# Find two numbers whose sum is 27 and product is 182.

Answer

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Hint: Here first we assume two variables and write the equations according to the condition given in question and solve the equations to get one of the values of the variable and substitute in the equation to get another variable.

Complete step-by-step answer:

First let us assume the two numbers are x and y

Then according to question, we will be having two equations those are

\[x + y = {\text{2}}7\] …… (1)

\[xy = 182\] …… (2)

Equation 1 can be written as

$\Rightarrow$ \[x = {\text{2}}7 - y\] …… (3)

On putting the value of x in equation (3), we get

$\Rightarrow$ $\left( {27 - y} \right)\left( y \right) = 182$

Above equation can also be written as

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

On splitting we get

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

$\Rightarrow$ $\left( {y - 13} \right)\left( {y - 14} \right) = 0$

From the equation above either y = 13 or y = 14.

Putting the value of y in equation(1) we get \[x = 14\] when \[y = 13\] and \[x = 13\] when \[y = 14\]

Hence the two numbers are 13 and 14.

Note: In these types of questions first assume the variables and then write the equation according to the condition given in the question then solve the equations to get the value of the variables.Students should take care while assuming the variables and writing the equations according to the question.

Complete step-by-step answer:

First let us assume the two numbers are x and y

Then according to question, we will be having two equations those are

\[x + y = {\text{2}}7\] …… (1)

\[xy = 182\] …… (2)

Equation 1 can be written as

$\Rightarrow$ \[x = {\text{2}}7 - y\] …… (3)

On putting the value of x in equation (3), we get

$\Rightarrow$ $\left( {27 - y} \right)\left( y \right) = 182$

Above equation can also be written as

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

On splitting we get

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

$\Rightarrow$ $\left( {y - 13} \right)\left( {y - 14} \right) = 0$

From the equation above either y = 13 or y = 14.

Putting the value of y in equation(1) we get \[x = 14\] when \[y = 13\] and \[x = 13\] when \[y = 14\]

Hence the two numbers are 13 and 14.

Note: In these types of questions first assume the variables and then write the equation according to the condition given in the question then solve the equations to get the value of the variables.Students should take care while assuming the variables and writing the equations according to the question.

Last updated date: 23rd Sep 2023

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