Question

the ages of A and B are in the ratio 9:4. Seven years hence, the ratio of their ages will be 5:3. Find their ages using simultaneous equations. please answer very quick. it is very important. ​

Answer 1

Given

$\bold{The \ ages \ of \ A \ and \ B \ are \ in \ the \ ratio \ 9:4.}$

$\sf Let \ their \ ages \ be \ x \ and \ y \ respectively.$

$\implies \sf \dfrac{x}{y} = \dfrac{9}{4}$

$\implies \sf 4x=9y$

$\implies \sf 4x-9y=0--(1)$

$\bold{Seven \ years \ hence, \ the \ ratio \ of \ their \ ages \ will \ be \ 5:3. }$

$\implies \sf \dfrac{x+7}{y+7}= \dfrac{5}{3}$

$\implies \sf 3x+21=5y+35$

$\implies \sf 3x-5y=35-21$

$\implies \sf 3x-5y=14--(2)$

$\sf Multiplying \ the \ first \ equation \ by \ 3, \ we \ get,$

$\implies \sf 12x-27y=0--(3)$

$\sf Multiplying \ the \ second \ equation \ by \ 4, \ we \ get,$

$\implies \sf 12x-20y = 56 -- (4)$

$\sf Solving (3) and (4), we \ get$

$\sf 12x-27y=0\\\sf \underline{ 12x-20y = 56}\\\underline{\underline{-7y=-56}}\\\implies y = 8$

$\implies x=18$

$\boxed{\boxed{\bold{Therefore, \ the \ ages \ of \ A \ and \ B \ are \ 18 \ and \ 8 \ \ respectively}}}}}}$

                                                                           

Answer 2

Answer:

Age of A is 18 years and B is 8 years.

Step-by-step explanation:

Method 1)

The ages of A and B are in the ratio 9:4.

Assume that the age of A is x and B is y.

→ x/y = 9/4

→ 4x = 9y

→ 4x - 9y = 0 ...............(1)

Seven years hence, the ratio of their ages will be 5:3.

Seven years hence:

• Age of A = (x + 7)
• Age of B = (y + 7)

As per given condition,

→ (x + 7)/(y + 7) = 5/3

→ 3(x + 7) = 5(y + 7)

→ 3x + 21 = 5y + 35

→ 3x - 5y = 14 ................(2)

On multiplying (1) with 3 & (2) with 4 we get,

→ 12x - 27y = 0

→ 12x = 27y ...........(3)

→ 12x - 20y = 56

→ 12x = 56 + 20y ........(4)

On comparing (3) & (4) we get,

→ 27y = 56 + 20y

→ 7y = 56

→ y = 8

Substitute value of y in (1)

→ 4x = 9(8)

→ 4x = 72

→ x = 18

Hence, the age of:

• A = x = 18 years
• B = y = 8 years

Method 2)

Assume that the age of A is 9x and B is 4x.

Seven years hence, the ratio of their ages will be 5:3.

Seven years hence:

• Age of A = (9x + 7) years
• Age of B = (4x + 7) years

As per given condition,

→ (9x + 7)/(4x + 7) = 5/3

→ 3(9x + 7) = 5(4x + 7)

→ 27x + 21 = 20x + 35

→ 7x = 14

→ x = 2

Hence, the age of :

• A = 9x = 9(2) = 18 years
• B = 4x = 4(2) = 8 years