Question

The ages of two brothers are in the ratio 8:3.In three years time their ages will be in the ratio 9:4.Find the difference between their ages

Answer 1

Given :

• The ages of two brothers are in the ratio 8:3.

• In three years time their ages will be in the ratio 9:4.

To find :

• The difference between their ages =?

Step-by-step explanation:

Let, The age of 1st brother be 8x .

Then, The age of 2nd brother be 3x.

After 3 years 1st brother age = 8x + 3

After 3 years 2nd brother age = 3x + 3.

According to the question :

➮ (8x + 3)/(3x + 3) = 9/4

➮ 4(8x + 3) = 9 (3x + 3)

➮ 32x + 12 = 27x + 27

➮ 32x - 27x = 27 - 12

➮ 5x = 15

➮ x = 15/5

➮ x = 3.

Therefore, We know that value of x = 3.

Hence,

1st brother age = 8x => 8 x 3 = 24 year's

And, 2nd brother age = 3x => 3 x 3 = 9 years

Now the difference between their ages ,

The difference between their ages = 1st brother age - 2nd brother age

Substituting the values, we get,

= 24 - 9

= 15.

Therefore, Difference between their ages = 15 years.

Answer 2

Let the present ages of the two brothers be $\displaystyle\sf {8x}$ and $\displaystyle\sf {3x.}$ Hence the difference in the ages will be $\displaystyle\sf {8x-3x=5x.}$ We have to find this.

Three years after, the ages will be $\displaystyle\sf {8x+3}$ and $\displaystyle\sf {3x+3,}$ which are in the ratio 9:4. Then we have,

$\displaystyle\longrightarrow\sf{\dfrac {8x+3}{3x+3}=\dfrac {9}{4}\quad\quad\dots (1)}$

What rule of componendo and dividendo says is that,

$\displaystyle\longrightarrow\sf{\dfrac {a}{b}=\dfrac {c}{d}\quad\implies\quad\dfrac {a+b}{a-b}=\dfrac {c+d}{c-d}}$

Then from (1), we have,

$\displaystyle\longrightarrow\sf{\dfrac {(8x+3)+(3x+3)}{(8x+3)-(3x+3)}=\dfrac {9+4}{9-4}}$

$\displaystyle\longrightarrow\sf{\dfrac {8x+3+3x+3}{8x+3-3x-3}=\dfrac {13}{5}}$

$\displaystyle\longrightarrow\sf{\dfrac {11x+6}{5x}=\dfrac {13}{5}}$

On multiplying each side by 5,

$\displaystyle\longrightarrow\sf{\dfrac {11x+6}{x}=13}$

$\displaystyle\longrightarrow\sf{11x+6=13x}$

$\displaystyle\longrightarrow\sf{13x-11x=6}$

$\displaystyle\longrightarrow\sf{2x=6}$

To obtain value of $\displaystyle\sf {5x,}$ we multiply both sides by $\displaystyle\sf {\dfrac {5}{2}.}$ Then,

$\displaystyle\longrightarrow\sf{2x\times\dfrac {5}{2}=6\times\dfrac {5}{2}}$

$\displaystyle\longrightarrow\sf {\underline {\underline {5x=15}}}$

Hence 15 is the answer.