# The cube of a number is 8 times the cube of another number. If the sum of the

cubes of numbers is 243, the difference of the numbers is:

(a) 3

(b) 4

(c) 6

(d) None of these

Last updated date: 24th Mar 2023

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Answer

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Hint- Taking cube root on both sides of the equation and use of cube roots of

number $8 = {2^3}$ and $27 = {3^3}$ .

Let a number be x and another number be y.

Now, we have to convert the question statement into a mathematical equation.

Cube of a number is 8 times the cube of another number, So we can write in math form.

${x^3} = 8{y^3}...........\left( 1 \right)$

The sum of the cubes of numbers is 243. So, we can write in math form like

${x^3} + {y^3} = 243..........\left( 2 \right)$

Put the value of ${x^3}$ in (2) equation.

$

\Rightarrow 8{y^3} + {y^3} = 243 \\

\Rightarrow 9{y^3} = 243 \\

\Rightarrow {y^3} = \dfrac{{243}}{9} \\

\Rightarrow {y^3} = 27 = {3^3} \\

\Rightarrow {y^3} = {3^3} \\

$

Taking cube roots on both sides

$ \Rightarrow y = 3$

Put the value of $y$ in (1) equation

$

\Rightarrow {x^3} = 8 \times {3^3} \\

\Rightarrow {x^3} = {2^3} \times {3^3} \\

$

Taking cube roots on both sides

$

\Rightarrow x = 2 \times 3 \\

\Rightarrow x = 6 \\

$

Now , we find the difference of two numbers that means we calculate $x - y$ .

So, $x - y = 6 - 3 = 3$

So, the correct option is (a).

Note-Whenever we face such types of problems we use some important points. Like let & #39;s

take two numbers (x and y) and use numbers to convert the question statement into a

mathematical equation then after solving some equation we get the required answer.

number $8 = {2^3}$ and $27 = {3^3}$ .

Let a number be x and another number be y.

Now, we have to convert the question statement into a mathematical equation.

Cube of a number is 8 times the cube of another number, So we can write in math form.

${x^3} = 8{y^3}...........\left( 1 \right)$

The sum of the cubes of numbers is 243. So, we can write in math form like

${x^3} + {y^3} = 243..........\left( 2 \right)$

Put the value of ${x^3}$ in (2) equation.

$

\Rightarrow 8{y^3} + {y^3} = 243 \\

\Rightarrow 9{y^3} = 243 \\

\Rightarrow {y^3} = \dfrac{{243}}{9} \\

\Rightarrow {y^3} = 27 = {3^3} \\

\Rightarrow {y^3} = {3^3} \\

$

Taking cube roots on both sides

$ \Rightarrow y = 3$

Put the value of $y$ in (1) equation

$

\Rightarrow {x^3} = 8 \times {3^3} \\

\Rightarrow {x^3} = {2^3} \times {3^3} \\

$

Taking cube roots on both sides

$

\Rightarrow x = 2 \times 3 \\

\Rightarrow x = 6 \\

$

Now , we find the difference of two numbers that means we calculate $x - y$ .

So, $x - y = 6 - 3 = 3$

So, the correct option is (a).

Note-Whenever we face such types of problems we use some important points. Like let & #39;s

take two numbers (x and y) and use numbers to convert the question statement into a

mathematical equation then after solving some equation we get the required answer.

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