Question

A mother is three times as old as her daughter. Ten years later, the mothers age will be two times as old as her daughter. The sum of their present ages in years is​

Answer 1

Given that, Mother is three times as old as her daughter. Ten years later, the mother's age will be two times as old as her daughter.

Let's say, that the daughter's present age be x years.

Then, mother's age = 3x years.

After ten years,

• Age of daughter = (x + 10) years
• Age of mother = (3x + 10) years

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$\underline{\bigstar\:{\pmb{\frak{According\; to \;the\; given \;Question :}}}}$

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• Ten years later, the mother's age will be two times as old as her daughter.

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$\dashrightarrow\sf Mother's\; age = 2\Big\{Daughter's\:age\Big\}\\\\\\\dashrightarrow\sf 3x + 10 = 2(x + 10) \\\\\\\dashrightarrow\sf 3x + 10 = 2x + 20 \\\\\\\dashrightarrow\sf 3x - 2x = 20 - 10\\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{x = 10}}}}\;\bigstar$

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Therefore,

• Present age of Daughter is, x = 10 years.
• Present age of mother is, 3x = (3) × (10) = 30 years.

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Hence, present age of Mother and Daughter is 30 years & 10 years respectively.

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¤ Now, the sum of their present ages (Mother's present age & daughter's present age) is —

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$\twoheadrightarrow\sf \Big\{Mother's\; age\Big\} + \Big\{Daughter's\; age\Big\} \\\\\\\twoheadrightarrow\sf 30\; years + 10\;years \\\\\\\twoheadrightarrow\underline{\boxed{\frak{\pmb{\purple{40\:years}}}}}\;\bigstar$

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$\therefore{\underline{\textsf{Hence, sum of their present ages is \textbf{40 years}.}}}$

Answer 2

Given : A mother is three times as old as her daughter & Ten years later, the mothers age will be two times as old as her daughter.

Exigency To Find : The sum of their present ages in years ?

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❍ Let's Assume Present age of daughter's be a yrs .

⠀⠀⠀⠀⠀Given that ,

⠀》A mother is three times as old as her daughter.

• Present age of mother is 3(a) yrs .

⠀⠀⠀⠀⠀☆ After 10 yrs their ages :

• Age of Daughter will be ( a + 10 ) yrs .
• Age of Mother will be ( 3a + 10 ) yrs .

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:According \: to \: \: the \: Question \::}}$

⠀⠀⠀━━━ Ten years later, the mothers age will be two times as old as her daughter.

$\qquad:\implies \sf Mother's \:age_{(\:After \:10\:yrs\:)}\:=\:2 \bigg( \:Daughter's \:age_{(\:After \:10\:yrs\:)}\:\bigg) \:\\\\\qquad:\implies \sf ( \: 3a + 10 \:) \:=\:2 \bigg( \:a + 10 \:\bigg) \:\\\\\qquad:\implies \sf \: 3a + 10 \:\:=\:2a + 20 \: \:\\\\\qquad:\implies \sf \: 3a \:\:=\:2a + 20 - 10 \: \:\\\\\qquad:\implies \sf \: 3a \:\:=\:2a + 10 \: \:\\\\\qquad:\implies \sf \: 3a - 2a \:\:=\: 10 \: \:\\\\\qquad:\implies \sf \: a \:=\:10 \: \:\\\\\qquad \therefore \pmb{\underline{\purple{\frak{\:a\: = 10\:yrs }}} }\:\:\bigstar$

Therefore,

• Present age of Daughter is , a= 10 yrs
• Present age of Mother is , 3a = 3(10) = 30 yrs

$\:\therefore \:\:\underline {\sf Present \:ages \:of \:Mother \:and \:Daughter \:are \: \bf 10 \:yrs \:\sf and \:\bf 30 \:yrs \: \sf ,\:respectively \:.}$

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$\qquad \dag \:\: \underline { \sf Finding \:Sum \:of \:their \:Present \:ages \: \:}:\:$

$\qquad \dashrightarrow \sf Sum \:of\:Present \:age _{(Mother \:\&\: Daughter \: )} =\:\bigg( Daughter's\:Present \:age\: \bigg) + \bigg( Mother's \:Present \:age\: \bigg)$

$\qquad \dashrightarrow \sf Sum \:of\:Present \:age _{(Mother \:\&\: Daughter\: )} =\:\bigg( a \: \bigg) + \bigg( 3a \bigg)$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: \: Value\:of\:a \::}}\\\\\qquad \dashrightarrow \sf Sum \:of\:Present\:age _{(Mother \:\&\: Daughter \: )} =\:\bigg( a \: \bigg) + \bigg( 3a \: \bigg) \\\\\qquad \dashrightarrow \sf Sum \:of\:Present \:age _{(Mother \:\&\: Daughter \: )} =\:\bigg( 10 \: \bigg) + \bigg( 3\times10\: \bigg) \\\\\qquad \dashrightarrow \sf Sum \:of\:Present\:age _{(Mother \:\&\: Daughter \: )} =\:\bigg( 10 \: \bigg) + \bigg( 30\: \bigg) \\\\\qquad \dashrightarrow \sf Sum \:of\:Present \:age _{(Mother \:\&\: Daughter \: )} =\: 10 \: + 30 \: \\\\\qquad \dashrightarrow \sf Sum \:of\:Present \:age _{(Mother \:\&\: Daughter \: )} =\: 40 \: \\\\\qquad \therefore \pmb{\underline{\purple{\frak{\:Sum \:of\:Present \:ages_{(Mother \:\&\: Daughter \: )} \: = 40\:yrs }}} }\:\:\bigstar$

$\therefore \underline {\sf Hence, \:The \:sum \:of \:their \:Present \:ages \:is \;\bf 40\:yrs \: .}$