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The sum of an uncle’s and his nephew’s ages is $46$. Three years hence the uncle will be $3$ times the nephew’s age. Find their present age.

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Last updated date: 13th Jun 2024
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Answer
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Hint: Assume, the age of the uncle is x years and age of his nephew is y years. Then according to the question, the sum of x and y will be equal to $46$ which will give an equation. Then add $3$ to their ages to find their ages after three years. Then form an equation according to the second statement of the question. Now solve the equations formed to get the value of x and y.

Complete step-by-step answer:
Given, the sum of an uncle’s and his nephew’s age is $46$. Let the age of the uncle be x years and the age of nephew be y years. Then, we can write-
$ \Rightarrow x + y = 46$ --- (i)
Now after three years the age of uncle =$x + 3$
And the age of the nephew be=$y + 3$
Then according to the question, the age of uncle is thrice the age of nephew.
$ \Rightarrow x + 3 = 3\left( {y + 3} \right)$
On simplifying, we get-
$ \Rightarrow x + 3 = 3y + 9$
On taking the variables on one side and constant on the other, we get-
$ \Rightarrow x - 3y = 9 - 3$
On solving, we get-
$ \Rightarrow x - 3y = 6$ --- (ii)
Now, on subtracting eq. (ii) from eq. (i) we get-
$ \Rightarrow x + y - \left( {x - 3y} \right) = 46 - 6$
On solving we get-
$ \Rightarrow x + y - x + 3y = 40$
On simplifying, we get-
$ \Rightarrow 4y = 40$
On division, we get-
$ \Rightarrow y = 10$
Now putting this value in eq. (i) we get-
$ \Rightarrow x + 10 = 46$
On transferring the constants on the right side, we get-
$ \Rightarrow x = 46 - 10$
On subtraction, we get-
$ \Rightarrow x = 36$
Now we had to find their present ages.
So the present age of the uncle is $36$ and the present age of his nephew is $10$.

Note: We can also use a substitution method to solve the question by putting $x = 46 - y$ in eq. (ii) and then –
$ \Rightarrow 46 - y - 3y = 6$
On solving, we get-
$ \Rightarrow 46 - 4y = 6$
On simplifying, we get-
$ \Rightarrow 4y = 46 - 6$
On subtraction, we get-
$ \Rightarrow 4y = 40$
On solving, we get-
$ \Rightarrow y = 10$