Question

the sum of ages of father and daughter is 52 years. six years ago, the product of their ages were 175 find their present ages

Answer 1

$\large \clubs \:  \bf Given :  -$

• The sum of ages of father and daughter is 52 years.

• Six years ago, the product of their ages was 175.

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$\large \clubs \:  \bf To \: Find :  -$

• Their Present Ages

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$\large \clubs \:  \bf Solution :  -$

Let,

• Present Age of Father = x years

• Present Age of Daughter = y years

✏ According To Question :

$\text{Sum of Their Ages = 52 years}$

$:\longmapsto \text{x + y = 52}$

$:\longmapsto \bf x = 52 - y \: - - - (1)$

《 6 Years Ago 》

• Age of Father = (x - 6) years

• Age of Daughter =(y - 6) years

✏ According To Question :

$\red{ (\text x - 6)(\text y - 6) = 175}$

$:\longmapsto\text x\text y - 6\text x - 6\text y + 36 = 175$

$:\longmapsto\text x\text y - 6\text x - 6\text y = 175 - 36$

$:\longmapsto \bf xy - 6x - 6y = 139 \: - - - (2)$

✏ Putting (1) in (2) :

$:\longmapsto(52 -\text y)\text y - 6(52 - \text y) - 6\text y = 139$

$:\longmapsto52\text y - \text y {}^{2} - 312 + \cancel{6\text y} - \cancel{6\text y} = 139$

$:\longmapsto52\text y - \text y {}^{2} = 139 + 312$

$\bf \red{:\longmapsto{{y}^{2} - 52y + 451 = 0 }}$

$:\longmapsto\text{y}^{2} - 11\text y - 41\text y + 451 = 0$

$:\longmapsto\text y(\text y - 11) - 41(\text y - 11) = 0$

$:\longmapsto(\text y - 11)(\text y - 41) = 0$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf y = 11 \: or \: y = 41} }}}$

☆ When y = 11 :-

$\text x = 52 - 11 \: \: \: \: \bf \{using \: (1) \}$

$\purple{ \Large :\longmapsto  \underline {\boxed{{\bf x = 41} }}}$

☆ When y = 41 :-

$\text x = 52 - 41$

$\purple{ \Large :\longmapsto  \underline {\boxed{{\bf x = 11} }}}$

As x is Age father and y is age of daughter

$\LARGE \orange{\bf\therefore\:\: x > y}$

Hence,

$\purple{ \large \underline {\boxed{{\bf x = 41 \: \: and \: \: y =11 } }}}$

Therefore,

$\pink{\begin{cases} \bf Age \: of \: Father = 41 \: years \\ \\ \bf Age \: of \: Daughter = 11years \end{cases}}$

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Answer 2

$\overline{\underline{\boxed{\sf GIVEN \darr}}}$

the sum of ages of father and daughter is 52 years. six years ago, the product of their ages were 175 find their present ages.

$\overline{\underline{\boxed{\sf SOLUTION \darr}}}$

• Here we are given that the age of father and his daughter is 52 which will be solved by simultaneous linear equation. Six years ago the product of their ages was 52 which will basically done by factorisation. After that putting the values in place the both the variables we can get their present ages.

Let us assume :

Father's age be a

His daughter's age be b

According to the question :

Sum of their ages is 52

It means that a and b is 52

So, the equation is :

• a + b = 52

• b = 52 - a ----- eq(1)

6 years ago :

• Father's age = a - 6

• Daughter's age = b - 6

Now, we know that the product of their ages is 175

• (a - 6)(b - 6) = 175

• a(b - 6) - 6(b - 6) = 175

• ab - 6a - 6b + 36 = 175

• ab - 6a - 6b = 175 - 36

• ab - 6a - 6b = 139 ------ eq(2)

Now, putting the values that we got from eq(1)

• a(52 - a) - 6a - 6(52 - a) = 139

• 52a - a² - 6a - 312 + 6a = 139

(+ 6a) and (- 6a) will be cancelled due to opposite signs

• 52a - a² - 312 = 139

Transposing - 312 to the other side we get 312

• 52a - a² = 139 + 312

• 52a - a² = 451

Now, again transposing 451 to the other side we get (- 451)

• 52a - a² - 451 = 0

Again, we need to factorise

• a² - 52a + 451 = 0

• a² - (41 + 11)a + 451 = 0

• a² - 41a - 11a + 451 = 0

Taking common

• a(a - 41) - 11(a - 41) = 0

Now, taking (a - 41) as common

• (a - 41)(a - 11) = 0

Here, we got 2nd Conditions

1st Condition :

⟼ a - 41 = 0

⟼ a = 41

2nd Condition :

⟼ a - 11 = 0

⟼ a = 11

Therefore,

If father's age = 11 then

b = 52 - 11

b = 41

Daughter's age is 41 but it isn't possible because we know that :

father's age must be more then that of his daughter's

So, 11 will be rejected because of the reason above and 41 will be accepted

So,

Father's age = a

Father's age = 41 years

Using eq(1) we will find daughter's age

• b = 52 - a

• b = 52 - 41

• b = 11

Hence, his daughter's age is 11 years