Question

The sum of three times the age of son and the age of his father is 110. If three times the age of his father is added to the age of son we get 170. Then, (a) What is the sum of their ages? (b) Find their ages

Answer 1

$\bf{\huge{\underline{\boxed{\rm{\blue{Answer:}}}}}}$

Given :-

• the sum of three times the age of son and the age of his father is 110.
• if three times the age of his father is added to the age of son we get 170.

To find :-

• the sum of their ages
• Their individual ages

Solution :-

Let the age of son be x years.

Let the age of father be y years.

As per the first condition,

• the sum of three times the age of son and the age of his father is 110,

We form the following equatio,

3x + y = 110 ---->1

As per the second condition now,

• if three times the age of his father is added to the age of son we get 170,

We get another equation as,

x + 3y = 170 ----> 2

We will solve both equations simultaneously, first, we will add both the equations and at second time we will subtract equations.

Let's start with adding the equations. Add, equation 1 to 2,

3x + y = 110 ---->1

x + 3y = 170 ---->2

-----------------------

4x + 4y = 280

As we can infer, that 4 is a coefficient common to both the variables x and y, we will figure it out as common,

4 ( x + y) = 280

x + y = $\bf\large\frac{280}{4}$

x + y = 70 ----> 3

Now, we are done with adding the equations. Let's move on to subtracting the equations.

Subtract equation 2 from 1,

.... +3x + y = +110 ----->1

- ( +x + 3y = +170 )----->2

___ -__ -____-______

2x - 2y = -60

Here, 2 is a common coefficient for both the variables,

2 ( x - y) = - 60

x - y = $\bf\large\frac{-60}{2}$

x - y = - 30 ----->4

Now, for the last time, we will add equations 3 and 4.

x + y = 70

x - y = -30

--------------------

2x = 40

x = $\bf\large\frac{40}{2}$

x = 20

Son's age = x = 20 years.

Substitute the value of x in equation 1,

3x + y = 110 ---->1

3 ( 20) + y = 110

60 + y = 110

y = 110 - 60

y = 50

Father's age = y = 50 years.

Sum of the ages of son and father,

Son's age = 20 years = x

Father's age = 50 years = y

To find :-

• x + y

Plug the values,

Sum =>20 + 50 = 70

$\bf{\huge{\underline{\boxed{\tt{\red{Verification:}}}}}}$

First case :-

• The sum of three times the age of son and the age of his father is 110.

Son's age = x = 20

Three times age of son = 3x

Three times age of son = 3 × 20 = 60 years = 3x

Age of father = y = 50

Sum of three times age of son and father = 110

3x + y = 110

3 ( 20) + y = 110

60 + 50 = 110

110 = 110

LHS = RHS

Hence first condition is satisfied

Second case :-

• If three times the age of his father is added to the age of son we get 170.

Father's age = 50 years

Three times age of father = 3y

Three times age of father = 3 (50) = 150 years

Age of son = 20 years

Sum of their ages = 170

x + 3y = 170

20 + 150 = 170

170 = 170

LHS = RHS

Hence the second condition too is verified!

Answer 2

Answer:

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