An examination consists of $160$ questions. One mark is given for every correct option. If a $\frac{1}{4}$ mark is deducted for each wrong option and a half mark is deducted for each unanswered question, then a student scores $79$. If half a mark is deducted for every wrong option and $\frac{1}{4}$ mark is deducted for every unanswered question, the person scores $76$. Find the number of correct answers he wrote.

Let the number of correct answers be $C$, wrong answers be $W$, and unanswered questions be $U$.

Total questions:

$C+W+U=160$ .....$(1)$

According to the first condition:

$C-\frac{W}{4}-\frac{U}{2}=79$ .....$(2)$

According to the second condition:

$C-\frac{W}{2}-\frac{U}{4}=76$ .....$(3)$

Subtract $(3)$ from $(2)$:

$\left(C-\frac{W}{4}-\frac{U}{2}\right)-\left(C-\frac{W}{2}-\frac{U}{4}\right)=79-76$

$-\frac{W}{4}+\frac{W}{2}-\frac{U}{2}+\frac{U}{4}=3$

$\frac{W}{4}-\frac{U}{4}=3$

$W-U=12$ .....$(4)$

From $(1)$,

$C=160-W-U$

Put this in $(2)$:

$160-W-U-\frac{W}{4}-\frac{U}{2}=79$

$160-\frac{5W}{4}-\frac{3U}{2}=79$

$\frac{5W}{4}+\frac{3U}{2}=81$

Multiply by $4$:

$5W+6U=324$ .....$(5)$

From $(4)$,

$W=U+12$

Put in $(5)$:

$5(U+12)+6U=324$

$5U+60+6U=324$

$11U=264$

$U=24$

So,

$W=24+12=36$

Now,

$C=160-36-24$

$C=100$