Two numbers are in the ratio $17 : 45$. One-third of the smaller is less than $\frac{1}{5}$ of the bigger by 15. The smaller number is

### Concept & Algebraic Formulation Use variables to express the numbers proportionally based on their ratio, then create an algebraic equation matching the statement given in the problem. ### Step-by-Step Solution 1. **Given:** Two numbers are in the ratio $17 : 45$. Let the smaller number be $17x$ and the bigger number be $45x$. 2. The problem states: $\frac{1}{5}$ of the bigger number $-$ $\frac{1}{3}$ of the smaller number $= 15$. 3. Substitute the expressions: $\frac{1}{5}(45x) - \frac{1}{3}(17x) = 15$. 4. Simplify the terms: $9x - \frac{17x}{3} = 15$. 5. Multiply the entire equation by 3 to clear the denominator: $27x - 17x = 45$. 6. Solve for $x$: $10x = 45 \implies x = 4.5 = \frac{9}{2}$. 7. Find the smaller number: $17x = 17 \times \frac{9}{2} = \frac{153}{2} = 76.5$ or $76\frac{1}{2}$. ### Exam Strategy & Shortcut When you reach $x = 4.5$, immediately multiply by 17. Knowing $17 \times 4 = 68$ and $17 \times 0.5 = 8.5$, adding them gives $76.5$ mentally, bypassing complex fraction operations. ### Common Pitfall Misreading the phrase "One-third of the smaller is less than $\frac{1}{5}$ of the bigger by 15" and reversing the subtraction order to $\frac{1}{3}(17x) - \frac{1}{5}(45x) = 15$, which yields a negative result. ### Final Answer Therefore, the correct answer is **$76\frac{1}{2}$**.