Question

the ratio of the ages of silvi and Jenny two years ago was 2 : 3. 4 years from now the ratio of their ages will be 3 : 4 find their present age​

Answer 1

Given :

• The ratio of the ages of silvi and Jenny two years ago was 2 : 3.

• 4 years from now the ratio of their ages will be 3 : 4.

To find :

• Their present ages =?

Step-by-step explanation :

Let, Silvi present age = x

Then, Jenny present age = y

1st Case :

2 years ago Silvy age = x - 2

2 years ago Jenny age = y - 2

According to the 1st case :

➮ x - 2 / y - 2 = 2 /3

➮ 3x - 6 = 2y - 4

➮ 3x - 2y - 2 = 0 .... (i)

2nd Case,

After 4 years Silvi age = x + 4

After 4 years Jenny age = y + 4

According to the 2nd case :

➮ x + 4 / y + 4 = 3/4

➮ 4x + 16 = 3y + 12

➮ 4x - 3y + 4 = 0 ..... (ii)

Now,

Multiply by 4 in equation (i), we get,

➮ 4( 3x - 2y - 2 = 0)

➮ 12x - 8y - 8 = 0 .... (iii)

Then, Multiply by 3 in equation (ii), we get,

➮ 3 ( 4x - 3y + 4 = 0)

➮ 12x - 9y + 12 = 0 ....(iv)

On subtracting equation (iii) from (iv), we get,

➮ 12x - 8y - 8 - ( 12 x - 9y + 12) = 0 - 0

➮ 12x - 8y - 8 - 12x + 9y - 12 = 0

➮ 12x - 12x - 8y + 9y - 8 - 12 = 0

➮ y - 20 = 0

➮ y = 20

Therefore, We get the value of y = 20.

Now,

Substituting the value of y = 20 in equation (i), we get,

➮ 3x - 2y - 2 = 0

➮ 3x - 2 × 20 - 2 = 0

➮ 3x - 40 - 2 = 0

➮ 3x - 42 = 0

➮ 3x = 42

➮ x = 42/3

➮ x = 14.

Therefore, We also get the value of x = 14.

Hence,

The present age of Silvi, x = 14 years.

Then, the present age of Jenny, y = 20 years.

Answer 2

GIVEN :-

• The ratio of the ages of Silvi and Jenny two years ago was 2 : 3.

• 4 years from now the ratio of their ages will be 3 : 4.

TO FIND :-

The present ages of Silvi and Jenny.

SOLUTION :-

Let the present age of Silvi be x and the present age of Jenny be y.

Two years ago,

Let the age of Silvi be (x - 2) years and the age of Jenny be (y - 2) years.

A/q,

$\displaystyle{\sf{\frac{x-2}{y-2}=\frac{ 2}{3} }}$

By cross multiplication,

$\displaystyle{\sf{\implies 3(x-2)=2(y-2) }}\\\\\displaystyle{\sf{\implies 3x-6=2y-4 }}\\\\\displaystyle{\sf{\implies 3x-2y=6-4 }}\\\\\displaystyle{\sf{\implies 3x-2y=2 }}\:\:\:\:\:\:\:\: \cdots(i)$

$\rule{100}{2}$

Four years later,

Let the age of Silvi be (x + 4) years and the age of Jenny be (y + 4) years.

A/q,

$\displaystyle{\sf{\frac{x+4}{y+4}=\frac{ 3}{4} }}$

By cross multiplication,

$\displaystyle{\sf{\implies 4(x+4)=3(y+4)}}\\\\\displaystyle{\sf{\implies 4x+16=3y+12}}\\\\\displaystyle{\sf{\implies 4x-3y=12-16}}\\\\\displaystyle{\sf{\implies 4x-3y=-4}}\:\:\:\:\:\:\:\: \cdots(ii)$

$\rule{100}{2}$

Solving eq.(i) and eq.(ii) :-

$\underline{(i):}\ \displaystyle{\sf{ 3x - 2y = 2}}$

$\displaystyle{\sf{\implies 3x = 2 + 2y}}$

$\displaystyle{\sf{\implies x=\frac{2+2y}{3} }}$

Putting the value of x in eq.(ii) :-

$\underline{(ii):}\ \displaystyle{\sf{4x - 3y = -4}}$

$\displaystyle{\sf{\implies 4\times\frac{2+2y}{3}-3y=-4 }}\\\\\displaystyle{\sf{\implies \frac{8+8y}{3}-3y=-4 }}\\\\\displaystyle{\sf{\implies \frac{8+8y-9y}{3}=-4 }}\\\\\displaystyle{\sf{\implies8-y=-12}}\\\\\displaystyle{\sf{\impliesy=8+12}}\\\\\Large\boxed{\displaystyle{\sf{\implies y=20\ years}}}$

$\rule{100}{2}$

Now,

$\displaystyle{\sf{\implies x=\frac{2+2y}{3} }}\\\\\displaystyle{\sf{\implies x=\frac{2+(2\times20)}{3} }}\\\\\displaystyle{\sf{\implies x=\frac{2+40}{3} }}\\\\\displaystyle{\sf{\implies x=\frac{42}{3} }}\\\\\Large\boxed{\displaystyle{\sf{\implies x=14\ years}}}$

Therefore, the present age of Silvi is 14 years and the present age of Jenny is 20 years.