Three numbers A, B and C are in the ratio 1 : 2 : 3. Their average is 600. If A is increased by 10% and B is decreased by 20%, then to get the average increased by 5%, C will be increased by

### Concept & Aggregate Net Change When dealing with averages of sets, remember that the Total Sum = Average $\times$ Count. Track the absolute numerical changes required in the total sum rather than dealing with complex interwoven percentage formulas. ### Step-by-Step Solution * **Step 1: Find the absolute values of A, B, and C.** * Ratio of A : B : C = $1 : 2 : 3$. Let them be $x$, $2x$, and $3x$. * Average of the three numbers = $600$. * Total Sum = $600 \times 3 = 1800$. * $x + 2x + 3x = 1800 \implies 6x = 1800 \implies x = 300$. * So, $A = 300$, $B = 600$, $C = 900$. * **Step 2: Calculate target total sum.** * The new average needs to be increased by $5\%$. * New Average = $600 \times 1.05 = 630$. * Target New Sum = $630 \times 3 = 1890$. * Required net increase in sum = $1890 - 1800 = +90$. * **Step 3: Calculate changes in A and B.** * $A$ is increased by $10\%$. Change in $A = +10\%$ of $300 = +30$. * $B$ is decreased by $20\%$. Change in $B = -20\%$ of $600 = -120$. * Current net change from A and B combined = $+30 - 120 = -90$. * **Step 4: Find the required change in C.** * Total required change = Change in $A$ + Change in $B$ + Change in $C$. * $+90 = -90 + \text{Change in } C$ * Change in $C = 90 + 90 = 180$. * Thus, $C$ must be increased by $180$. ### Exam Strategy & Shortcut Focus purely on the absolute changes. The sum needs to go up by $5\%$ of $1800$, which is $+90$. $A$ ($300$) goes up $10\%$ $\implies +30$. $B$ ($600$) goes down $20\%$ $\implies -120$. Right now, you are down $90$ ($30 - 120 = -90$). To reach a target of $+90$, you must overcome the $-90$ deficit AND add another $90$. So, $C$ must increase by $90 + 90 = 180$. ### Common Pitfall A common mistake is trying to calculate percentage changes on $C$ relative to $A$ or $B$, or confusing "decrease by 20%" as finding the remaining $80\%$ without correctly tracking the net deficit on the total sum. ### Final Answer Therefore, the correct answer is **180**.