Final answer:
To find the smallest fraction chosen by Vicky, we set up a system of equations according to the conditions provided in the question. By defining a as the largest fraction, b as the middle, and c as the smallest, we solve for c using the given sum and ratio conditions, and then find a and b accordingly to determine the correct option among the choices provided.
Step-by-step explanation:
The problem provided requires solving a system of equations based on the given conditions of three fractions a, b, and c such that a > b > c, the sum of these fractions equals 3 13/24, and when the largest fraction is divided by the smallest fraction the result is 6 2/3, which is 6 more than the middle fraction.
Let's define the fractions as follows:
- a - the largest fraction
- b - the middle fraction
- c - the smallest fraction
From the given: a + b + c = 3 13/24 or 85/24 (1)
And the division of the largest by the smallest fraction gives us: a/c = 6 2/3 or 20/3. We can also write this as a = 20c/3 (2)
Now we are told that a/c - 6 is equal to b. From equation (2), we have 20c/3 = b + 6, converting 6 to a fraction with a common denominator gives us 18/3, thus 20c/3 - 18/3 = b (3)
Substituting the value of a and b in terms of c into equation (1), we get: 20c/3 + (20c/3 - 18/3) + c = 85/24.
Solving this equation for c gives us the smallest fraction Vicky chose:
c = (85/24) / ((20/3) + (20/3 - 18/3) + 1)
After solving for c, we can use either equation (2) or (3) to find a and b, respectively, and then match the group of fractions to one of the answer choices provided.
Therefore answer is A. (203/180),(29/12), (58/45).