**Problem 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.

**Solution : **

Let 'x' and 'y' be the present ages of Martin and Luther respectively.

**Given :** Martin is four times as old as his brother Luther at present.

Then,

x = 4y ----(1)

**Given :** After 10 years, Martin will be twice the age of his brother Luther.

Then,

x + 10 = 2(y + 10)

x + 10 = 2y + 20

Subtract 10 from each side.

x = 2y + 10 ----(2)

From (1) and (2),

4y = 2y + 10

Subtract 2y from each side.

2y = 10

Divide each side by 2.

y = 5

Substitute 5 for y in (1).

(1)----> x = 4(5)

x = 20

So, the present ages of martin and Luther are 20 years and 5 years respectively.

**Problem 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' and 'y' be the present ages of father and son respectively.

**Given :** A father is 30 years older than his son

Then,

x = y + 30 ----(1)

**Given :** One year ago, father was four times as old as his son.

Then,

x - 1 = 4(y - 1)

x - 1 = 4y - 4

Add 1 to each side.

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

From (1) and (2),

y + 30 = 4y - 3

Subtract y from each side.

30 = 3y - 3

Add 3 to each side.

33 = 3y

Divide each side by 3.

11 = y

Substitute 5 for y in (1).

(1)----> x = 11 + 30

x = 41

So, the present ages of father and son are 41 years and 11 years respectively.

**Problem 3 :**

The ages of Abraham and Adam are in the ratio 5 : 7. Four years from now, the ratio of their ages will be 3 : 4. Find the present ages of them.

**Solution : **

**Given :** The ages of Abraham and Adam are in the ratio

5 : 7

Then,

age of Abraham = 5x

age of Adam = 7x

Four years from now,

age of Abraham = 5x + 4

age of Adam = 7x + 4

**Given :** Four years from now, the ratio of their ages will be

3 : 4

Then,

(5x + 4) : (7x + 4) = 3 : 4

(5x+4) / (7x+4) = 3 / 4

4(5x + 4) = 3(7x + 4)

20x + 16 = 21x + 12

Subtract 20x from each side.

16 = x + 12

Subtract 12 from each side.

4 = x

Then

5x = 5(4) = 20

7x = 7(4) = 28

So, the present ages of Abraham and Adam are 20 years and 28 years.

**Problem 4 :**

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

**Solution : **

Let 'x' and 'y' be the present ages of Mother and Airi respectively.

**Given :** Airi's mother is four times a old as Airi.

Then,

x = 4y ----(1)

**Given :** After five years Airi's mother will be three times as old as Airi will be then.

Then,

x + 5 = 3(y + 5)

x + 5 = 3y + 15

Subtract 5 from each side.

x = 3y + 10 ----(2)

From (1) and (2),

4y = 3y + 10

Subtract 3y from each side.

y = 10

Substitute 10 for y in (1).

(1)----> x = 4(10)

x = 40

So, the present ages of Airi's mother and Airi are 40 years and 10 years respectively.

**Problem 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 : **

Let 'x' and 'y' be the present ages of Kiran and Kate respectively.

**Given :** The ratio of the present ages of Kiran and Kate is

7 : 8

Then,

present age of Kiran = 7x

present age of Kate = 8x

**Given :** The sum of the present ages of Kiran and Kate is 60 years.

Then,

7x + 8x = 60

15x = 60

Divide each side by 15.

x = 4

Then,

7x = 7(4) = 28

8x = 8(4) = 32

So, the present ages of Kiran and Kate are 28 years and 32 years respectively.

**Problem 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 'y' be the present ages of Andrea and Anu respectively.

**Given :** Andrea is three times as old as her sister Anu.

x = 3y ----(1)

**Given :** Three years ago, Andrea was two years less than four times the age of her sister Anu.

Then,

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

x - 3 = 4y - 12 - 2

x - 3 = 4y - 14

Add 3 to each side.

x = 4y - 11 ----(2)

From (1) and (2),

3y = 4y - 11

Subtract 3y from each side.

0 = y - 11

Add 11 to each side.

11 = y

Substitute 11 for y in (1).

(1)----> x = 3(11)

x = 33

So, the present ages of Andrea and Anu are 33 years and 11 years respectively.

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