If three times the larger of the two numbers is divided by the smaller one, we get 4 as quotient and 3 as the remainder. Also, if seven times the smaller number is divided by the larger one, we get 5 as quotient and 1 as remainder. The numbers are

Let the larger number be x and smaller one be y. We know that 
Dividend = (Divisor × Quotient) + Remainder ….(i)
When 3x is divided by y, we get 4 as quotient and 3 as remainder. Therefore, by using (i), we get $3\mathrm{x}=4\mathrm{y}+3\Rightarrow 3\mathrm{x}-4\mathrm{y}-3=0$  ….(ii)
When 7y is divided by x, we get 5 as quotient and 1 as remainder. Therefore, by using (i), we get $7\mathrm{y}=5\mathrm{x}+1\Rightarrow 5\mathrm{x}-7\mathrm{y}+1=0$  ….(iii)
Solving equations (ii) and (iii), by cross – multiplication, we get $\frac{\mathrm{x}}{-4-21}=\frac{-\mathrm{y}}{3+15}=\frac{1}{-21+20}\Rightarrow \mathrm{x}=25\text{ and }\mathrm{y}=18$ 
Hence, the required numbers are 25 and 18.