**Problem 1 :**

The sum of the age of a man and his son is 35 years and the
product of their ages is 150. Find their ages.

**Solution :**

Let x and y be the age of a man and his son respectively.

Sum of the age of a man and his son is 35.

x + y = 35

y = 35 – x ------ (1)

Product of their ages is 150.

x ⋅ y = 150 ---- (2)

By applying the value of y in (2), we get

x(35 – x) = 150

35x – x^{2} = 150

x - 35x + 150 = 0

x^{2} - 30x –
5x + 150 = 0

(x – 5) (x - 30) = 0

x - 5 = 0 x = 5 |
x - 30 = 0 x = 30 |

Here x represent the man’s age. So it should not be 5.

y = 35 – 30

y = 5

Therefore the age of the man is 35 and his son is 5.

**Problem 2 :**

The product of the man’s age 5 years ago and 5 years later is 600. Find his present age.

**Solution :**

Let x be the present age of man.

Age of man 5 years ago = (x – 5)

Age of man 5 years after = (x + 5)

The product of the man’s age 5 years ago and 5 years later is 600.

(x – 5) (x + 5) = 600

x^{2} – 5^{2} = 600

x^{2} – 25 = 600

x^{2 } = 600 + 25

x^{2 } = 625

x = √625

x = 25

Therefore the present age of the man is 25 years.

**Problem 3 :**

1 year ago a father was 8 times as old as his son. Now his age is square of his son’s age. Find the present age.

**Solution :**

Here the age of father is compared by his son age.

So x and y be the ages of his father and son.

One year ago his father’s age = x – 1

(x - 1) = 8(y – 1) ----- (1)

x = y² ------ (2)

By applying (2) in (1), we get

y^{2} - 8y – 1 + 8 = 0

y^{2} - 8y + 7 = 0

(y - 1) (y – 7) = 0

y - 1 = 0 y = 1 If y = 1 x = 1² x = 1 |
y - 7 = 0 y = 7 If y = 7 x = 7 x = 49 |

Therefore the present age of father and son are 49 years and 7 years respectively.

**Problem 4 :**

The sum of the age of a father and his son is 45 years. Five years ago the product of their ages was 124. Determine their present age.

**Solution :**

Let x and y be the present age of father and his son respectively.

The sum of the age of a father and his son is 45 years.

x + y = 45

y = 45 – x ------(1)

Five years ago his father’s age = x - 5

Five years ago son’s age = y – 5

Five years ago the product of their ages was 124.

(x – 5) (y – 5) = 124

(x – 5) (45 – x – 5) = 124

(x – 5) (40 – x) = 124

40x – x^{2} – 200 + 5x = 124

45x – x² – 200 = 124

x² – 45x + 124 + 200 = 0

x² – 45x + 324 = 0

(x – 9) (x – 36) = 0

x - 9 = 0 x = 9 |
x - 36 = 0 x = 36 |

Here x represent the present age of father. The possible age of father is 36.

To find the value of y, we may apply the value of x in (1), we get

y = 45 - 36

y = 9

Therefore the present age of father = 36 years.

Present age of son = 9 years.

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