Question

The difference between age of a mother and her daughter is 24 .The sum of the reciprocal of their ages is 1/9 .Find the mother age

Answer 1

Answer:

The age of the mother is 36 and the age of the daughter is 12.

Step-by-step explanation:

Let the age of the mother be $x$ and the age of the daughter be $y$. Since the difference between their ages is 24, we have

$x-y=24$

Also, reciprocal of their ages is $\frac{1}{9}$ . So

$\dfrac{1}{x}+\dfrac{1}{y} = \dfrac{1}{9}$

We can rewrite the above equation as,

$\implies \dfrac{y+x}{xy} = \dfrac{1}{9}\\\\\implies 9(x+y)=xy$

From the first equation, we obtain

$x=y+24$

Substituting this in the second equation,

$\implies 9(24+y+y)=(24+y)y\\\\\implies 216+18y=24y+y^2\\\\\implies y^2+24y-18y-216=0\\\\\implies y^2+6y-216=0$

We can factorize this as

$\implies (y+18)(y-12)=0\\\\\implies y+18=0\ \ \ \ or \ \ \ \ y-12=0\\\\\implies y=-18\ \ \ \ or \ \ \ \ y=12$

Since age cannot be negative, we can take the value of $y$ as 12.

So the age of the daughter is 12.

So the age of the mother is,

$x=y+24=12+24=36$

Answer 2

Assumption

• Assume that let the age of mother be x

• Let the age of daughter be y

Given That

• The difference between theirs age is 24 years

So

$\rm \longrightarrow \: x - y = 24 \: \: \: ..(1)$

We can also write this as

$\rm \longrightarrow \: x = y + 24$

We know that

• The sum of the reciprocal of their ages is 1/9

So :

$\rm \implies \: \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{9}$

$\rm \implies \: \dfrac{x + y}{x \: y} = \dfrac{1}{9}$

$\rm \implies \: 9 \: (x + y )= x \: y$

• Now From equation (1)

$\rm \implies \: 9 \: (y + 24+ y )= (y + 24) \: y$

$\rm \implies \: 9 \: (2 \: y + 24 )= {y}^{2} + 24 \: y$

$\rm \implies \: 18 \: y +216= {y}^{2} + 24 \: y$

$\rm \implies {y}^{2} +24y-18y-216=0$

$\rm\implies y^2+6y-216=0$

• By using Factorization method we get

$\rm \implies \: {y}^{2} + 18\: y - 12 \: y - 216= 0$

$\rm \implies \: y \: (y - 12) + 18\: (y - 12) = 0$

$\rm \implies (y+18)(y-12)=0$

$\rm \implies y+18=0\ \ \ \ or \ \ \ \ y-12=0$

$\rm \implies y= - 18 \: or \: y=12 \:$

Note :

• The age can not be negative

$\bf\therefore\: y=12 \:$

Now :

➡ By using substitution method place value of y = 12 in x = y +24

$\rm \implies \: x = (12) + 24$

$\bf\therefore\: x=36 \:$

Therefore :

➡ The age of the mother is 36 years

Verification :

• It is mentioned in question that the difference between age of a mother and her daughter is 24

$\rm \implies \: Difference \: of \: their \: age = \:Age \: of \: Mother - Age \: of \: daughter$

$\rm \implies 24 = \:x - y$

• Substitute the obtained values of x and y

$\rm \implies 24 = \:36- 12$

$\rm \implies 24 = 24$

$\rm \implies L.H.S = R.H.S$

Hence Proved ❤️