The average of five numbers is $306.4$. The average of the first two numbers is $431$ and the average of the last two numbers is $214.5$. What is the third number?

### Concept & Strategy This follows the standard subgroup subtraction principle. By finding the sum of all components and subtracting the parts we know, we can isolate the unknown central item. $$\text{Missing Number} = \text{Total Sum of All} - (\text{Sum of First Part} + \text{Sum of Last Part})$$ ### Step-by-Step Solution * **Given:** * Total numbers = $5$, Overall average = $306.4$. * First $2$ numbers, Average = $431$. * Last $2$ numbers, Average = $214.5$. * **Calculation:** 1. Calculate the total sum of all $5$ numbers: $5 \times 306.4$. * $5 \times 300 = 1500$ * $5 \times 6.4 = 32$ * Total sum = $1532$. 2. Calculate the total sum of the first $2$ numbers: $2 \times 431 = 862$. 3. Calculate the total sum of the last $2$ numbers: $2 \times 214.5 = 429$. 4. The 3rd number is the only number not included in the first two or the last two. 5. Third number = Total Sum - (Sum of First 2 + Sum of Last 2) 6. Third number = $1532 - (862 + 429) = 1532 - 1291 = 241$. 7. Looking at the given options: (a) 52, (b) 108, (c) 321, (d) Cannot be determined. The answer $241$ is not present. ### Exam Strategy & Shortcut **Unit Digit Check:** Let's check the unit digits of the sums. Total sum ($5 \times 306.4$) ends in a $0$ (since $5 \times .4 = 2.0$, technically the integer part ends in $2$). Let's be precise: $1532$ (ends in $2$). First two sum ($2 \times 431$) ends in $2$. Last two sum ($2 \times 214.5$) ends in $9$ (since $214.5 \times 2 = 429$). Sum of parts ends in $2 + 9 = 11 \rightarrow$ unit digit $1$. Third number unit digit = $2 - 1 = 1$. None of the numerical options (52, 108, 321) logically fit if you quickly check approximations (1500 - 1300 = 200). $321$ has the right unit digit but is too large. Hence, "None of these". ### Common Pitfall Selecting "Cannot be determined" simply because the answer isn't among the numerical options. If a precise unique value *can* be calculated (in this case, $241$), the answer is "None of these", not "Cannot be determined". ### Final Answer **Therefore, the correct answer is None of these.**