Imperfect proofreaders -- How many mistakes were missed by both?
How many mistakes were missed by both?
Proofreader "A" finds 240 mistakes in a document.
Proofreader "B", working independently, finding 392 mistakes in the same document.
One hundred twenty eight mistakes of each's list of mistakes were in common. In other words, both "A" and "B" found the same mistakes.
"A" and "B" had different efficiencies at finding mistakes and their efficiencies were no different over the entire document and over the mistakes that they both found.
The answer is 231.
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1 solution
The total number of mistakes was not given in the problem. In fact, that is the actual problem to be solved.
t is the total number of mistakes. a is the rate of "A" at finding mistakes. b is the rate of "B" at finding mistakes.
Therefore the equation set to solved is t a = 2 4 0 ∧ t b = 3 9 2 ∧ t a b = 1 2 8
Which can be rewritten as t t 2 4 0 t 3 9 2 = 1 2 8 .
Which can be rewritten as t = 1 2 8 2 4 0 × 3 9 2 ⇒ 7 3 5
The number of missed mistakes is the total number of mistakes less the sum of the mistakes found by "A" and "B" plus the mistakes found in common because otherwise those mistakes would have been subtracted twice.
7 3 5 + 1 2 8 − ( 2 4 0 + 3 9 2 ) ⇒ 2 3 1 . Q.E.D.