If 3A = 2B = 4C, what is the simplified ratio A : B : C expressed in whole numbers?

Introduction / Context:
This algebraic ratio question asks us to determine the relationship between three variables A, B and C given that 3A, 2B and 4C are all equal to the same quantity. It tests the ability to interpret compound equalities and then express a three term ratio in the simplest whole number form.

Given Data / Assumptions:

• We are given that 3A = 2B = 4C.• All variables A, B and C are assumed to be non zero real numbers.• We must find the ratio A : B : C.

Concept / Approach:
If 3A, 2B and 4C are all equal to some common value, say k, then we can express A, B and C individually in terms of k. Specifically, A = k / 3, B = k / 2, and C = k / 4. Once we have expressions for A, B and C in terms of the same constant, we can form the ratio A : B : C and clear denominators by multiplying all terms by the least common multiple of the denominators. This produces a clean integer ratio.

Step-by-Step Solution:
Step 1: Let the common value be k. Then we have:• 3A = k, so A = k / 3.• 2B = k, so B = k / 2.• 4C = k, so C = k / 4.Step 2: Now write the ratio A : B : C in terms of k:A : B : C = (k / 3) : (k / 2) : (k / 4).Step 3: Because k is common and positive, it cancels out from the ratio. So we focus on 1 / 3 : 1 / 2 : 1 / 4.Step 4: To convert this to whole numbers, find the least common multiple of the denominators 3, 2 and 4, which is 12.Step 5: Multiply each term by 12:• (1 / 3) * 12 = 4.• (1 / 2) * 12 = 6.• (1 / 4) * 12 = 3.Step 6: Therefore, the simplified ratio A : B : C is 4 : 6 : 3.

Verification / Alternative check:
We can verify by assigning any convenient value to k and confirming that 3A, 2B and 4C become equal. For example, let k = 12. Then A = 12 / 3 = 4, B = 12 / 2 = 6, and C = 12 / 4 = 3. Check the equalities: 3A = 3 * 4 = 12, 2B = 2 * 6 = 12, and 4C = 4 * 3 = 12. All are equal to 12, which matches k. This confirms that A : B : C = 4 : 6 : 3 is consistent with the original condition.

Why Other Options Are Wrong:
• 3 : 2 : 4: This ratio does not satisfy 3A = 2B = 4C when tested with actual values.• 6 : 4 : 3: While similar in shape, interchanging positions leads to inconsistent equalities for 3A, 2B and 4C.• 2 : 3 : 4: This again rearranges the terms incorrectly and fails to make 3A, 2B and 4C equal.

Common Pitfalls:
Many learners mistakenly take the coefficients 3, 2 and 4 directly as the ratio A : B : C without inverting them. However, 3A = k actually implies A is k / 3, not 3k. Others forget to clear denominators and leave the ratio in fractional form, which makes comparison difficult. It is also easy to confuse the meaning of 3A = 2B = 4C and equate only two at a time without considering the third. Adopting the common value k technique simplifies the reasoning and avoids these mistakes.

Final Answer:
The correct simplified ratio is A : B : C = 4 : 6 : 3.