Question

the sum of ages of two friends is 20 years 5years ago the product of their ages in years was 40 is the situation possible if so determine the present ages​

Answer 1

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The situation is not possible. Therefore the present age cannot be determined.

$\bold{\underline{\underline{\large{\sf{StEp\:by\:stEp\:explanation </p><p>:}}}}}$

GiVeN :

• The sum of ages of two friends is 20 years
• 5 years ago the product of their ages in years was 40

To FiNd :

• If the given situation is possible.
• If the given situation is possible, determin the present age.

SoLuTiOn :

Let the present age of first friend be x years.

Let the present age of second friend be y years.

$\sf{\underline{\underline{As\:PeR\:tHe\:FiRsT\:cOnDiTiOn:}}}$

• Sum of the ages of two friends is 20

Constituting it mathematically,

$\hookrightarrow$ $\sf{x\:+\:y\:=\:=20}$

$\hookrightarrow$ $\sf{y\:=\:20\:-\:x}$ ---> (1)

$\sf{\underline{\underline{As\:PeR\:tHe\:S</p><p>eCoNd\:cOnDiTiOn:}}}$

• 5 years ago the product of their ages in years was 40

Ages 5 years ago :

Age of 1st friend

• x - 5 years

Age of 2nd friend

• y - 5 years

Product of ages

• 40

Constituting it mathematically,

$\hookrightarrow$ $\sf{(x-5)(y-5)\:=\:40}$

From equation 1, we can write 20 - x instead of y,

$\hookrightarrow$ $\sf{(x-20)(20-x-5)\:=\:40}$

$\hookrightarrow$ $\sf{(x-20)(20-5-x)\:=\:40}$

$\hookrightarrow$ $\sf{(x-5)(15-x)\:=\:40}$

$\hookrightarrow$ $\sf{x(15-x) -5(15-x)\:=\:40}$

$\hookrightarrow$ $\sf{15x\:-x^2\:-75\:+\:5x\:=\:40}$

$\hookrightarrow$ $\sf{-x^2\:-\:75\:+15x\:+\:5x\:=40}$

$\hookrightarrow$ $\sf{-x^2\:-75\:+20x\:=\:40}$

$\hookrightarrow$ $\sf{40\:=\:-x^2\:-\:75\:\:+\:20x}$

$\hookrightarrow$ $\sf{40\:+x^2\:-\:20x\:=\:-\:75}$

$\hookrightarrow$ $\sf{x^2\:-\:20x\:=\:-\:75\:-\:40}$

$\hookrightarrow$ $\sf{x^2\:-\:20x\:=\:-\:115}$

$\hookrightarrow$ $\sf{x^2\:-\:20\:+\:115\:=\:0}$

$\hookrightarrow$ $\sf{(x-20)\:x\:+\:115\:=\:0}$

$\hookrightarrow$ $\sf{x^2\:+\:115\:=\:20x}$

$\hookrightarrow$ $\sf{(x-10)^2\:+\:15\:=\:0}$

For calculation, refer to the website :

• Wolframalpha

Now further solving the equation will give us complex roots. Hence the situation is not possible.

•°• We can not determine the present age of the two friends.