The sum of two numbers is $27$ and product is $182$ . The numbers are: $\newline$ $10$ and $15$ $\newline$ $18$ and $14$ $\newline$ $12$ and $15$ $\newline$ $11$ and $24$

Full solution

Q. The sum of two numbers is

27

and product is

182

. The numbers are:

\newline

10

and

15

\newline

18

and

14

\newline

12

and

15

\newline

11

and

24

Denote numbers $x$ and $y$ : Let's denote the two numbers as $x$ and $y$ . According to the problem, we have two equations: $\newline$ $1$ ) $x + y = 27$ (Sum of the two numbers) $\newline$ $2$ ) $xy = 182$ (Product of the two numbers) $\newline$ We need to find the values of $x$ and $y$ that satisfy both equations.
Express $y$ in terms of $x$ : We can express $y$ in terms of $x$ using the first equation: $y = 27 - x$ .
Substitute $y$ into second equation: Now we substitute $y = 27 - x$ into the second equation to find $x$ : $x(27 - x) = 182$ Expanding this, we get: $27x - x^2 = 182$
Rearrange to form quadratic equation: Rearrange the equation to form a quadratic equation: $\newline$ $x^2 - 27x + 182 = 0$
Factor quadratic equation: We can factor this quadratic equation to find the values of $x$ : $(x - 14)(x - 13) = 0$
Find possible values for x: Setting each factor equal to zero gives us the possible values for x: $\newline$ $x - 14 = 0$ or $x - 13 = 0$ $\newline$ So, $x = 14$ or $x = 13$
Calculate corresponding values for y: If $x = 14$ , then $y = 27 - x = 27 - 14 = 13$ . If $x = 13$ , then $y = 27 - x = 27 - 13 = 14$ .
Check product for both pairs: We check the product for both pairs to ensure there's no math error: $\newline$ For $x = 14$ and $y = 13$ : $14 \times 13 = 182$ $\newline$ For $x = 13$ and $y = 14$ : $13 \times 14 = 182$ $\newline$ Both pairs give the correct product, so there is no math error.