Question

A mother said that, her age is one year less than thrice her daughter's age. The daughter is 9 years less than the difference between their present ages. Find the sum of their ages (in years).

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Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

A mother said that, her age is one year less than thrice her daughter's age.

Let assume that Daughter age be 'x' years.

So,

Mother age be 3x - 1 years.

So, we have

$\red{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{Daughter \: age \: = \: x \: years} \\ \\ &\sf{Mother \: age \: = \: 3x - 1 \: years} \end{cases}\end{gathered}\end{gathered}}$

Again, Given that,

The daughter is 9 years less than the difference between their present ages.

$\rm :\longmapsto\:3x - 1 - x - 9 = x$

$\rm :\longmapsto\:2x - 10= x$

$\rm :\longmapsto\:2x - x= 10$

$\rm :\longmapsto\:x= 10$

$\bf\implies \:\boxed{ \tt{ \: \: x \: = \: 10\: \: }}$

Hence,

$\red{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{Daughter \: age \: = \: 10\: years} \\ \\ &\sf{Mother \: age \: = \: 29 \: years} \end{cases}\end{gathered}\end{gathered}}$

So, Sum of ages = 10 + 29 = 39 years

Alternative Method :-

Given that

A mother said that, her age is one year less than thrice her daughter's age.

Let assume that,

$\red{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{Daughter \: age \: = \: x\: years} \\ \\ &\sf{Mother \: age \: = \: y \: years} \end{cases}\end{gathered}\end{gathered}}$

So,

$\rm :\longmapsto\:\boxed{ \tt{ \: x = 3y - 1 }}- - - (1)$

Also, given that,

The daughter is 9 years less than the difference between their present ages.

$\rm :\longmapsto\:x - y = 9 + y$

$\rm :\longmapsto\:3y - 1 - y = 9 + y$

$\rm :\longmapsto\:2y - 1 = 9 + y$

$\rm :\longmapsto\:2y - y = 9 + 1$

$\rm :\longmapsto\:y = 10$

$\bf\implies \:\boxed{ \tt{ \: \: y \: = \: 10\: \: }}$

Hence,

$\red{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{Daughter \: age \: = \: 10\: years} \\ \\ &\sf{Mother \: age \: = \: 29 \: years} \end{cases}\end{gathered}\end{gathered}}$

So,

• Sum of ages = 10 + 29 = 39 years.

Answer 2

$\bf\huge\blue{ANSWER}$

$\small \text{We \: have \: to \: assume \: the \: following}$

$\bf{Let's \: the \: daughter \: age \: be \: x \: years}$

$\bf \huge \red{Given}$

$\small \text{Mother \: present \: age = (3x - 1) \: years}$

$\bf\huge\pink{Therefore}$

$\bf \orange{x = (3x - 1) - x - 9}$

$\bf \green{x = 3 \: x - 1 - x - 9}$

$\bf \purple{x = 2x - 10}$

$\bf\huge \boxed{x = 10}$

$\bf\huge\orange{Therefore}$

$\small \text{Mother's age =3(10) - 1= 29 years }$

$\small \text{Sum of mother and daughter ages}$

$\bf{= 29 + 10= 39 \: years}$