Question

7 years ago Varun's age was 5 times the age of Swati's
age after 3 years Swati's age will be 2/5 th of Varun's
age. Find the present ages of both.​

Answer 1

Answer:

• Present age of Varun = 13 yrs
• Present age of Swati = 37 yrs

Explanation:

Let, Present age of Varun = a

and, Present age of Swati = b

then, 7 years ago

Varun's age = a - 7  yrs

and Swati's age = b - 7  yrs

As given,

→ 7 yrs ago Varun's age = 5 ( 7 yrs ago Swati's age )

→ a - 7 = 5 ( b - 7 )

→ a - 7 = 5 b - 35

→ a = 5 b - 28     ____equation(1)

Now,

after three yrs from Now

age of Varun = a + 3

and, age of Swati = b + 3

as given,

→ Age of Swati after 3 yrs = 2/5 ( Age of Varun after 3 yrs )

→ b + 3 = 2/5  ( a + 3 )

[ multiplying by 5 both sides ]

→ 5 b + 15 = 2 ( a + 3 )

→ 5 b + 15 = 2 a + 6

[ using equation (1) ]

→ 5 b + 15 = 2 ( 5 b - 28 ) + 6

→ 5 b + 15 = 10 b - 56 + 6

→ 5 b - 10 b = -56 + 6 - 15

→ - 5 b = -65

→ b = 13  

Putting value of b in equation (1)

→ a = 5 b - 28

→ a = 5 (13) - 28

→ a = 37

Therefore,

• Present age of Varun is 13 yrs., and
• Present age of Swati is 37 yrs.

Answer 2

Answer :-

$\:\: ➮\:\underline{\boxed{\sf{Understanding\: the \: concept \:}}}$

Here the concept of Linear Equations in Two Variables has been used. According to this, if we make the value of one variable depend on other, we can find the values of both. Standard form of Linear Equations in Two Variables is given as :-

⟹ ax + by + c = 0

⟹ px + qy + d = 0

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★ Question :-

7 years ago Varun's age was 5 times the age of Swati's age after 3 years Swati's age will be 2/5 the of Varun's age. Find the present ages of both.

★ Solution :-

Given,

» 7 years ago, Varun's age is 5 times Swati's age.

» After 3 years, Swati's age will be 2/5 times the age of Varun.

Now,

• Let the present age of Varun be 'x' years.

• Let the present age of Swati be 'y' years.

Then, according to the question,

~ Case I (For 7 years ago) :-

➥ x - 7 = 5(y - 7)

➥ x - 7 = 5y - 35

➥ x = 5y - 35 + 7

➥ x = 5y - 28 ... (i)

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~ Case II (For 3 years, hence) :-

➥ y + 3 = 2/5(x + 3)

➥ 5(y + 3) = 2(x + 3)

➥ 5y + 15 = 2x + 6

➥ 2x - 5y = 15 - 6 .... (ii)

From equations (i) and equations (ii), we get,

⟹ 2(5y - 28) - 5y = 15 - 6

⟹ 10y - 56 - 5y = 9

⟹ 5y = 9 + 56

⟹ 5y = 65

$\: \: \longrightarrow \: \: \huge{\bold{y \: = \: \dfrac{65}{5}}}$

⟹ y = 13

Now from equation (i) and value of x, we get,

⟹ x = 5y - 28

⟹ x = 5(13) - 28

⟹ x = 65 - 28

⟹ x = 37

$\: \: ➮ \: \: \underline{\boxed{\rm{Hence,\: the \: age \: of \: Varun \: is \: \underline{37 \: years} \: and \: the \: age \: of \: Swathy \: is \: \underline{13 \: years}}}}$

______________________________

$\: \: ➮ \: \: \underline{\boxed{\it{Confused?, \: Don't \: worry \: let's \: verify \: it \: }}}$

For verification we must simply, apply the values we got into equations we formed.

~ Case I :-

=> x = 5y - 28

=> 37 = 5(13) - 28

=> 37 = 65 - 28

=> 37 = 37

Clearly, LHS = RHS.

~ Case II :-

=> 2x - 5y = 15 - 6

=> 2(37) - 5(13) = 9

=> 74 - 65 = 9

=> 9 = 9

Clearly, LHS = RHS.

Here both the conditions are satisfied.

So our answer is correct.

Hence, Verified.

_______________________________

$\: \: ➮ \: \: \underline{\boxed{\bf{Refer \: here \: for \: more \: to \: know \: }}}$

• Linear Equations is the equation formed using constant and linear terms.

• Linear Equations in Two Variables is the form of Linear equation where we deal with two equations simultaneously.

• Linear Equations in One Variable are the linear equations where we find the value of one variable using constant terms.