GATE Civil 2020 Set 1 | GA | Question: 5

Question

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  ______.

• A. $80$
• B. $85$
• C. $90$
• D. $95$

Answer 1

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 $(D).$

Comments

Or after writing the equation (x-5).(y-5) = 0, you can simply conclude that either x=5 or y=5. That makes the other number = 95 as x+y=100.

Answer 2

Let +ve no be x and y

x + y = 100

(x-5) (y-5) = 0

Zero rule of product :-   if a*b = 0 then a = 0 or b = 0 or both a = b= 0

Case 1: a = 0

(x – 5) = 0

x = 5, y = 95

Case 2: b = 0

(y – 5) = 0

y = 95, x= 5

So, the no. should be 5, 95

Answer 3

the answer is 95 and the other is 5. therefore the answer is choice D