Question

the sum of the two numbers is 100 and their product is 900. number equals ?

Answer 1

$\sf{\underline{Answer\::}}$

The two numbers considering both cases of value of y are,

1. x = 90,y = 10
2. x = 10, y = 90

$\sf{\underline{Explanation\::}}$

The sum of two numbers is 100.

Also,the product of the exact same two numbers is 900.

Let's make assumptions to simplify the problem.

Let the greater number of the two numbers be x.

Let the smaller number of the two numbers be y.

Number = 10x + y.

Moving ahead with the condition forming equation,

=> $\sf{x+y=100}$

=> $\sf{x=100-y.....(1)}$

We are done with the sum of the numbers. Next forming equation based on product of the numbers.

=> $\sf{xy=900}$

=> $\sf{(100-y)y=900}$

=> $\sf{100y-y^2=900}$

=> $\sf{-y^2+100y=900}$

=> $\sf{y^2-100y=-900}$

=> $\sf{y^2-100y+900}$

=> $\sf{y^2-90y-10y+900}$

=> $\sf{y(y-90)-10(y-90)}$

=> $\sf{(y-90)\:\:or\:\:(y-10)=0}$

=> $\sf{y=90\:\:\:or\:\:y=10}$

Here,we have two positive values of our smaller number,y.

This creates two possibilities,

1. Two numbers when value of y is 90.
2. Two numbers when value of y is 10.

Possibility 1, y = 90 :

=> $\sf{x=100-y}$

=> $\sf{x=100-90}$

=> $\sf{x=10}$

Possibility 2, y = 10 :

=> $\sf{x=100-y}$

=> $\sf{x=100-10}$

=> $\sf{x=90}$

Answer 2

Given ,

The sum of two number is 100 and their product is 900

Let us assume that ,

The two numbers be " x " and " y "

Thus ,

x + y = 100 --- (i)

xy = 900 --- (ii)

Solving (i) and (ii) , we get

x(100 - x) = 900

(x)² - 100x + 900 = 0

(x)² - 10x - 90x + 900 = 0

x(x - 10) - 90(x - 10) = 0

(x - 90)(x - 10)

x = 90 or x = 10

Put the value of x in eq (i) , we get

90 + y = 100 or 10 + y = 100

y = 10 or y = 90

$\therefore \bold{ \sf \underline{The \: two \: numbers \: are \: 90 \: or \: 10 \: and \: 10 \: or \: 90</p><p>}}$