There are three positive numbers. One third of the average of all the three numbers is 8 less than the value of the highest number. The average of the lowest and the second lowest number is 8. What is the highest number?

### Concept & Logic To solve this, we translate the word problem into algebraic equations using the basic formula for averages: $$Average = \frac{\text{Sum of terms}}{\text{Number of terms}}$$ Let the three positive numbers in ascending order be $x$, $y$, and $z$, where $z$ is the highest number. ### Step-by-Step Solution * **Given:** The average of the lowest ($x$) and second lowest ($y$) number is 8. * **Calculation:** $$\frac{x + y}{2} = 8$$ $$x + y = 16$$ (This is our first equation) * **Given:** One-third of the average of all three numbers is 8 less than the highest number ($z$). * **Calculation:** $$\frac{1}{3} \times \left(\frac{x + y + z}{3}\right) = z - 8$$ $$\frac{x + y + z}{9} = z - 8$$ * **Deduction:** Substitute the value of $(x + y)$ from our first equation into this new equation: $$\frac{16 + z}{9} = z - 8$$ $$16 + z = 9(z - 8)$$ $$16 + z = 9z - 72$$ $$16 + 72 = 9z - z$$ $$88 = 8z$$ $$z = 11$$ ### Exam Strategy & Shortcut Instead of writing out full variables, use chunking. You know the sum of the first two numbers is $2 \times 8 = 16$. Let the highest number be $H$. The sum of all three is $(16 + H)$. The average of all three is $\frac{16 + H}{3}$. One third of this is $\frac{16 + H}{9}$. Set this equal to $H - 8$ and solve directly. This mental chunking saves you from writing multiple $x$ and $y$ variables. ### Common Pitfall A very common mistake is misreading "one third of the average" as just "one third of the sum". This leads to the incorrect equation $\frac{x+y+z}{3} = z - 8$, which will result in $z = 10$. Always read the fractional relationships carefully! ### Final Answer **Therefore, the correct answer is 11.**