If 3A = 6B = 9C, then what is the ratio A : B : C?

Introduction / Context:
This question is a straightforward algebraic ratio problem. We are told that three expressions 3A, 6B and 9C are all equal to each other, and asked to find the ratio A : B : C. Such questions test the ability to manipulate proportional relationships and express variables in terms of a common constant.

Given Data / Assumptions:
• 3A = 6B = 9C. • A, B and C are positive real numbers. • We must find the simplified ratio A : B : C.

Concept / Approach:
When three expressions are all equal to one common value, we can represent that value with a constant, say k. Thus 3A = 6B = 9C = k. From each equality, we solve for A, B and C in terms of k. Then we form the ratio A : B : C and simplify it. This is a systematic way to handle equal products or equal multiples in ratio problems.

Step-by-Step Solution:
Step 1: Let 3A = 6B = 9C = k. Step 2: From 3A = k, we get A = k / 3. Step 3: From 6B = k, we get B = k / 6. Step 4: From 9C = k, we get C = k / 9. Step 5: Now A : B : C = (k / 3) : (k / 6) : (k / 9). Step 6: Factor out k to see that it cancels: A : B : C = 1/3 : 1/6 : 1/9. Step 7: Multiply each term by the least common multiple of denominators 3, 6 and 9, which is 18, to convert to whole numbers: (1/3) * 18 = 6, (1/6) * 18 = 3, (1/9) * 18 = 2. Step 8: Therefore, A : B : C = 6 : 3 : 2.

Verification / Alternative Check:
Choose a simple value for k, for example k = 18. Then A = 18 / 3 = 6, B = 18 / 6 = 3 and C = 18 / 9 = 2. Now compute 3A, 6B and 9C. We have 3A = 18, 6B = 6 * 3 = 18 and 9C = 9 * 2 = 18. All three expressions are equal, as required. This confirms that the ratio A : B : C is 6 : 3 : 2.

Why Other Options Are Wrong:
• 6 : 3 : 1 would imply C is half the correct value compared with the derived ratio. • 9 : 3 : 6 and 9 : 3 : 1 do not satisfy 3A = 6B = 9C when we substitute example values based on these ratios.

Common Pitfalls:
A common mistake is to misread 3A = 6B = 9C as three separate equalities without introducing a common constant, which can lead to incorrect algebra. Another error is forgetting to cancel the common factor k when forming the ratio, or not clearing the denominators properly. Using the constant k method and then clearing denominators systematically is the cleanest approach.

Final Answer:
The required ratio is A : B : C = 6 : 3 : 2.