Question

Genius Where Are you ?


Q.1} The age of a man is 4 times the sum of the ages of his two sons. Ten years hence, his age will be double of the sum of the ages of his sons. The father's present age is - ?

Q.2} The sum of the digits of a two-digit number is 9. When the digits are reversed, the number is increased by 9. Find the number.

Answer 1

$\huge{\bf{\underline{\pink{Question - 1)}}}}$

Given,

• The age of a man is 4 times the sum of the ages of his two sons.
• Ten years after his age will be double of the sum of the ages of his sons.

To Find,

• The Present age of Father

Solution,

⇒Suppose the Present age of the Father be a

And , Suppose the sum of ages of two sons  be b

According to the First Condition :-

• The age of a man is 4 times the sum of the ages of his two sons.

$\implies \boxed{\sf{a = 4b}}$

According to the Second Condition :-

• Ten years after his age will be double of the sum of the ages of his sons.

$\implies \sf{a + 10 = 2(b + 20)}$

$\implies \sf{a + 10 = 2b + 40}$

$\implies \sf{a - 2b = 40 - 10}$

$\implies \sf{a - 2b = 30}$

$\implies \sf{4b - 2b = 30}$

(Put the value of a From the First Condition)

$\implies \sf{3b = 30}$

$\implies \sf{b = \dfrac{30}{2}}$

$\implies \boxed{\sf{b = 15}}$

Now Put the value of b in First Condition :-

$\implies \sf{a = 4b}$

$\implies \sf{a = 4 \times 15}$

$\implies \boxed{\sf{a = 60 }}$

Therefore,

$\boxed{\sf{\red{Present \ age \ of \ the \ Father = a \ years = 60 \ years}}}}$

$\rule{200}1$

$\huge{\bf{\underline{\pink{Question -2)}}}}$

Given,

• Sum of the two digits of a two digit number is 9.
• When the digits are reversed then the number is increased by 9.

To Find,

• Two Digit Number

Solution,

⇒Suppose the digit at the tens's place be x

And, Suppose the digit at the one's place be y

Therefore ,

• Two Digit Number = 10x + y
• Reversed Number = 10y + x

According to the First Condition :-

• Sum of the two digits of a two digit number is 9.

$\implies \sf{x + y = 9}$

$\implies \boxed{\sf{x = 9 - y}}$

According to the Second Condition :-

• When the digits are reversed then the number is increased by 9.

$\implies \sf{10x + y + 9 = 10y + x}$

$\implies \sf{10x - x + 9 = 10y - y }$

$\implies \sf{9x + 9 = 9y}$

$\implies \sf{9(9 - y) + 9 =9y}$

(Put the Value of x From the First Condition)

$\implies \sf{81 - 9y + 9 = 9y}$

$\implies \sf{90 = 9y + 9y}$

$\implies \sf{18y = 90}$

$\implies \sf{y = \dfrac{90}{18}}$

$\implies \boxed{\sf{y = 5}}$

Now Put the Value of y in First Condition :-

$\implies \sf{x = 9 - y}$

$\implies \sf{x = 9 - 5}$

$\implies \boxed{\sf{x = 4}}$

Therefore,

$\boxed{\sf{\red{The \ Two \ Digit \ Number = 10x + y = 10(4) + 5 = 45}}}$

$\rule{200}2$

Answer 2

Question 1 :

Given :-

• Age of a man = 4 times sum of ages of two sons.
• Ten years hence, his age will be double of the sum of ages of two sons.

To find :-

• Present age of father

Solution :-

Let present age of father be a years & ages of two sons be b & c years respectively.

According to Question :

⇒ a = 4(b + c)

⇒ a = 4b + 4c ------ Equation (i)

Case 2 :

⇒ (a + 10) = 2(b + 10 + c + 10)

⇒ a + 10 = 2(b + c + 20)

⇒ a + 10 = 2b + 2c + 40

⇒ a = 2b + 2c + 40 - 10

⇒ a = 2b + 2c + 30 ------ Equation (ii)

Comparing both equations :-

⇒ 4b + 4c = 2b + 2c + 30

⇒ 4b + 4c - 2b - 2c = 30

⇒ 2b + 2c = 30

⇒ 2(b + c) = 30

⇒ b + c = 30/2

⇒ b + c = 15

Putting value in Equation (i)

⇒ a = 4(b + c)

⇒ a = 4 × 15

⇒ a = 60

Therefore,

Present age of father = 60 years (Ans.)

_________________________

Question 2 :

Given :-

• Sum of digits = 9
• Digits reversed, new number increases by 9

To find :-

• Original number

Solution :-

Let the digit at unit's place be r & digit at ten's place be m.

◗ Original number = r + 10m

◗ Reversed number = m + 10r

Case 1 :

⇒ r + m = 9

⇒ r = 9 - m ------- Equation (i)

Case 2 :

⇒ m + 10r = r + 10m + 9

⇒ m + 10r - r - 10m = 9

⇒ 9r - 9m = 9

⇒ 9(r - m) = 9

⇒ r - m = 9/9

⇒ r - m = 1

• Putting value from Equation (i)

⇒ 9 - m - m = 1

⇒ -2m = 1 - 9

⇒ -2m = -8

⇒ m = -8/-2

⇒ m = 4

Now finding original number :

⇒ Original number = r + 10m

⇒ Original number = (9 - m) + 10(4)

⇒ Original number = (9 - 4) + 40

⇒ Original number = 5 + 40

⇒ Original number = 45

Therefore,

Original two-digit number = 45 (Ans.)