What is the fourth proportional to the numbers 2, 3 and 6, that is, the value x such that 2 : 3 = 6 : x?

Introduction / Context:
This problem asks for the fourth proportional to three given numbers. If a, b, c and d are in proportion such that a : b = c : d, then d is called the fourth proportional to a, b and c. These problems are fundamental in ratio and proportion and help reinforce cross multiplication techniques and the underlying structure of proportional relationships.

Given Data / Assumptions:
- First number a = 2.
- Second number b = 3.
- Third number c = 6.
- We seek d such that 2 : 3 = 6 : d.

Concept / Approach:
From the definition of proportion, a/b = c/d. Therefore, d can be found by rearranging the equation to d = (b * c) / a. Once we know a, b and c, we simply apply this formula. Alternatively, we can express the ratio as a fraction and use cross multiplication to solve for d directly. Both methods are equivalent and give the same final result.

Step-by-Step Solution:
Step 1: Write the proportion 2 : 3 = 6 : x, where x is the fourth proportional. Step 2: Express this as a fraction: 2 / 3 = 6 / x. Step 3: Cross multiply: 2x = 3 * 6. Step 4: Compute 3 * 6 = 18. Step 5: Solve for x: 2x = 18, so x = 18 / 2 = 9. Step 6: Therefore, the fourth proportional to 2, 3 and 6 is 9.

Verification / Alternative check:
As a quick check, substitute x = 9 back into the ratio and verify equality. The ratio 2 : 3 is equal to 2/3, and 6 : 9 simplifies to 2 : 3 as well since both 6 and 9 can be divided by 3. Therefore 6/9 also equals 2/3. Since both ratios match, x = 9 is the correct fourth proportional.

Why Other Options Are Wrong:
10, 7 and 8 do not maintain the proportion. For instance, 6 : 10 simplifies to 3 : 5, which is not equal to 2 : 3. Similarly, 6 : 7 and 6 : 8 do not match the original 2 : 3 ratio. Only 9 ensures that 2 : 3 = 6 : 9 when simplified.

Common Pitfalls:
Some learners confuse third and fourth proportionals and may incorrectly use c^2/a instead of b*c/a. Others may perform cross multiplication incorrectly, perhaps writing 2/3 = x/6 or swapping numerator and denominator. Writing out the proportion clearly and cross checking the simplified ratios helps avoid these mistakes.

Final Answer:
The fourth proportional to 2, 3 and 6 is 9.