Average of $a$, $b$, and $c$ is 11; average of $c$, $d$ and $e$ is 17; average of $e$ and $f$ is 22 and average of $e$ and $c$ is 17. Find out the average of $a$, $b$, $c$, $d$, $e$, and $f$.

### Concept & Logic To find the average of a combined set of variables, you first need to find their total sum. We can extract the sum of different subsets from their given averages using the fundamental formula: $$\text{Sum} = \text{Average} \times \text{Number of elements}$$ Then, we strategically combine these subset sums to isolate the grand total $(a+b+c+d+e+f)$. ### Step-by-Step Solution * **Extracting Sums from Given Averages:** 1. Average of $a, b, c = 11 \rightarrow a + b + c = 11 \times 3 = 33$ 2. Average of $c, d, e = 17 \rightarrow c + d + e = 17 \times 3 = 51$ 3. Average of $e, f = 22 \rightarrow e + f = 22 \times 2 = 44$ 4. Average of $e, c = 17 \rightarrow e + c = 17 \times 2 = 34$ * **Combining the Subsets:** Notice that if we add equations (1), (2), and (3) together, we get almost all the variables we need, but with some duplicates. Let's add them: $(a + b + c) + (c + d + e) + (e + f) = 33 + 51 + 44$ $a + b + 2c + d + 2e + f = 128$ * **Isolating the Target Sum:** We want the sum $(a + b + c + d + e + f)$. Notice that our combined equation above can be rewritten as: $(a + b + c + d + e + f) + c + e = 128$ We already know from equation (4) that $(e + c) = 34$. Let's substitute this in: $(a + b + c + d + e + f) + 34 = 128$ $(a + b + c + d + e + f) = 128 - 34$ Sum of all 6 variables = 94 * **Calculating the Final Average:** $$\text{Average} = \frac{\text{Sum}}{6}$$ $$\text{Average} = \frac{94}{6} = \frac{47}{3}$$ Converting to a mixed fraction: $\frac{47}{3} = 15\frac{2}{3}$ ### Exam Strategy & Shortcut Look for the "overlapping" variables. The problem provides three continuous groups: $(a,b,c)$, $(c,d,e)$, and $(e,f)$. When you sum these groups, the boundary variables ($c$ and $e$) are counted twice. The problem conveniently provides the exact sum of those specific overlapping variables ($c+e$) at the end. Mentally: Total Sum = $(33 + 51 + 44) - (34) = 128 - 34 = 94$. Average = $94 / 6 = 15\frac{2}{3}$. This can be done in under 30 seconds without writing algebra. ### Common Pitfall A common trap is attempting to solve for the individual values of $a, b, c, d, e,$ and $f$. We have 6 variables but only 4 equations, meaning it is mathematically impossible to find their individual values. You must solve for the *group* sum as a single entity. ### Final Answer **Therefore, the correct answer is $15\frac{2}{3}$.**