**Example 1 :**

Martin is four times as old as his brother Luther at present. After 10 years he will be twice the age of his brother. Find their present ages.

(A) 10 years , 3 years (B) 5 years , 10 years

(C) 20 years , 15 years (D) 8 years , 4 years

**Solution :**

Let "x" be the present age of Luther

Martin's age = 4 ⋅ Luther's age

Martin's age = 4x

After 10 years :

Age of Luther = x + 10

Age of Martin = 4x + 10

After 10 years he will be twice the age of his brother.

4 x + 10 = 2 (x + 10)

4 x + 10 = 2 x + 20

4x - 2x = 20 - 10

2x = 10

x = 5

Age of Luther = 5 years

Age of Martin = 4 (5) ==> 20 years

So, the correct option is (B) 5 years and 20 years.

**Example 2 :**

A father is 30 years older than his son,and one year ago he was four times as old as his son. Find the present ages of his father and his son.

**Solution :**

Let x be the present age of son

Father's present age = x + 30

One year ago son's age = x - 1

One year ago father's age = x + 30 - 1 ==> x + 29

father's age = 4 (Age of Son)

x + 29 = 4 (x - 1)

x + 29 = 4x - 4

x - 4x = - 4 - 29

-3x = -33

x = 33/3

x = 11

Present age of son = 11 years

present age of father = x + 30

= 11 + 30

= 41 years

So, the correct option is (A) 11 years and 4 years.

**Example 3 :**

The ages of Abraham and Adam are in the ratio 5:7. If Abraham was 9 years older and Adam 9 years younger, the age of Abraham would have been twice of Adam. Find the present ages of them.

(A) 12 years , 15 years (B) 8 years , 15 years

(C) 15 years , 21 years (D) 4 years , 14 years

**Solution :**

5x and 7x are the ages of Abraham and Adam

If Abraham was 9 years older then,

Abraham's age = 5x + 9

If Adam was 9 years younger then,

Adam's age = 7x - 9

The age of Abraham would have been twice of Adam.

5x + 9 = 2 (7x - 9)

5x + 9 = 14x - 18

5x - 14x = -18 - 9

-9x = -27

x = 3

5x = 5(3) ==> 15 years

7x = 7(3) = 21 years

Hence the present ages of Abraham and Adam are 15 years and 21 years respectively.

So, the correct option is (C) 15 years , 21 years

**Example 4 :**

Liza's mother is four times as old as Liza. After five years her mother will be three times as old as she will be then .What is their present ages.

(A) 10 years , 40 years (B) 8 years , 32 years

(C) 5 years , 20 years (D) 4 years , 16 years

**Solution :**

Let x and 4x be the age of Liza and her mother respectively.

After 5 years :

Age of Liza = x + 5

Liza's mother = 4x + 5

Her mother will be three times as old as her.

So,

4x + 5 = 3 (x + 5)

4x + 5 = 3x + 15

4x - 3x = 15 - 5

x = 10

Present age of Liza = 10 years

Present age of Liza's mother = 4(10) ==> 40 years

So, the correct option is (A) 10 years , 40 years

**Example 5 :**

The sum of the present ages of kiran and Kate is 60 years. If the ratio of their present ages be 7:8, find their present age.

**Solution :**

Sum of their ages = 60

7x and 8x be the ages of Kiran and Kate respectively.

7x + 8x = 60

15x = 60

x = 60/15

x = 4

Age of Kiran = 7x => 7 (4) ==> 28 years

Age of Kate = 8x => 8 (4) ==> 32 years

So, the correct option is (A) 28 years , 32 years

**Example 6 :**

Andrea is three times as old as her sister Anu. Three years ago, she was two years less than four times the age of her sister. Find their present ages.

**Solution :**

Let x and 3x be the age of Anu and Andrea respectively.

Age of Anu = x - 3

Age of Andrea = 3x - 3

Age of Andrea = two years less than four times the age of Anu

3x - 3 = 4(x - 3) - 2

3x - 3 = 4x - 12 - 2

3x - 3 = 4x - 14

3x - 4x = -14 + 3

-x = - 11

x = 11

3x = 33

So, the correct option is (C) 33 years , 11 years.

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