Question 1194368: Student A, Student B, and Student C have been hired to help scientists develop a new flavour of juice. There are 4200 samples to test. Each sample either contains
blueberry or does not. Each student is asked to taste each sample and report whether or not they think it contains blueberry. Student A reports correctly on exactly 90% of the samples containing blueberry and reports correctly on exactly 88% of the samples that do not contain blueberry. The results for all three students are shown below.
Student A Student B Student C
%samples containing blueberry 90% 98% (2m)%
%samples not containing blueberry 88% 86% (4m)%
Student B reports 315 more samples as containing blueberry than Student A. For
some positive integers m, the total number of samples that the three students report as containing blueberry is equal to a multiple of 5 between 8000 and 9000. What is the sum of all such values of m.
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! Certainly, let's break down the problem and find the sum of all possible values of 'm'.
**1. Define Variables:**
* Let 'x' be the number of samples containing blueberries.
* Let 'y' be the number of samples not containing blueberries.
**2. Set up Equations**
* **Total Samples:** x + y = 4200
* **Student B Reports More Blueberries:**
* 0.98x - 0.90x = 315
* 0.08x = 315
* x = 3937.5
* **Since the number of samples must be whole numbers:**
* x = 3938
* y = 4200 - 3938 = 262
**3. Determine the Number of Samples Reported as Containing Blueberries by Each Student:**
* **Student A:** 0.90x = 0.90 * 3938 = 3544.2 ≈ 3544
* **Student B:** 0.98x = 0.98 * 3938 = 3861.24 ≈ 3861
* **Student C:** (2m/100) * x = (2m/100) * 3938 = 78.76m ≈ 78.76m
**4. Find the Total Number of Samples Reported as Containing Blueberries by All Students:**
* Total = Student A + Student B + Student C
* Total = 3544 + 3861 + 78.76m
* Total = 7405 + 78.76m
**5. Determine the Range for Total Samples Containing Blueberries**
* The total number of samples reported as containing blueberries is a multiple of 5 between 8000 and 9000.
* Therefore:
* 8000 ≤ 7405 + 78.76m ≤ 9000
* 595 ≤ 78.76m ≤ 1595
**6. Find Possible Values of 'm'**
* 595 / 78.76 ≤ m ≤ 1595 / 78.76
* 7.55 ≤ m ≤ 20.23
* Since 'm' is a positive integer:
* Possible values of 'm': 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
**7. Calculate the Sum of All Possible Values of 'm'**
* Sum = 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20
* Sum = 153
**Therefore, the sum of all possible values of 'm' is 153.**
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