Let the number of check's in $\(50 & 100\)$ denominations be a \& b respectively.



As a total of 30 travelers checks are purchased, we get $\(a+b=30 \ldots$ (1)\) \& as the total worth of the checks is $\( 1800\)$, we get $\(50 \mathrm{a}+100 \mathrm{~b}=1800\)$, which gives $\(\mathrm{a}+2 \mathrm{~b}=36\)$.

Solving equations (1) \& (2), we get $\mathrm{a}\(=24 \)\& \(\mathrm{~b}=6\)$

We need to find the number of $ \(50\)$ checks which should be spent so that the average value of the checks comes out to be $ \(80\)$.

Let the number of $ \(50\)$ checks spent be k , so the remaining number of $\(50\)$ checks is $\((24-\mathrm{k})\)$.

Now, the average value of all the remaining checks is $\(\frac{(24-\mathrm{k}) \times 50+6 \times 100}{(24-\mathrm{k})+6}=80\)$ which when simplified gives $\(1200-50 k+600=2400-80 k \Rightarrow k=20\)$


Hence the number of $ \(50\)$ checks which should be spent so the average of the remaining checks becomes $ \(80\)$ is 20 , so the answer is (D).