The average of two numbers is $XY$. If one number is $X$, the other is

### Concept & Formula The average of two numbers is their sum divided by $2$. We can reverse this relationship to find an unknown number if we know the average and the other number. The total sum is always the average multiplied by the number of items. $$ \text{Sum} = \text{Average} \times \text{Total Count} $$ ### Step-by-Step Solution * **Given:** The average of the two numbers is $XY$. The first number is $X$. Let the unknown second number be $A$. * **Calculation:** Apply the average formula for the two numbers: $\frac{X + A}{2} = XY$ Multiply both sides by $2$ to isolate the sum on the left: $X + A = 2XY$ Subtract $X$ from both sides to find the value of the unknown number $A$: $A = 2XY - X$ ### Exam Strategy & Shortcut **Direct Summation Method:** Memorize the rule that for two numbers, $\text{Sum} = 2 \times \text{Average}$. Since the average is $XY$, the sum is immediately $2XY$. If one piece of the sum is $X$, the remaining piece must be the total minus the known piece, giving $2XY - X$ directly without writing out algebraic steps. ### Common Pitfall A common error is confusing the sum with the average and selecting $XY - X$, or mistakenly dividing by $2$ instead of multiplying, which might lead a student to guess $\frac{Y}{2}$. Always remember that the sum is larger than the average when dealing with positive integers. ### Final Answer **Therefore, the correct answer is 2XY - X.**