Question

The ratio of Rayan's age to duke's age is 6:7.After 12 years, the ratio of their ages will be 12:13.Find their present age​

Answer 1

GIVEN :

• The ratio of the present age of Rayan and Duke is 6 : 7
• After a duration of 12 years they turns to the ratio of 12 : 13 years

TO FIND :

• The present age of Rayan and Duke.

SOLUTION :

Let us consider that their present age is :

6x and 7x respectively.

Hence , after 12 years ratio their of age will be : (6x + 12) and (7x + 12) years respectively.

Which is also equal to the ratio of 12 : 13

Therefore,

The equation formed is as follows :

$\implies \bf \: (6x + 12) \ratio(7x + 12) = 12 \ratio 13$

According to the question :

$\sf \implies \frac{6x + 12}{7x + 12} = \frac{12}{13}$

$\sf \implies 13(6x + 12) = 12(7x + 12)$

$\sf \implies \: 48x + 156 = 84x + 144$

$\sf \implies 156 - 144 = 84x - 48x$

$\sf \implies \: 12 = 6x$

$\sf \implies \:x = \frac{ \cancel{12}}{\cancel6}$

${ \underline{ \boxed{ \blue{\bf \therefore \: x = 2 \: years}}}}$

Thus,

${ \underline{ \boxed{ \blue{ \bf{ \therefore \: Required \: answer : }}}}}$

${ \underline{ \boxed{ \blue{ \bf{ \therefore \: Present \: age \:of \: Rayan \: is \: 6x = 6 \times 2 = 12 \: years { \boxed{ \red { \checkmark} }}}}}}}$

${ \underline{ \boxed{ \blue{ \bf{ \therefore \: Present \: age \:of \: duke \: is \: 7x = 7 \times 2 = 14 \: years { \boxed{ \red { \checkmark} }}}}}}}$

Verification :

It is told here that their ages will be in the ratio 12 : 13 after 12 years duration

Hence,

After 12 years their ages :

✥ Rayan's age = 12 + 12 = 24 years.

✥ Duke's age = 14 + 12 = 26 years.

Forming in ratio :

24 : 26

$\longmapsto \sf \frac{ \cancel{24}}{\cancel{26}} \\ \sf \longmapsto \: \frac{12}{13} \\ \longmapsto \bf 12 \ratio13$

Afterall we get the same ratio as told in the question i.e., 12 : 13 between the duration of 12 years.

$\bold\gray\dag{ \underline{ \boxed{ \bf{ \green {Hence}, \: \pink v \red e \purple r \green i \blue f \orange i \gray e d}}}}\bold\gray\dag$

Answer 2

$\huge{\bf{\red{AnSweR-}}}$

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$\large{\bf{\green{Given-}}}$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

• Ratio of Rayan's age to duku's age is 6:7

• After 12 years the ratio will be 12:13

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

$\large{\bf{\green{To\:Find-}}}$

• The present ages of rayan and Duke

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

$\large{\bf{\green{ConSideRinG-}}}$

• Rayan's age = 6x

• Duke's age = 7x

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

$\large{\bf{\green{SoluTioN-}}}$

• According to question , it is said that after 12 years the ratio of their ages will be 12:13

• So , the age of rayan after 12 years will be 6x + 12 and the age of duke is 7x + 12

• But we know that the ratio of ages after 12 years is 12 : 13 So , by putting it together we get :

$\;\;\;\;\;\;\;\;\;\;\;\;$

$\large{\bf{\boxed{\frac{6x\:+\:12}{7x\:+\:12}}}}$ = $\large{\bf{\boxed{\frac{12}{13}}}}$

$\;\;\;\;\;\;\;\;\;\;\;$

by solving the above equation , we can determine the value of x and from there we can find present ages of rayan and Duke

by solving the equation we get,

➜ $\large{\bf{\frac{6x+12}{7x+12}}}$ - $\large{\bf{\frac{12}{13}}}$

$\;\;\;\;\;\;\;\;\;\;\;$

➜ $\large{\bf{\frac{6x+12}{7x+12}}}$ × 13 = ${\bf{\frac{12}{\cancel{13}}}}$ × $\cancel{13}$

$\;\;\;\;\;\;\;\;\;\;\;$

➜ $\large{\bf{78x\:+\:156\:=\:12(7x+12)}}$

$\;\;\;\;\;\;\;\;\;\;\;$

➜ $\large{\bf{78x+156\:=\:84x+144}}$

➜ 12 = 6x

➜ X = $\large{\bf{\cancel{\frac{12}{6}}}}$

➜ $\large{\bf{\boxed{x\:=\:2}}}$

$\;\;\;\;\;\;\;\;\;\;\;$

$\large{\bf{\red{\boxed{\:age\:of\:rayan\:=\:6x\:=\:2×6=12}}}}$

$\large{\bf{\red{\boxed{\:age\:of\:Duke\:=\:7x\:=\:2×7=14}}}}$

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