Depth · one concept from first principles · step 2 of 4

Write the overall average as a weighted average

$50$ students took an exam, and $20$ of them failed. The lowest passing score was $5$ points below the average of all $50$ students, $30$ points below the average of the students who passed, and $3$ points below twice the average of the students who failed. Find the lowest passing score.

ExpressionsData & probability

Your answer

How to solve

Strategy Convert to Algebra — Let the two group averages be unknowns and express the overall average from them, weighting each group by its size. The three descriptions of the cutoff then give two equations.

1STEP 1

Write the overall average

Let the passers' average be x and the failers' average be y. The 30 passers scored 30x in total and the 20 failers 20y.

overall average = (30x + 20y)/50 = (3x + 2y)/5

2STEP 2

Set the three descriptions equal

All three expressions name the same cutoff score.

(3x + 2y)/5 - 5 = x - 30 = 2y - 3

3STEP 3

Clear fractions

From the first equality, multiply by 5: 3x + 2y - 25 = 5x - 150. The second equality rearranges directly.

2x - 2y = 125, x - 2y = 27

4STEP 4

Solve and read off the cutoff

Subtracting removes y at once. The cutoff is 30 below the passers' average.

x = 98, 98 - 30 = 68

Answer

68 points

From x - 2y = 27, y = 35.5. Overall average = (30 × 98 + 20 × 35.5)/50 = (2940 + 710)/50 = 73, and 73 - 5 = 68. Also 2 × 35.5 - 3 = 68. All three descriptions agree.

Takeaway

To combine averages, go back to totals: average times the number of people.

Write the overall average
Set the three descriptions equal
Clear fractions
Solve and read off the cutoff

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