The sum of two positive numbers is $100$. After subtracting $5$ from each number, the product of the resulting numbers is $0$. One of the original numbers is  ______.

1. $80$
2. $85$
3. $90$
4. $95$

Let the two positive numbers be $x$ and $y.$

According to the question, $x + y = 100 \quad \rightarrow (1)$

And, $(x-5) \cdot (y – 5) = 0$

$\implies xy -5x – 5y + 25 = 0$

$\implies x(100 – x) – 5(100) + 25 = 0 \quad [\because \text{From equation (1)}]$

$\implies 100x – x^{2} – 500 + 25 = 0$

$\implies -x^{2} + 100x – 475 = 0$

$\implies x^{2} - 100x + 475 = 0$

$\implies x^{2} – 95x – 5x + 475 = 0$

$\implies x(x-95)-5(x-95) = 0$

$\implies (x-95)(x-5) = 0$

$\implies (x-95) = 0 \; \& \;(x-5) = 0$

Now, if $x = 95 \implies y = 5,$ and if $x = 5 \implies y = 95.$

$\therefore$ One of the original number is $95.$

$\textbf{PS:}$ If one number is non-zero, then another number is $5,$ so that when we subtract $5$ from each the product will be $0.$

So, the correct answer is $(D).$
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the answer is 95 and the other is 5. therefore the answer is choice D
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