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A says to B “I was twice as old as you were when I was as old as you are”. The sum of their present ages is 49 years. Find their present ages.

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Last updated date: 20th Jun 2024
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Answer
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Hint: In these type of age’s questions the age’s should be taken as a variable and should form two linear equations from the given data in the question, and these linear equations should be solved to get the values of the two considered variables, in most cases $x$ and $y$ are taken as the required variables.

Complete answer:
Given A’s age is twice as B’s when A’s age is equal to B’ age, and the sum of their present ages is 49 years.
Let us consider the present ages of A be $x$ and present age of B be $y$.
From the given data,
\[ \Rightarrow \]$x + y = 49 - - - - - (1)$,
From the given sentence A says “I was twice as old as you were when I was as old as you are”.
A’s age $ \Rightarrow 2 \times $(B’s age when A’s age was equal to B’s age)
A’s age $ \Rightarrow 2 \times $(B’s age when A’s age was y)
Now as we considered A’s age be$x$ and B’s age be $y$, so from the above given data we get,
$ \Rightarrow x = 2\left[ {y - \left( {x - y} \right)} \right]$
Now simplifying we get,
$ \Rightarrow x = 2y - 2x + 2y$
Now adding the like terms on L.H.S we get,
$ \Rightarrow x = 4y - 2x$
Now bring like terms to one side we get,
$ \Rightarrow x + 2x = 4y$
Now adding like terms on R.H.S we get,
$ \Rightarrow 3x = 4y$
Now bringing all terms to one side we get,
\[ \Rightarrow \]$3x - 4y = 0 - - - - - (2)$
Now adding (1) and (2) we get,
\[ \Rightarrow \]$x + y = 49 - - - - - (1)$
\[ \Rightarrow \]$3x - 4y = 0 - - - - (2)$
Now multiplying (1) with 4 to get equal $y$ terms in both the equations we get,
\[ \Rightarrow \]$x + y = 49 - - - - - (1) \times 4$
\[ \Rightarrow \]$3x - 4y = 0 - - - - (2)$
After multiplying (1) by 4 we get,
\[ \Rightarrow \]$4x + 4y = 196 - - - - - (1)$
\[ \Rightarrow \]$3x - 4y = 0 - - - - (2)$
Now eliminating the $y$ terms as have equal $y$ constant and adding $x$ terms we get,
\[ \Rightarrow \]$7x = 196$,
Now dividing both sides with 7 we get,
\[ \Rightarrow \]$\dfrac{{7x}}{7} = \dfrac{{196}}{7}$,
Now simplifying we get,
\[ \Rightarrow \]$x = 28$,
Now substituting the $x$ value in (1) we get,
\[ \Rightarrow \]$x + y = 49$,
Substituting the value of $x$ we get,
\[ \Rightarrow \]$28 + y = 49$,
Subtracting 28 on both sides we get,
\[ \Rightarrow \]$28 + y - 28 = 49 - 28$
Now after subtraction we get,
\[ \Rightarrow \]$y = 21$,
$\therefore $Present age of A = 28 years,
$\therefore $Present age of B = 21 years.

The present age of A will be 28 years and the present age of B will be 21 years.

Note:
An equation is said to be linear equation in two variables if it is written in the form of \[ax + by = c\], where\[a\], \[b\] and \[c\] are real numbers and the coefficients of x and y, i.e., \[a\] and \[b\] respectively, are not equal to zero.
For example, \[10x + 4y = 3\] and \[ - x + 5y = 2\]- are linear equations in two variables.
The solution for such an equation is a pair of values, one for\[x\] and one for \[y\] which further makes the two sides of an equation equal.
The solution of linear equations in two variables, \[ax + by = c\] is a particular point in the graph, such that when x-coordinate is multiplied by \[a\] and y-coordinate is multiplied by \[b\], then the sum of these two values will be equal to. Basically, for a linear equation in two variables, there are infinitely many solutions.