Question

The ages of Ranu and Anu are is the ratio 3 : 4. Twelve years later, the ratio of their ages will be 5 : 6.
Find their present ages.
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Answer 1

Step-by-step explanation:let the age of ranu and Anu be x then

3x+12+4x+12=5x+6x then,

3x+4x+12+12=11x7x+24=11x24=11x-7x=4x

x=24/4=6x=6 then 6*3=18 and

6*4=24then ranu's age is 18 and anu's age is 24

Answer 2

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{The \ present \ ages \ of \ Ranu \ and \ Anu}$

$\sf{are \ 18 \ and \ 24 \ years \ respectively. }$

$\sf\orange{Given:}$

$\sf{\implies{The \ ages \ of | Ranu \ and \ Anu \ are}}$

$\sf{in \ the \ ratio \ of 3:4.}$

$\sf{\implies{12 \ years \ later, \ the \ ratio \ of \ their}}$

$\sf{Ages \ will \ be \ 5:6}$

$\sf\pink{To \ find:}$

$\sf{Present \ ages \ of \ Ranu \ and \ Anu.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{Method(I) \ By \ using \ two \ variables}$

$\sf{Let \ the \ age \ of \ Ranu \ be \ x \ and \ age}$

$\sf{of \ Anu \ be \ y.}$

$\sf{According \ to \ first \ condition}$

$\sf{x:y=3:4}$

$\sf{\therefore{\frac{x}{y}=\frac{3}{4}}}$

$\sf{\therefore{4x=3y}}$

$\sf{\therefore{4x-3y=0...(1)}}$

$\sf{According \ to \ second \ condition}$

$\sf{\frac{x+12}{y+12}=\frac{5}{6}}$

$\sf{\therefore{6(x+12)=5(y+12)}}$

$\sf{6x+72=5y+60}$

$\sf{\therefore{6x-5y=-12...(2)}}$

$\sf{Multiply \ equation(1) \ by \ 3 \ we \ get,}$

$\sf{12x-9y=0...(3)}$

$\sf{Multiply \ equation(2) \ by \ 2 \ we \ get,}$

$\sf{12x-10y=-24...(4)}$

$\sf{Subtract \ equation(4) \ from \ equation(3)}$

$\sf{12x-9y=0}$

$\sf{-}$

$\sf{12x-10y=-24}$

_________________________

$\boxed{\sf{\therefore{y=24}}}$

$\sf{Substitute \ y=24 \ in \ equation(1) \ we \ get,}$

$\sf{4x-3(24)=0}$

$\sf{4x-72=0}$

$\sf{4x=72}$

$\sf{x=\frac{72}{4}}$

$\boxed{\sf{\therefore{x=18}}}$

$\sf{Ranu's \ age=x=18 \ years.}$

$\sf{Anu's \ age=y=24 \ years.}$

$\sf\purple{\tt{\therefore{The \ present \ ages \ of \ Ranu \ and \ Anu}}}$

$\sf\purple{\tt{are \ 18 \ and \ 24 \ years \ respectively. }}$

_______________________________________

______________________________________________________________________________

$\sf{Method(II) \ By \ single \ variable.}$

$\sf{Let \ the \ common \ factor \ be \ x.}$

$\sf{According \ to \ first \ condition}$

$\sf{Ratio \ of \ their \ present \ ages=\frac{3x}{4x}}$

$\sf{\therefore{Ranu's \ age=3x}}$

$\sf{Anu's \ age=4x}$

$\sf{According \ to \ second \ condition}$

$\sf{\frac{3x+12}{4x+12}=\frac{5x}{6x}}$

$\sf{\therefore{6x(3x+12)=5x(4x+12)}}$

$\sf{\therefore{18x^{2}+72x=20x^{2}+60x}}$

$\sf{\therefore{18x^{2}-20x^{2}+72x-60x=0}}$

$\sf{\therefore{-2x^{2}+12x=0}}$

$\sf{\therefore{-2x(x-6)=0}}$

$\sf{\therefore{-2x=0 \ or \ x-6=0}}$

$\sf{\therefore{x=0 \ or \ x=6}}$

$\sf{But, \ x \ can't \ be \ 0}$

$\boxed{\sf{\therefore{x=6}}}$

$\sf{Ranu's \ age=3(6)=18 \ years.}$

$\sf{Anu's\ age=4(6)=24 \ years.}$

$\sf\purple{\tt{\therefore{The \ present \ ages \ of \ Ranu \ and \ Anu}}}$

$\sf\purple{\tt{are \ 18 \ and \ 24 \ years \ respectively. }}$