How many liters of oil at Rs. 40 per liter should be mixed with 240 liters of the second variety of oil at Rs.60 per liter so as to get a maximum whose cost is Rs.52 per liter?

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