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# word problem

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In Mr. Patterson's class, the average score among students who studied for an exam was 78. The average among students who did not study was 54. The overall class average was 70. What portion of the class did not study? Express your answer as a common fraction.

Jul 4, 2024

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Let S be the number of students who studied and NS be the number of students who did not study

We can represent the total number of students (T) with the following equation

T = S + N

We are given the average scores for each group and the overall class average:

Average score for those who studied (AS): AS = 78

Average score for those who did not study (ANS): ANS = 54

Overall class average (Avg): Avg = 70

We can express the total score for all students using the average scores and the number of students in each group:

Total score for those who studied (TS): TS = AS * S

Total score for those who did not study (TNS): TNS = ANS * NS

Total score for the entire class (Total): Total = Avg * T (since Avg represents the average score per student)

Since the total score for the entire class is the sum of the scores from each group:

Total = TS + TNS

We can substitute the expressions for each score:

Avg * T = AS * S + ANS * NS

Now we can substitute the values we are given:

70 * (S + NS) = 78 * S + 54 * NS

We can rearrange the equation to isolate NS (the number of students who did not study):

20 * (S + NS) = 24 * S

20S + 20NS = 24S

-4S = 20NS

NS = (-4/20) * S (we can divide both sides by -4 as long as S is not 0, which we can safely assume since there must be at least one student who studied)

NS = -(1/5) * S

Since NS represents a number of students, it cannot be negative. Therefore, we can flip the signs to get a positive value for the fraction representing the portion of the class that did not study:

NS/T = (1/5) * S/T

We are interested in the portion of the class that did not study, so we can express this as:

Fraction of class who did not study = NS/T = 1/5.

Jul 4, 2024
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To find the portion of the class that did not study, we can set up an equation using the given information: Let x be the proportion of students who studied and (1 - x) be the proportion of students who did not study. The equation can be set up as follows: \(78x + (1 - x) 54 = 70\) . Solving this equation, we get:

1. \(78x + 54 - 54x = 70\)
2. \(24x + 54 = 70\)
3. \(24x = 16\)
4. \(x = 16 / 24\)
5. \(x = 2 / 3\)

Therefore, the portion of the class that did not study is \(1 - 2/3 = 1/3\).

Jul 5, 2024