1. Given relation

We have $A=\{1,2,3,4,5\}$ and relation $R$ on $A$ defined by $xRy \iff 4x\le 5y.$

We need:

• $m =$ number of ordered pairs in $R$
• $n =$ minimum number of ordered pairs to be added to make $R$ symmetric
• then compute $m+n$

2. Find all pairs in $R$

For each fixed $x\in A$, we find all $y\in A$ such that $4x\le 5y \iff y\ge \frac{4x}{5}.$

Let us check each $x$:

(i) $x=1$

$4(1)=4,\quad 4\le 5y$ $y\ge \frac45$ So all $y\in\{1,2,3,4,5\}$ work.

Pairs: $(1,1),(1,2),(1,3),(1,4),(1,5)$ Count = 5

(ii) $x=2$

$4(2)=8,\quad 8\le 5y$ $y\ge \frac85=1.6$ So $y=2,3,4,5$ work.

Pairs: $(2,2),(2,3),(2,4),(2,5)$ Count = 4

(iii) $x=3$

$4(3)=12,\quad 12\le 5y$ $y\ge \frac{12}{5}=2.4$ So $y=3,4,5$ work.

Pairs: $(3,3),(3,4),(3,5)$ Count = 3

(iv) $x=4$

$4(4)=16,\quad 16\le 5y$ $y\ge \frac{16}{5}=3.2$ So $y=4,5$ work.

Pairs: $(4,4),(4,5)$ Count = 2

(v) $x=5$

$4(5)=20,\quad 20\le 5y$ $y\ge 4$ So $y=4,5$ work.

Pairs: $(5,4),(5,5)$ Count = 2

Therefore, $m=5+4+3+2+2=16.$

3. Find minimum additions to make $R$ symmetric

A relation is symmetric if whenever $(x,y)\in R$, then $(y,x)\in R$ also.

So we list all pairs in $R$ and check whether their reverse is already present.

Current relation: $R=\{(1,1),(1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5),(4,4),(4,5),(5,4),(5,5)\}$

Now check non-diagonal pairs:

• $(1,2)\in R$, but $(2,1)\notin R$ $\Rightarrow$ add $(2,1)$
• $(1,3)\in R$, but $(3,1)\notin R$ $\Rightarrow$ add $(3,1)$
• $(1,4)\in R$, but $(4,1)\notin R$ $\Rightarrow$ add $(4,1)$
• $(1,5)\in R$, but $(5,1)\notin R$ $\Rightarrow$ add $(5,1)$
• $(2,3)\in R$, but $(3,2)\notin R$ $\Rightarrow$ add $(3,2)$
• $(2,4)\in R$, but $(4,2)\notin R$ $\Rightarrow$ add $(4,2)$
• $(2,5)\in R$, but $(5,2)\notin R$ $\Rightarrow$ add $(5,2)$
• $(3,4)\in R$, but $(4,3)\notin R$ $\Rightarrow$ add $(4,3)$
• $(3,5)\in R$, but $(5,3)\notin R$ $\Rightarrow$ add $(5,3)$
• $(4,5)\in R$, and $(5,4)\in R$ already, so nothing needed.

Diagonal pairs $(1,1),(2,2),\dots,(5,5)$ automatically satisfy symmetry.

Hence the minimum number of pairs to be added is $n=9.$

4. Compute $m+n$

$m+n=16+9=25.$

5. Check options

The correct option is: $\boxed{25}$ which is Option C.

6. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So they agree.