Introduction / Context:
This question is a standard example of working with compound ratio expressions where a single quantity is given equal to multiple expressions involving three variables. Such questions test your ability to rewrite relationships like A = 2B = 4C into a clean ratio A : B : C. This is very useful in algebra and in many quantitative aptitude tests.
Given Data / Assumptions:
- We are given A = 2B = 4C.
- We assume A, B and C are positive real numbers.
- The task is to express A : B : C in simplest integer form.
Concept / Approach:
When we have an expression like A = 2B = 4C, we introduce a common value, say k, to represent the equal quantity. That is, we set A = k, 2B = k and 4C = k. This allows us to express B and C in terms of k and then find the ratio A : B : C. To convert the resulting fractional expressions into whole numbers, we multiply all terms by a suitable common multiple of the denominators.
Step-by-Step Solution:
1) Let A = 2B = 4C = k for some positive constant k.
2) From A = k, we have A = k.
3) From 2B = k, we get B = k / 2.
4) From 4C = k, we get C = k / 4.
5) So A : B : C = k : (k / 2) : (k / 4).
6) To clear denominators, multiply all three terms by 4.
7) This gives A : B : C = 4k : 2k : 1k.
8) Therefore, the required simplest ratio is 4 : 2 : 1.
Verification / Alternative check:
We can pick a specific value for k to verify. Let k = 4. Then A = 4, B = 4 / 2 = 2, C = 4 / 4 = 1. Check the original relationship: 2B = 2 * 2 = 4 and 4C = 4 * 1 = 4, both equal A = 4. So 4 : 2 : 1 is consistent with A = 2B = 4C. This numerical example confirms that our derived ratio is correct.
Why Other Options Are Wrong:
Option B (1 : 2 : 4) reverses the pattern and does not satisfy A = 2B = 4C when tested with actual values. Option C (8 : 4 : 1) correctly preserves proportions but is not in simplest form, since it can be reduced by dividing all terms by 2 to obtain 4 : 2 : 1. Option D (16 : 4 : 1) and option E (2 : 1 : 4) also fail to meet the original condition when substituted back into the relationship A = 2B = 4C.
Common Pitfalls:
A frequent error is inverting the relationships, such as assuming A : B : C = 1 : 2 : 4 without carefully solving for each variable. Another mistake is failing to clear denominators correctly, which leads to fractional ratios that are not simplified. Always introduce a common constant k, express each variable in terms of k, and then carefully reduce the resulting ratio to its simplest integer form.
Final Answer:
The simplest ratio of A : B : C that satisfies A = 2B = 4C is
4 : 2 : 1.
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