Question

The sum of the reciprocals of Anjali's age 3 year ago and 5 year from now is 1/3. Find the present age of Anjali.

Answer 1

$\boxed{ \boxed{ \bf SOLUTION}}$

Let present age of Anjali be x year.

$\therefore \: \rm \: Anjali's \: age \: 3 \: year \: ago = (x - 3) \: year$

$\rm \: and \: Anjali's \: age \: 5 \: year \: from \: now = (x + 5) \: year$

According to the question,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \: \frac{1}{x - 3} + \frac{1}{x + 5} = \frac{1}{3} \implies \: \frac{x + 5 + x - 3}{(x - 3)(x + 5)} = \frac{1}{3}$

$\implies \: \rm \: \frac{2x - 2}{ {x}^{2} - 3x + 5x - 15} = \frac{1}{3}$

$\implies \: \rm \: 3(2x + 2) = {x}^{2} + 2x - 15$

$\implies \: \rm \: 6x + 6 = x {}^{2} + 2x - 15$

$\implies \: \rm \: {x}^{2} + 2x - 15 - 6x - 6 = 0 \implies \: {x}^{2} - 4x - 21 = 0,$

which is the required quadratic equation.

Now, by factorisation method, we get

$\rm \: {x}^{2} - 7x + 3x - 21 = 0$

$\implies \: \rm \: x(x - 7) + 3(x - 7) = 0 \implies \: (x - 7)(x + 3) = 0$

$\implies \: \rm \: x - 7 = 0 \: or \: x + 3 = 0 \implies \: x = 7 \: or \: x = - 3$

But x = –3 is not possible because age cannot be negative.

$\therefore \: \rm \: \: x = 7$

$\rm \: Hence, \: Anjali's \: present \: age \: is \: \underline{ \bf{ \red{7 \: year}}}.$

Answer 2

Answer:

Let present age of Anjali be x year.

\begin{gathered} \therefore \: \rm \: Anjali's \: age \: 3 \: year \: ago = (x - 3) \: year \\ \end{gathered}

∴Anjali

′

sage3yearago=(x−3)year

\rm \: and \: Anjali's \: age \: 5 \: year \: from \: now = (x + 5) \: yearandAnjali

′

sage5yearfromnow=(x+5)year

According to the question,

\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \: \frac{1}{x - 3} + \frac{1}{x + 5} = \frac{1}{3} \implies \: \frac{x + 5 + x - 3}{(x - 3)(x + 5)} = \frac{1}{3}

x−3

1

+

x+5

1

=

3

1

⟹

(x−3)(x+5)

x+5+x−3

=

3

1

\implies \: \rm \: \frac{2x - 2}{ {x}^{2} - 3x + 5x - 15} = \frac{1}{3}⟹

x

2

−3x+5x−15

2x−2

=

3

1

\implies \: \rm \: 3(2x + 2) = {x}^{2} + 2x - 15⟹3(2x+2)=x

2

+2x−15

\implies \: \rm \: 6x + 6 = x {}^{2} + 2x - 15⟹6x+6=x

2

+2x−15

\begin{gathered} \implies \: \rm \: {x}^{2} + 2x - 15 - 6x - 6 = 0 \implies \: {x}^{2} - 4x - 21 = 0, \\ \end{gathered}

⟹x

2

+2x−15−6x−6=0⟹x

2

−4x−21=0,

which is the required quadratic equation.

Now, by factorisation method, we get

\rm \: {x}^{2} - 7x + 3x - 21 = 0x

2

−7x+3x−21=0

\begin{gathered} \implies \: \rm \: x(x - 7) + 3(x - 7) = 0 \implies \: (x - 7)(x + 3) = 0 \\ \end{gathered}

⟹x(x−7)+3(x−7)=0⟹(x−7)(x+3)=0

\begin{gathered} \implies \: \rm \: x - 7 = 0 \: or \: x + 3 = 0 \implies \: x = 7 \: or \: x = - 3 \\ \end{gathered}

⟹x−7=0orx+3=0⟹x=7orx=−3

But x = –3 is not possible because age cannot be negative.

\therefore \: \rm \: \: x = 7∴x=7

\rm \: Hence, \: Anjali's \: present \: age \: is \: \underline{ \bf{ \red{7 \: year}}}.Hence,Anjali

′

spresentageis

7year

.

Step-by-step explanation:

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