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Mathematics

If (a+b)3(a−b)3=6427\dfrac{(a + b)^3}{(a - b)^3} = \dfrac{64}{27}

(a) Find a+ba−b\dfrac{a + b}{a - b}

(b) Hence using properties of proportion, find a : b.

Ratio Proportion

ICSE 2024

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Answer

(a) Solving,

⇒(a+b)3(a−b)3=6427⇒(a+b)3(a−b)3=4333⇒(a+ba−b)3=(43)3⇒a+ba−b=43\Rightarrow \dfrac{(a + b)^3}{(a - b)^3} = \dfrac{64}{27} \\[1em] \Rightarrow \dfrac{(a + b)^3}{(a - b)^3} = \dfrac{4^3}{3^3} \\[1em] \Rightarrow \Big(\dfrac{a + b}{a - b}\Big)^3 = \Big(\dfrac{4}{3}\Big)^3 \\[1em] \Rightarrow \dfrac{a + b}{a - b} = \dfrac{4}{3} \\[1em]

Hence, a+ba−b=43.\dfrac{a + b}{a - b} = \dfrac{4}{3}.

(b) Solving further,

⇒3(a+b)=4(a−b)⇒3a+3b=4a−4b⇒4a−3a=3b+4b⇒a=7b⇒ab=71⇒a:b=7:1.\Rightarrow 3(a + b) = 4(a - b) \\[1em] \Rightarrow 3a + 3b = 4a - 4b \\[1em] \Rightarrow 4a - 3a = 3b + 4b \\[1em] \Rightarrow a = 7b \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{7}{1} \\[1em] \Rightarrow a : b = 7 : 1.

Hence, a : b = 7 : 1.

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