Let $a_k$ be the number of $k$-cent coins used to make $A$ cents with $B$ coins. 

\[a_1+a_2+a_5+a_{10}+a_{20}+a_{50}+a_{100}=B\]

\[a_1+2a_2+5a_5+10a_{10}+20a_{20}+50a_{50}+100a_{100}=A\]

Now, we use $ka_k$ $\frac{100}{k}$-cent coins for each possible value of $k$.

The number of coins used $=a_1+2a_2+5a_5+10a_{10}+20a_{20}+50a_{50}+100a_{100}=A$

The amount of cents $=100(a_1+a_2+a_5+a_{10}+a_{20}+a_{50}+a_{100})=100B=B\text{ dollars}$

$\therefore$ We can use $A$ coins to form $B$ dollars.