Introduction / Context:
This question is another example of solving compound proportional relationships, similar to earlier problems but with a different numeric structure. You are given that 2A, 3B and 8C are all equal to one common value. The task is to express A : B : C as a simple integer ratio. Such problems strengthen your ability to translate equalities into ratios and to handle denominators correctly.
Given Data / Assumptions:
- We are told 2A = 3B = 8C.
- A, B and C are all assumed to be positive real numbers.
- We want to find A : B : C in simplest integer form.
Concept / Approach:
Introduce a common constant k such that 2A = 3B = 8C = k. Then express each variable in terms of k. From there, A, B and C are written as fractions involving k. To convert these fractional expressions into an integer ratio, multiply through by the least common multiple of the denominators. Finally, simplify the resulting numbers to get the simplest form of A : B : C.
Step-by-Step Solution:
1) Let 2A = 3B = 8C = k for some positive constant k.
2) From 2A = k, A = k / 2.
3) From 3B = k, B = k / 3.
4) From 8C = k, C = k / 8.
5) So A : B : C = (k / 2) : (k / 3) : (k / 8).
6) Cancel k from all terms, giving 1 / 2 : 1 / 3 : 1 / 8.
7) To remove denominators, multiply each term by the LCM of 2, 3 and 8, which is 24.
8) This gives (1 / 2) * 24 : (1 / 3) * 24 : (1 / 8) * 24 = 12 : 8 : 3.
9) Therefore, the simplest ratio A : B : C is 12 : 8 : 3.
Verification / Alternative check:
Select k = 24 as a convenient value. Then A = 24 / 2 = 12, B = 24 / 3 = 8 and C = 24 / 8 = 3. Check the original condition: 2A = 2 * 12 = 24, 3B = 3 * 8 = 24 and 8C = 8 * 3 = 24. All three expressions equal 24, which confirms that 2A = 3B = 8C. Therefore 12 : 8 : 3 is consistent with the given equation and is the correct ratio.
Why Other Options Are Wrong:
Option A (8 : 3 : 2) and option B (8 : 4 : 3) do not satisfy the relationship 2A = 3B = 8C when you test them with any common multiple. Option C (2 : 3 : 8) reverses the structure and clearly does not meet the required equality. Option E (3 : 8 : 12) is simply the reverse of the correct order and again fails to meet the original condition when substituted. Only 12 : 8 : 3 produces equal values for 2A, 3B and 8C.
Common Pitfalls:
Typical mistakes include directly writing A : B : C as 2 : 3 : 8 or 1 : 2 : 4 without carefully forming the equations. Another common error is forgetting to use the least common multiple when clearing denominators, which can lead to incorrect or non-simplified ratios. Always express each variable in terms of a common constant, cancel that constant and then scale up to remove fractions.
Final Answer:
The simplest ratio A : B : C that satisfies 2A = 3B = 8C is
12 : 8 : 3.
Discussion & Comments