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The problem states: "Three numbers are in the ratio 2:3:4. The sum of their cubes is 3391285. Find the numbers."

Let's denote the three numbers as \( 2x \), \( 3x \), and \( 4x \) respectively, where \( x \) is a common factor.

Given:
\[
(2x)^3 + (3x)^3 + (4x)^3 = 3391285
\]

Now, let's calculate each term:

\[
(2x)^3 = 8x^3
\]
\[
(3x)^3 = 27x^3
\]
\[
(4x)^3 = 64x^3
\]

So, the equation becomes:
\[
8x^3 + 27x^3 + 64x^3 = 3391285
\]

Combine the terms:
\[
99x^3 = 3391285
\]

Now, solve for \( x^3 \):
\[
x^3 = \frac{3391285}{99} = 34255
\]

Taking the cube root of both sides to find \( x \):
\[
x = \sqrt[3]{34255} \approx 32.9995
\]

Since \( x \) should be an integer, we'll approximate \( x \) as \( 33 \).

Thus, the numbers are:
\[
2x = 2 \times 33 = 66
\]
\[
3x = 3 \times 33 = 99
\]
\[
4x = 4 \times 33 = 132
\]

The three numbers are **66, 99, and 132**.

Would you like more details or have any questions?

Here are some related questions:
1. What is the general approach to solving ratio problems?
2. How do you simplify cube roots when dealing with larger numbers?
3. Can the same method be applied to different ratios, like 3:5:7?
4. How do you verify the solution by recalculating the sum of the cubes?
5. What if the ratio was a fraction, e.g., \( \frac{1}{2} : \frac{3}{2} : \frac{5}{2} \)?

**Tip:** When working with ratios, assigning a variable to represent the common factor helps in setting up equations to solve for unknowns effectively.

Learn how to find three numbers in the ratio 2:3:4 whose sum of cubes equals 3391285. This problem involves solving equations based on the ratio of numbers and their cubes. Suitable for high school students.

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