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)
Solve the above equation to get the answer.

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
Solve the above equation.

Correct Option: C

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,
Solve the equation and get the answer.

Correct Option: D

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

Let us assume the first part is A then second part will be 50 - A.
According to question,
Sum of their reciprocals is 1/2.
1/A + 1/(50 - A) = 1/12
Solve the equation.

Correct Option: C

Let us assume the first part is A then second part will be 50 - A.
According to question,
Sum of their reciprocals is 1/2.
1/A + 1/(50 - A) = 1/12
⇒ (50 - A + A)/(50 - A)A = 1/12
⇒ (50)/(50 - A)A = 1/12
⇒ (50) x 12 = 1 x (50 - A)A
⇒ 600 = (50 - A)A
⇒ 600 = 50A - A²
⇒ A² - 50A + 600 = 0
⇒ A² - 30A - 20A + 600 = 0
⇒ A(A - 30) - 20(A - 30) = 0
⇒ (A - 30)(A - 20) = 0
⇒ (A - 30) = 0 or (A - 20) = 0
⇒ A = 20 , 30

40 notebooks at the rate of ₹ 18 per notebook and 55 pencils at the rate of ₹ 8 per pencil.
Amount paid = ₹ (40 x 18 +55 x 8)

Correct Option: B

According to question,
40 notebooks at the rate of ₹ 18 per notebook and 55 pencils at the rate of ₹ 8 per pencil.
Amount paid = ₹ (40 x 18 + 55 x 8)
Amount paid = ₹ (720 + 440)
Amount paid = ₹ 1160

Let the number be 10x + y where x > y
According to the question.
x + y = 15
and x - y = 3
Solving both the equation.

Correct Option: C

Let the number be 10x + y where x > y
According to the question.
x + y = 15 ............(1)
and x - y = 3 .........(2)
Add the Equation (1) and (2) , We will get
x + y + x - y = 15 + 3
⇒ 2x = 18
⇒ x = 9
Put the value of x in equation (1). We will get
9 + y = 15
⇒ y = 15 - 9
⇒ y = 6
So we have digits of number x = 9, y = 6
∴ x * y = 9 * 6 = 54