Math Problem Solver

The numbers are $28$ and $7$.

The difference between the numbers is $21$, and their quotient is $4$. This translates to the two equations: $$\begin{aligned}n_{1} - n_{2} &= 21\\\frac{n_{1}}{n_{2}} &= 4\\\end{aligned}$$ Solving the second equation for $n_{1}$ in terms of $n_{2}$ produces the formula $n_{1} = 4 n_{2}$. Substituting the formula for $n_{1}$ into the first equation yields:$$\begin{gather*}4 n_{2} - n_{2} = 21\\3 n_{2} = 21\end{gather*}$$ So, $n_{2} = 7$. Substituting $7$ for $n_{2}$ in the first equation results in $n_{1} - 7 = 21$, so $n_{1} = 28$.