Introduction / Context:
This question is yet another variant of equal expressions involving three variables, where multiple scaled versions of the variables are equal. This time the coefficients are 5, 13 and 7, which are all different primes. The aim is to determine the ratio A : B : C. These problems help strengthen your algebraic manipulation skills and understanding of proportional relationships.
Given Data / Assumptions:
- We have the equality 5A = 13B = 7C.
- Variables A, B and C are positive real numbers.
- We need to find A : B : C in simplest integer form.
Concept / Approach:
Introduce a common equal value k so that 5A = 13B = 7C = k. Then express each variable in terms of this common value. The resulting expressions will involve fractions with denominators 5, 13 and 7. To convert these into whole numbers in a ratio, use the least common multiple of those denominators. This standard method ensures that A, B and C are expressed with a common scaling, giving an integer ratio.
Step-by-Step Solution:
1) Let 5A = 13B = 7C = k.
2) From 5A = k, A = k / 5.
3) From 13B = k, B = k / 13.
4) From 7C = k, C = k / 7.
5) So A : B : C = (k / 5) : (k / 13) : (k / 7).
6) Cancel k from each term to obtain 1 / 5 : 1 / 13 : 1 / 7.
7) The denominators are 5, 13 and 7, whose LCM is 5 * 13 * 7 = 455.
8) Multiply each term by 455 to clear denominators.
9) A term: (1 / 5) * 455 = 91.
10) B term: (1 / 13) * 455 = 35.
11) C term: (1 / 7) * 455 = 65.
12) Thus, A : B : C = 91 : 35 : 65.
Verification / Alternative check:
Choose k = 455 as a convenient common value. Then A = 455 / 5 = 91, B = 455 / 13 = 35 and C = 455 / 7 = 65. Check the original equation: 5A = 5 * 91 = 455, 13B = 13 * 35 = 455 and 7C = 7 * 65 = 455. All three are equal, confirming that 5A = 13B = 7C. This verifies our ratio 91 : 35 : 65 is correct and consistent with the given condition.
Why Other Options Are Wrong:
Option B (65 : 35 : 91) and option C (35 : 91 : 65) are permutations of the numbers that do not match the positions of A, B and C as per the given equality. Option D (7 : 13 : 5) and option E (13 : 5 : 7) incorrectly treat the coefficients in the equality as direct ratio terms, which is not valid. Only 91 : 35 : 65 gives values of A, B and C that satisfy 5A = 13B = 7C.
Common Pitfalls:
A frequent mistake is to wrongly assume that A : B : C is simply 5 : 13 : 7 or its inverse, without doing the full conversion process. Another error is miscomputing the least common multiple of 5, 13 and 7, which are all prime and therefore multiply directly. Always carefully express each variable in terms of the common constant and then clear denominators properly to get the correct ratio.
Final Answer:
The simplest ratio A : B : C that satisfies 5A = 13B = 7C is
91 : 35 : 65.
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