Two numbers are such that if we divide the $3$ times the greater number by the smaller number we obtained the quotient is $4$ and remainder $3$ and if $7$ times the smaller number is divided by bigger number then quotient and remainder are $5$ and $1$ respectively. Find the numbers.

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Solution

Let $x$ and $y$ are two numbers such that $x > y$
$3 x = 4 \times y + 3$
$\Rightarrow 3 x - 4 y = 3$ ........(i)
$7 y = 5 x + 1$
$\Rightarrow 7 y - 5 x = 1$ .......(ii)
Multiplying (i) by $5$ and (ii) by $3$
$15 x - 20 y = 15$
$21 y - 15 x = 3$
and then adding,
we get
$y = 18$
Putting the value of y in (i),
we get
$3 x - 4 \times 18 = 3$
$\Rightarrow 3 x - 72 = 3$
$\Rightarrow 3 x = 75$
$\Rightarrow x = 25$
Hence, bigger number is $25$ and smaller number is $18.$

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