Question

The product of two numbers is 154. If the difference between the sum of two numbers is 3, find the numbers

Answer 1

ANSWER :

The two numbers are -11 or 14 and -14 or 11.

EXPLANATION :

GIVEN :-

• Product of two numbers is 154.
• Difference between two numbers is 3.

TO FIND :

• The two numbers.

SOLUTION :

Let the two numbers be x and y.

Product of two numbers= 154

Difference between two numbers=3

According to the question,

$\sf{xy=154...........(i)}$

$\sf{x-y=3}$

$\implies\sf{x=3+y.........(ii)}$

★Taking (i) no. equation★

$\sf{\:\:\:\:xy=154}$

† Putting x=3+y from the eq.(ii) †

$\implies\sf{(3+y)y=154}$

$\implies\sf{3y+y^2=154}$

$\implies\sf{y^2+3y-154=0}$

$\implies\sf{y^2+(14-11)y-154=0}$

$\implies\sf{y^2+14y-11y-154=0}$

$\implies\sf{y(y+14)-11(y+14)=0}$

$\implies\sf{(y+14)(y-11)=0}$

Either,

y+14 = 0

→y = -14

Or,

y-11 = 0

→y = 11

† Putting y = -14 and y =11 in eq.(ii) †

➪

x = 3+y

→ x = 3-14

→ x = -11

➪

x = 3+ y

→ x = 3+11

→ x = 14

x = 14 or -11

y = 11 or -14

Therefore, the two numbers are 14 or (-11) and 11 or (-14).

Answer 2

AnsWer :

X = 14 and Y = 11.

Or, X = -11 and Y = -14.

Solution :

Let the two number be x and y.

Case 1.

"The product of two number is 154."

• XY = 154.

Case 2.

"If the difference between the two number is 3."

• X - Y = 3.

Let's try to form equation,

$\tt xy = 154 - - - (1)$

$\tt x - y = 3. - - - (2)$

Taking equation, 2.

$\tt x = 3 + y. - - - (3)$

Putting the equation (3) in equation (1), we get.

$\tt \implies xy = 154.$

$\tt \implies (3 + y)y = 154.$

$\tt \implies 3y + {y}^{2} = 154.$

$\tt \implies {y}^{2} + 3y = 154.$

$\tt \implies {y}^{2} + 3y - 154 = 0.$

$\tt \implies {y}^{2} + 14y - 11y - 154 = 0.$

$\tt \implies y(y + 14) - 11(y + 14) = 0.$

$\tt \implies (y + 14) (y - 11) = 0.$

Either,

$\tt \implies y + 14 = 0.$

$\tt \implies y = - 14 .$

And

$\tt \implies y - 11 = 0.$

$\tt \implies y = 11.$

Putting the value of y in equation (3), we get.

When, Y = -14.

$\tt \implies3 + y = x.$

$\tt \implies3 + ( - 14) = x.$

$\tt \implies x = - 11.$

When, Y = 11.

$\tt \implies3 + y = x.$

$\tt \implies3 + (11) = x.$

$\tt \implies x = 14.$

$\tt Either\begin{cases} \sf{X = + 14.} \\ \sf{Y = + 11.} \end{cases}$

$\tt Or\begin{cases} \sf{X = - 11.} \\ \sf{Y = - 14.} \end{cases}$

Therefore, the two numbers are x = -11 and y = -14, Or x = 14 and y =11.