Question

the sum of the ages of a father and his son is 45 years . 5 years ago the product of their ages was 34 .find the present age​

Answer 1

Gɪᴠᴇɴ :-

• Sum of the ages = 45 years
• Product of ages = 34 ( 5-years ago)

Tᴏ Fɪɴᴅ :-

• Present ages of Father and Son

Sᴏʟᴜᴛɪᴏɴ :-

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

then,

➣According to 1st conditions :-

• Sum of ages = 45 years
• Father's age + Son's age = 45

• x + y = 45. ---(1)

➣According to 2nd condition :-

• Product of ages = 34 years (5-years ago)

• Age of Father ( 5 years ago) = (x - 5) yrs.

• Age of Son ( 5-years ago) = (y - 5) yrs

Now,

➩(x - 5)(y - 5) = 34

➩ xy - 5x - 5y +25 = 34

➩ xy - 5 (x + y) = 34 - 25

➩ (45 - y)y - 5(45) = 9. ---(using-1)

➩ 45y - y² - 225 = 9

➩ 45y - y² - 234 = 0

➩ y² - 45y + 234 = 0

➩ y² - 39y - 6y + 234 = 0

➩ y(y - 39) -6 (y - 39) = 0

➩ (y - 6)(y - 39) = 0

➩ y = 6 or y = 39

As we know that father's age must be greater than son , so we take the smaller value of y.

Put y = 6 in (1) we get,

➩ x + y = 45

➩ x + 6 = 45

➩ x = 45 - 6

➩ x = 39

Hence,

• Father's age = x = 39 years.
• Son's age = 6 years.

Answer 2

Given :-

Sum of age of father and son = 45

Product of their age = 34

To Find :-

Their present age

Solution :-

Let the present age of father be x and sons age be y

$\sf \: x + y = 45(eq \: 1)$

Now,

Product of their ages 5 years ago = 34

Age of father = (x - 5)

Age of son = (y - 5)

$\sf \: (x - 5)(y - 5) = 34$

$\sf \: xy - 5x - 5y + 25= 34$

$\sf \: xy - 5(x + y) + 25 = 34$

$\sf \: xy - 5(x + y) = 34 - 25$

$\sf \: xy - 5(x + y) = 9$

(45 - y)y - 5(45) = 9. (Eq-1)

45y - y² - 225 = 9

45y - y² - 234 = 0

y² - 45y + 234 = 0

y² - 39y - 6y + 234 = 0

y(y - 39) -6 (y - 39) = 0

(y - 6)(y - 39)

Age of son = 6 or 39

But, Son is smaller so, Age is 6 years

Now,

x + y = 45

x + 6 = 45

x = 39

Father's age = 39 years