Question

The cube of a number is 8 times the cube of another number. If the sum of the cubes of the numbers is 243, then find the difference between the numbers.

Answer 1

Answer:

x=24 and y=3

Step-by-step explanation:

Let one number be x and the other number be y.

Given that,

Cube of one number is 8 times the cubes of other→x³=8y³.................(1)

Also,

Sum of the cube of these numbers is 243.→x³+y³=243

Now,

Solving the equation,

x³+y³=243

→y³+8y³=243 [Using (1)]

→9y³=243

→y³=27

→ y=3

Substituting y=3 in (1),

x³ = 8y³

x³ = 8(3)³

x³ = 8(27)

x³ = 216

x = 6

Now,

Difference of the numbers:

x-y

=6-3

=3

•Verification:

x³+y³

=(3)³+(6)³

=27+216

=243

=RHS

Here,LHS=RHS=243

Hence, verified

Answer 2

Let us suppose the two numbers as x and y.

So, by the condition given in our question, we can make two equations;

Equation 1: x³ = 8y³

Equation 2: x³ + y³ = 243

So, as we have these equations, let's find the values of x and y;

First, we'll find the value of "y" by the substitution method;

We know the variable value of x³ as 8y³. So, keeping this in mind, let's start substituting;

x³ + y³ = 243

Plugging the variable value of x³;

8y³ + y³ = 243

Simplify;

9y³ = 243

As we have to isolate "y³", we'll divide both sides by 9;

$\dfrac{\cancel{9}\text{y}^3}{\cancel{9}}=\dfrac{243}{9}$

Simplify;

y³ = 27

To isolate "y", let's cube root both the sides;

$\sqrt[3]{\text{y}^3}=\sqrt[3]{27}$

Simplify;

y = 3

Now as we got the real value of y, let's substitute it to the equation 1;

x³ = 8·(3)³

Simplify;

x³ = 8 × 27 = 216

Now, cube root both sides;

$\sqrt[3]{\text{x}^3}=\sqrt[3]{216}$

Simplify;

x = 6

Now, as we have got the value of x and y, let's find their difference;

x - y = 6 - 3 = 3

Therefore, the difference between the numbers is 3.