Two numbers are in the ratio 2 ratio 3. The difference of their cubes is 9,728. Find the numbers.​

2 thoughts on “Two numbers are in the ratio 2 ratio 3. The difference of their cubes is 9,728. Find the numbers.​”

1. ☯GIVEN :

   • Two numbers are in the ratio 2:3.

   • Difference of their cubes = 9728

   ☯ TO FIND :

   • Required numbers .

   ➲SOLUTION :

   Let the required numbers are 2x and 3x.

   • Difference of their cubes = 9728

   [tex] \implies \sf{ (Second \: Number)^3 – (First \:Number)^3 =9728 } \\ \\[/tex]

   [tex] \implies \sf{(3x)^3 -(2x)^3 = 9728 } \\ \\[/tex]

   [tex] \implies \sf{27x^3 -8x^3 = 9728} \\ \\[/tex]

   [tex] \implies \sf{19x^3 = 9728} \\ \\[/tex]

   [tex]\implies \sf{x^3 = \cancel {\dfrac{9728}{19}}} \\ \\ [/tex]

   [tex]\implies \sf{x^3 = 512} \\ \\ [/tex]

   [tex]\implies \sf{x = \sqrt[ 3]{512} } \\ \\[/tex]

   [tex] \implies\sf{x = \sqrt[ 3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} } \\ \\ [/tex]

   [tex]\implies \sf{x = \sqrt[ 3]{2^3 \times 2^3 \times 2^3} } \\ \\[/tex]

   [tex] \implies\sf{x = 2 \times 2 \times 2} \\ \\[/tex]

   [tex]\implies\underline { \huge{ \boxed{\bf{x = 8}}}} [/tex]

   [tex]\huge { \pink{\therefore}}[/tex]First Number = [tex]\bf {2 \times 8 = 16}[/tex]

   [tex]\huge { \red{\therefore}}[/tex] Second Number = [tex]\bf {3 \times 8 = 24}[/tex]

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2. Step-by-step explanation:

   Let the numbers be 2x and 3x

   According to question,

   [tex] {(3x)}^{3} – {(2x)}^{3} = 9728[/tex]

   [tex]27 {x}^{3} – 8 {x}^{3} = 9728[/tex]

   [tex]19 {x}^{3} = 9728[/tex]

   [tex] {x}^{3} = \frac{9728}{19} [/tex]

   [tex] {x}^{3} = 512[/tex]

   [tex]x = \sqrt[3]{512} [/tex]

   [tex]x = 8[/tex]

   Numbers =

   2x = 2 × 8 = 16

   3x = 3 × 8 = 24

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