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
Answer
Verified
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.
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