Everything is symmetrical in its own way

Algebra Level 1

Find A : S : K \color{#3D99F6}{A}:\color{#D61F06}{S}:\color{#20A900}{K} if A + S 2 = S + K 3 = A + K 4 . \dfrac{\color{#3D99F6}{A}+\color{#D61F06}{S}}{2}=\dfrac{\color{#D61F06}{S}+\color{#20A900}{K}}{3}=\dfrac{\color{#3D99F6}{A}+\color{#20A900}{K}}{4}.

4 : 9 : 16 4:9:16 4 : 3 : 2 4:3:2 2 : 3 : 4 2:3:4 3 : 1 : 5 3:1:5

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Chew-Seong Cheong
Oct 28, 2015

Let A + S 2 = S + K 3 = A + K 4 = X \dfrac{A+S}{2} = \dfrac{S+K}{3} = \dfrac{A+K}{4} = \color{#3D99F6}{X} . Then we have:

{ A + S = 2 X . . . ( 1 ) S + K = 3 X . . . ( 2 ) A + K = 4 X . . . ( 3 ) ⇒ { ( 3 ) − ( 1 ) : K − S = 2 X . . . ( 4 ) ( 4 ) + ( 2 ) : 2 K = 5 X ⇒ K = 5 2 X ( 2 ) : S = 3 X − K = 1 2 X ( 1 ) : A = 2 X − S = 3 2 X \begin{cases} A+S = 2X &...(1) \\ S+K = 3X &...(2) \\ A+K = 4X &...(3) \end{cases} \quad \Rightarrow \begin{cases} (3)-(1): & K-S = 2X & ...(4) \\ (4)+(2): & 2K = 5X & \Rightarrow K = \dfrac{5}{2}X \\ (2): & S = 3X - K = \dfrac{1}{2}X \\ (1): & A = 2X - S = \dfrac{3}{2}X \end{cases}

⇒ A : S : K = 3 : 1 : 5 \Rightarrow A:S:K = \boxed{3:1:5}

18 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution sir, as usual! Thanks for sharing it!

Sandeep Bhardwaj - 5 years, 7 months ago

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