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5 years ago a man was 7 times as old as his son. After 5 years the father will be 3 time as old as his son. Find their present age.

$5$ years ago a man was $7$ times as old as his son. After $5$ years the father will be $3$ time as old as his son. Find their present age.

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Pythagorean Theorem and its converse

Full solution

Q.

5

years ago a man was

7

times as old as his son. After

5

years the father will be

3

time as old as his son. Find their present age.

Set up equations: Step $1$ : Set up the equations based on the age relationships given in the problem. $\newline$ Let the current age of the father be $F$ and the son be $S$ . $\newline$ $5$ years ago, the father was $7$ times the age of his son: $F - 5 = 7(S - 5)$ $\newline$ After $5$ years, the father will be $3$ times as old as his son: $F + 5 = 3(S + 5)$
Simplify equations: Step $2$ : Simplify both equations. $\newline$ From the first equation: $F - 5 = 7S - 35$ $\newline$ $\Rightarrow F = 7S - 30$ $\newline$ From the second equation: $F + 5 = 3S + 15$ $\newline$ $\Rightarrow F = 3S + 10$
Solve for S: Step $3$ : Set the simplified expressions for F equal to each other to solve for S. $\newline$ $7S - 30 = 3S + 10$ $\newline$ $\Rightarrow 7S - 3S = 10 + 30$ $\newline$ $\Rightarrow 4S = 40$ $\newline$ $\Rightarrow S = 10$
Find F: Step $4$ : Substitute the value of $S$ back into one of the original equations to find $F$ . Using $F = 7S - 30$ : $F = 7(10) - 30$ $\Rightarrow F = 70 - 30$ $\Rightarrow F = 40$

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