Question

the difference of the ages of son and father is 28 years the difference of square of their ages is 1848.Find the ages of son and father
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Answer 1

Answer:

Father is 47 years old and Son is 19 years old.

Step-by-step explanation:

Let the ages of father and son be x and y respectively.

According to the first condition,

Difference between the ages of father and son = 28

⇒ x - y = 28

⇒ x = 28 + y ------- (Eq. 1)

__________________

Second condition:

Difference of square of their ages is 1848.

⇒ (x)² - (y)² = 1848

⇒ (28 + y)² - y² = 1848 (Substitute Eq.1)

⇒ [(28)² + y² + 2(28)(y)] - y² = 1848 ∵ [(a + b)² = a² + b² + 2ab]

⇒ 784 + y² + 56y - y² = 1848

⇒ 56y + 784 = 1848

⇒ 56y = 1848 - 764

⇒ 56y = 1064

⇒ y = 1064/56

⇒ y = 19

Son's age = 19 years.

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★ Father's age :

⇒ x = 28 + y

⇒ x = 28 + 19

⇒ x = 47

Father's age = 47 years

Therefore, Father is 47 years old and Son is 19 years old.

Answer 2

❍ Let's Consider the age of father and son be a yrs & b yrs , respectively.

⠀⠀⠀⠀ CASE I : The difference of the ages of son and father is 28 years .

$\qquad :\implies \sf Age \:of \:Father \: - \: Age \:of \: Son \: = 28 \:$

$\qquad :\implies \sf a \: - \: b \: = 28 \:$

$\qquad :\implies \bf a \: = \: 28 + b \:\qquad \qquad \bigg\lgroup \sf{ Equation \: 1 \:\:}\bigg\rgroup$

⠀⠀⠀⠀CASE II : The difference of square of their ages is 1848 .

$\qquad :\implies \sf ( Age \:of \:Father)^2 \: - \: (Age \:of \: Son)^2 \: = 28 \:$

$\qquad :\implies \sf ( a)^2 \: - \: (b)^2 \: = 28 \:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: Eq.1 \:\: \::}}$

$\qquad :\implies \sf ( a)^2 \: - \: (b)^2 \: = 28 \:$

$\qquad :\implies \sf ( 28 + b )^2 \: - \: (b)^2 \: = 28 \:$

$\dag\:\:\it{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\bigg\lgroup \sf{ Algebraic \:Indentity \:\:\:: (a + b)^2 = a^2 + b^2 + 2ab }\bigg\rgroup$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Applying \: this \: Algebraic \: Indentity \::}}$

$\qquad :\implies \sf ( 28 + b )^2 \: - \: (b)^2 \: = 1848 \:$

$\qquad :\implies \sf 28^2 + b^2 + 2 \times 28 \times b \: - \: (b)^2 \: = 1848 \:$

$\qquad :\implies \sf 28^2 \cancel{+ b^2} + 2 \times 28 \times b \: \cancel{- \: (b)^2} \: = 1848\:$

$\qquad :\implies \sf 28^2 + 2 \times 28 \times b \: \: = 1848 \:$

$\qquad :\implies \sf 784 + 2 \times 28 \times b \: \: = 1848 \:$

$\qquad :\implies \sf 784 + 56b \: \: = 1848 \:$

$\qquad :\implies \sf 56b \: \: = 1848 - 784\:$

$\qquad :\implies \sf 56b \: \: = 1064\:$

$\qquad :\implies \sf b \: \: = \cancel{\dfrac{1064}{56}}\:$

$\qquad :\implies \frak{\underline{\purple{\:b= 19 \:yrs }} }\bigstar$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \:value \:of \:b \:in \:Eq.1 \: \:\: \::}}$

$\qquad :\implies \bf a \: = \: 28 + b \:\qquad \qquad \bigg\lgroup \sf{ Equation \: 1 \:\:}\bigg\rgroup$

$\qquad :\implies \sf a \: = \: 28 + 19 \:$

$\qquad :\implies \frak{\underline{\purple{\:a = 47 \:yrs }} }\bigstar$

Therefore,

• Son's age is : b = 19 yrs
• Father's age is : a = 47 yrs

⠀⠀⠀⠀⠀$\therefore {\underline{ \sf \:Hence,\:The \:age \:of\:Son \:and \:Father \:are\:\bf 19 \:yrs \: \& \: 47 \:yrs \: \sf , respectively . \:}}$

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