There are three positive numbers. One third of the average of all the three…

Concept — For any collection of n numbers, average = (sum of the numbers) ÷ n, so the sum can always be recovered from the average as sum = average × n. A phrase of the form “P is k less than Q” translates to the equation P = Q − k. Applying these two ideas turns every verbal statement about an average into a linear equation, and the unknown is then obtained by solving those equations together.

Application — Let the three positive numbers, written in increasing order, be a ≤ b ≤ c, so that c is the highest number and a is the lowest.

1. The average of all three numbers is (a + b + c)/3, so one third of that average is (a + b + c)/9.

2. “8 less than the value of the highest number” is c − 8, which gives the first equation: (a + b + c)/9 = c − 8.

3. The average of the lowest and the second lowest number is 8, so (a + b)/2 = 8, which gives a + b = 16.

4. Substituting a + b = 16 into the first equation gives (16 + c)/9 = c − 8.

5. Multiplying both sides by 9 clears the denominator: 16 + c = 9c − 72.

6. Collecting the c terms on one side and the constants on the other: 16 + 72 = 9c − c, that is 88 = 8c.

7. Dividing both sides by 8 gives c = 11, so the highest number is 11.

Cross-check — With a + b = 16 and c = 11 the three numbers total 27, so their average is 9 and one third of that average is 3; the highest number less 8 is 11 − 8 = 3, and the two sides agree. The result is also consistent with the stated ordering and positivity — for example the numbers 6, 10 and 11 satisfy every condition in the question.

Common slips — Dropping the words “one third of” and reading the condition as (16 + c)/3 = c − 8 gives c = 20, while reversing the comparison into 8 − c gives 16 + c = 72 − 9c and c = 5.6. Neither value can be reached from a correct translation of the sentence, so the phrase must be converted exactly as written.