If $2\frac{1}{2}$ is added to a number and the sum multiplied by $4\frac{1}{2}$ and $3$ is added to the product and the sum is divided by $1\frac{1}{5}$, the quotient becomes $25$. What is the number?

### Concept & Strategy This is a classic "working backwards" or direct algebraic translation problem. Converting mixed fractions to improper fractions or decimals is crucial for clean calculation. Let the unknown number be $x$. ### Step-by-Step Solution * **Preparation:** Convert all mixed fractions to decimals for easier manipulation: $2\frac{1}{2} = 2.5$ $4\frac{1}{2} = 4.5$ $1\frac{1}{5} = 1.2$ * **Calculation (Working Forward algebraically):** Step 1: Add $2.5$ to $x \rightarrow (x + 2.5)$ Step 2: Multiply by $4.5 \rightarrow 4.5(x + 2.5)$ Step 3: Add $3 \rightarrow 4.5(x + 2.5) + 3$ Step 4: Divide by $1.2$, and the result is $25$. $$ \frac{4.5(x + 2.5) + 3}{1.2} = 25 $$ Now, solve for $x$: Multiply by $1.2$: $$ 4.5(x + 2.5) + 3 = 25 \times 1.2 $$ $$ 4.5(x + 2.5) + 3 = 30 $$ Subtract $3$: $$ 4.5(x + 2.5) = 30 - 3 $$ $$ 4.5(x + 2.5) = 27 $$ Divide by $4.5$: $$ x + 2.5 = \frac{27}{4.5} $$ $$ x + 2.5 = 6 $$ Subtract $2.5$: $$ x = 6 - 2.5 $$ $$ x = 3.5 $$ Convert back to a mixed fraction: $x = 3\frac{1}{2}$. ### Exam Strategy & Shortcut **Working Backwards** is highly efficient here and skips setting up a massive equation. Start at the end: $25$. It was divided by $1.2 \rightarrow$ multiply by $1.2 = 30$. $3$ was added to it $\rightarrow$ subtract $3 = 27$. It was multiplied by $4.5 \rightarrow$ divide by $4.5 = 6$. $2.5$ was added to it $\rightarrow$ subtract $2.5 = 3.5$. The answer is $3.5$ or $3\frac{1}{2}$. ### Common Pitfall Attempting to keep everything as improper fractions (like $\frac{9}{2}$ and $\frac{6}{5}$) can lead to messy, error-prone stacked fractions. Switching to decimals ($4.5, 1.2$) makes the arithmetic visually simpler and faster. ### Final Answer **Therefore, the correct answer is $3\frac{1}{2}$.**