The sum of two numbers is $\frac{28}{25}$ of the first number. The second number is what percent of the first?

### Concept & Formula When a relationship between variables is given as a fraction, setting up a direct algebraic equation lets you find the ratio between the variables. Once you have the ratio, calculating the percentage is straightforward. $$ \text{Percentage} = \left( \frac{\text{Second Number}}{\text{First Number}} \right) \times 100 $$ ### Step-by-Step Solution * Let the first number be $x$ and the second number be $y$. * **Given:** The sum of the two numbers ($x + y$) is $\frac{28}{25}$ of the first number ($x$). * Set up the equation: $$ x + y = \frac{28}{25}x $$ * Isolate $y$ to find its relationship with $x$: $$ y = \frac{28}{25}x - x $$ $$ y = \left( \frac{28}{25} - 1 \right)x $$ $$ y = \left( \frac{28 - 25}{25} \right)x $$ $$ y = \frac{3}{25}x $$ * The question asks for the second number ($y$) as a percentage of the first number ($x$). $$ \text{Required Percentage} = \left( \frac{y}{x} \right) \times 100 $$ $$ \text{Required Percentage} = \left( \frac{\frac{3}{25}x}{x} \right) \times 100 $$ $$ \text{Required Percentage} = \frac{3}{25} \times 100 $$ $$ \text{Required Percentage} = 3 \times 4 = 12\% $$ ### Exam Strategy & Shortcut Use ratio units directly. The phrase "sum ... is $\frac{28}{25}$ of the first" means if the first number is $25$ units, the total sum is $28$ units. Therefore, the second number is simply the difference: $28 - 25 = 3$ units. The required percentage is $\frac{3}{25}$. Since $\frac{1}{25} = 4\%$, then $\frac{3}{25} = 3 \times 4\% = 12\%$. ### Common Pitfall Students sometimes calculate the second number as a percentage of the *sum* instead of the *first number*, leading to $\frac{3}{28} \approx 10.7\%$, which is incorrect based on the phrasing "what percent of the first". ### Final Answer **Therefore, the correct answer is 12%.**