If the arithmetic mean of $0$, $5$, $4$, $3$ is $a$, that of $-1$, $0$, $1$, $5$, $4$, $3$ is $b$ and that of $5$, $4$, $3$ is $c$, then the relation between $a$, $b$ and $c$ is

### Concept & Strategy To find the relationship between multiple variables defined as averages, you must systematically calculate the exact numerical value of each variable using the basic average formula, then test those values against the given options. $$ \text{Average} = \frac{\text{Sum of terms}}{\text{Number of terms}} $$ ### Step-by-Step Solution * **Given:** Three separate sets of numbers whose averages define variables $a$, $b$, and $c$. * **Calculation:** Step 1: Calculate the value of $a$. Sum = $0 + 5 + 4 + 3 = 12$ Count = $4$ $a = \frac{12}{4} = 3$ Step 2: Calculate the value of $b$. Sum = $-1 + 0 + 1 + 5 + 4 + 3 = 12$ Count = $6$ $b = \frac{12}{6} = 2$ Step 3: Calculate the value of $c$. Sum = $5 + 4 + 3 = 12$ Count = $3$ $c = \frac{12}{3} = 4$ Step 4: Test the calculated values ($a=3$, $b=2$, $c=4$) against the options. (a) $3 = 2 = 4$ (Incorrect) (b) $a : b : c \rightarrow 3 : 2 : 4$ (Correct, this matches our values perfectly) (c) $4(3) = 5(2) = 4 \rightarrow 12 = 10 = 4$ (Incorrect) (d) $3 + 2 + 4 = 12 \rightarrow 9 = 12$ (Incorrect) ### Exam Strategy & Shortcut **Observation of Constant Sum:** A sharp eye will notice immediately that the sum of the numbers in all three sets is identical! Set A sum = $12$. Set B sum = $12$ (the $-1$ and $+1$ cancel). Set C sum = $12$. Since Sum is constant, the Average is strictly inversely proportional to the Count ($n$). $a = \frac{12}{4}$, $b = \frac{12}{6}$, $c = \frac{12}{3}$. This immediately yields the values $3, 2, 4$ without doing heavy addition, making the ratio $3:2:4$ obvious. ### Common Pitfall Ignoring the zero. When calculating $a$, students sometimes divide by $3$ instead of $4$, thinking $0$ doesn't count as a data point. A zero is a valid observation and must be included in the denominator count. ### Final Answer **Therefore, the correct answer is a : b : c = 3 : 2 : 4.**