Question

2 years ago the age of father was 3and half times of the age of his daughter 6 years hence the age of father will be 10 years more than the twice of the age of his daughter. find their present ages.​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

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$\green{\underline \bold{\tt{Given :-}}}$

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• 2 years ago the age of father was 3 and half times of the age of his daughter

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• 6 years hence the age of father will be 10 years more than the twice of the age of his daughter

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$\red{\underline \bold{\tt{To \: Find :-}}}$

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• Their present ages

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$\large{\orange{\underline{\tt{Solution :-}}}}$

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• Let the present age of father be "f"

• Let the present age of daughter be "d"

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$\underline{\bold{\texttt{Father's age 2yrs ago :}}}$

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⭕ f - 2

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$\underline{\bold{\texttt{Daughter's age 2yrs ago :}}}$

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⭕ d - 2

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$\purple{\underline \bold{According \: to \: the \ question :}}$

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2 years ago the age of father was 3 and half times of the age of his daughter

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$\bf\boxed { 3 \ and \ half = 3\dfrac {1 } { 2 } = \dfrac { 7 } { 2 }}$

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$\sf \longmapsto f - 2 = \dfrac { 7 } { 2 } \times (d - 2)$

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$\sf \longmapsto f - 2 = \dfrac { 7d } { 2 } - 7$

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$\sf \longmapsto f - 2 = \dfrac { 7d - 14 } { 2 }$

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$\sf \longmapsto 2f - 4 = 7d - 14$

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$\sf \longmapsto 2f - 7d = -10$ -----(1)

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$\underline{\bold{\texttt{6 yrs hence age of father :}}}$

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⭕ f + 6

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$\underline{\bold{\texttt{6 yrs hence age of daughter :}}}$

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⭕ d + 6

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6 years hence the age of father will be 10 years more than the twice of the age of his daughter

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$\sf \longmapsto f + 6 = 2(d + 6) + 10$

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$\sf \longmapsto f + 6 = 2d + 12 + 10$

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$\sf \longmapsto f + 6 = 2d + 22$

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$\sf \longmapsto f - 2d = 16$ -------(2)

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$\underline{\bold{\texttt{Multiplying equation (2) by 2 }}}$

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$\sf \longmapsto 2f - 4d = 32$ ------(3)

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$\underline{\bold{\texttt{Equation (3) - (1) }}}$

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$\sf \longmapsto 2f - 4d -( 2f - 7d ) = 32 - (-10)$

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$\sf \longmapsto 3d = 42$

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$\sf \longmapsto d = \dfrac { 42 } { 3 }$

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➳ d = 14

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$\underline{\bold{\texttt{Putting d = 14 in (2) }}}$

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➳ f - 2d = 16

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➳ f - 2(14) = 16

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➳ f - 28 = 16

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➳ f = 44

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• Hence the the present ages of daughter and father is 14 & 44 respectively