Question

b)
Two years ago, the father's age was the square of his son's age. Two years
hence, his age will be four times the age of his son. Find their present ages.​

Answer 1

Answer:

The present age of the father = 38

The present age of the son  = 8

Step-by-step explanation:

Given,

Two years ago, the father's age was the square of his son's age

After Two years, the father's age will be four times the age of his son.

To find,

The present age of father and son

Solution:

Let 'x' be the present age of the father and 'y' be the present age of the son

Two years ago

Father's age = x -2 and Son's age = y-2

Two years later

Father's age = x+2 and son's age = y+2

Since Two years ago, the father's age was the square of his son's age, we have

x-2 = (y-2)²

x-2 = y² - 4y +4

x =  y² - 4y +6 -------------------(1)

Since After Two years, the father's age will be four times the age of his son, we have

x+2 = 4(y+2)

x+2 = 4y+8

x = 46 +8 -2 = 4y+6

x = 4y+6 -----------------------(2)

Comparing equations (1) and (2) we get

y² - 4y +6 =  4y+6

y² - 4y +6 -4y- 6 = 0

y² - 8y  = 0

y(y-8) = 0

y = 0, or y = 8

Since y cannot be zero, we have y =8

The present age of the son = 8

The present age of father = x = 4×8+6 = 32+6 = 38

∴The present age of the father = 38 and the Present age of the son  = 8

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Answer 2

Answer:

The present age of the son is 8 years and the present age of the father is 38 years.

Step-by-step explanation:

Step 1 of 2

Let the present age of father be $x$.

And the present age of son be $y$.

According to the question,

$2$ years ago, the father$'s$ age was the $square$ of his son$'s$ age.

⇒ (x - 2) = (y - 2)²

Simplify the equation as follows:

⇒ x - 2 = y² + 4 - 4y

⇒ x = y² + 4 - 4y + 2

⇒ x = y² - 4y + 6 . . . . . (i)

Also,

$2$ years later, the father$'s$ age will be $4$ times the age of his son.

⇒ (x + 2) = 4 $\times$ (y + 2)

⇒ x $+$ 2 = 4y $+$ 8

⇒ x = 4y + 6 . . . . . (ii)

Step 2 of 2

Solve the equation (i) and (ii).

Equate the equation (i) and (ii), we get

y² - 4y + 6 = 4y + 6

y² - 4y + 6 - 4y -6 = 0

y² - 8y = 0

y(y - 8) = 0

y = 0 or y = 8

Notice that the age cannot be 0. So, the case y = 0 is not possible.

Substitute the value of $y$ in equation (ii), we get

x = 4(8) + 6

x = 32 + 6

x = 38

Thus, the present age of the son is 8 years.

And the present age of the father is 38 years.

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