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The cube of a number is 8 times the cube of another number. If the sum of thecubes of numbers is 243, the difference of the numbers is:(a) 3(b) 4(c) 6(d) None of these

Last updated date: 17th Jul 2024
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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.