If $a : b : c = 2 : 3 : 4$ and $2a - 3b + 4c = 33$, then the value of $c$ is (S.S.C., 2007)

### Concept & Absolute Value from Proportions When a ratio is given alongside an equation equal to a specific, concrete number, you cannot just substitute the base ratio values directly. You must introduce a constant of proportionality (like $k$) to convert the relative ratios into absolute values. ### Step-by-Step Solution - **Given Ratio:** $a : b : c = 2 : 3 : 4$. - **Introduce Constant:** Let $a = 2k$, $b = 3k$, and $c = 4k$, where $k$ is an unknown constant. - **Given Equation:** $2a - 3b + 4c = 33$. - **Substitute Variables with k:** $2(2k) - 3(3k) + 4(4k) = 33$ - **Simplify:** $4k - 9k + 16k = 33$ $(4 - 9 + 16)k = 33$ $11k = 33$ - **Solve for k:** $k = \frac{33}{11} = 3$. - **Find c:** We defined $c = 4k$. $c = 4(3) = 12$. ### Exam Strategy & Shortcut Calculate the "ratio unit value" directly in the equation: $2(2) - 3(3) + 4(4) = 4 - 9 + 16 = 11$ units. The equation says 11 units $= 33$, so 1 unit $= 3$. Since $c$ represents 4 units, $c = 4 \times 3 = 12$. This skips writing out the algebraic $k$. ### Common Pitfall Students often forget to find the final value of $c$ and mistakenly select the value of the constant multiplier $k$ if it is available in the options, or they make simple arithmetic errors when combining the negative and positive terms ($4k - 9k + 16k$). ### Final Answer Therefore, the correct answer is **$12$**.