Question

Find two number whose sum is 27and product is 182

Answer 1

GIVEN:

• The sum of two numbers is 27.
• The product of of two numbers is 182.

TO FIND:

• What are the numbers ?

SOLUTION:

Let one number be 'x' and another number be 'y'.

CASE:- 1

➣ The sum of two numbers is 27.

According to question:-

【One number + other number = 27 】

$\large\bf{\dashrightarrow x + y = 27....1)}$

$\rm{\dashrightarrow x = 27-y }$

CASE:- 2

➣ The product of of two numbers is 182.

According to question:-

【One number $\times$ other number = 182 】

$\large\bf{\dashrightarrow x \times y = 182 }$

Put the value of 'x' from equation 1) in equation 2)

$\rm{\dashrightarrow (27-y)y = 182 }$

$\rm{\dashrightarrow 27y - y^2 = 182 }$

$\rm{\dashrightarrow 0 = y^2 - 27y + 182 }$

$\rm{\dashrightarrow 0 = y^2 -(14+13)y + 182 }$

$\rm{\dashrightarrow 0 = y^2 -14y -13y + 182 }$

$\rm{\dashrightarrow 0 = y(y -14) -13(y -14)}$

$\rm{\dashrightarrow 0 = (y-13)(y-14) }$

$\bf{\dashrightarrow y = 13 }$

$\bf{\dashrightarrow y = 14 }$

Taking y = 13

Put the value of 'y' in equation 1)

$\rm{\dashrightarrow x + 13 = 27 }$

$\rm{\dashrightarrow x = 27-13 }$

$\bf{\dashrightarrow x = 14 }$

Take y = 14

$\rm{\dashrightarrow x + 14 = 27 }$

$\rm{\dashrightarrow x = 27-14 }$

$\bf{\dashrightarrow x = 13 }$

❝ Hence, the two numbers are 13 and 14 ❞

______________________

Answer 2

Given that ,

The sum of two numbers is 27 and their product is 182

Let , the two numbers are x and y

x + y = 27 ---- (i)

xy = 182

We know that ,

[ (a - b) ² = (a + b)² - 4ab ]

Thus ,

(x - y)² = (27)² - 4 × 182

(x - y)² = 729 - 728

(x - y)² = 1

Taking square root on both sides , we get

x - y = √1

x - y = 1 ---- (ii)

From eq (i) and eq (ii) , we get

2y = 26

y = 13

Put the value of y = 13 in eq (i) , we get

x + 13 = 27

x = 14

$\therefore \sf \underline{The \: two \: numbers \: are \: 14 \: and \: 13}$