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
