4 with equation (2), we will get.
32x - 20y = -360 ............(4)
Subtracts the Equation (4) from Equation (3). we will get,
35x - 20y - (32x - 20y) = 0 - (-360)
? 35x - 20y - 32x + 20y = 360
? 3x = 360
? x = 120
Put the value of x in equation (1) to get the value of y.
7x - 4y = 0
? 7(120) - 4y = 0
? 840 - 4y = 0
? 4y = 840
? y = 210
? Average of the numbers = (x + y)/2
put the vale of x and y.
? Average of the numbers = (120 + 210)/2
? Average of the numbers = 330/2
? Average of the numbers = 165

Given, 6x - 10y = 10 ..........(i)
and x/(x + y) = 5/7
? 7x = 5x + 5y
? 2x - 5y = 0 ...(ii)

On multiplying Eq. (ii) by 2 and subtracting from Ed.(i), we get
6x - 10y = 10
4x - 10y = 0
---------------------
2x = 10
? x = 5
Putting the value of x in Eq. (i), we get
30 - 10y = 10
? 10y = 20
? y = 2
? (x - y) = 5 - 2 = 3