If three times the larger of the two numbers is divided by the smaller one, we get $4$ as quotient and $3$ as remainder. Also, if seven times the smaller number is divided by the larger one, we get $5$ as quotient and $1$ as remainder. Find the numbers.

### Concept & Logic Translate word problems into a system of linear equations using the division formula: $\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$. ### Step-by-Step Solution **Given:** Let the larger number be $x$ and the smaller number be $y$. **Calculation / Deduction:** * Formulate the first equation based on: "Three times the larger... divided by the smaller... quotient $4$, remainder $3$": $$ 3x = 4y + 3 $$ $$ 3x - 4y = 3 \quad \text{--- (Eq 1)} $$ * Formulate the second equation based on: "Seven times the smaller... divided by the larger... quotient $5$, remainder $1$": $$ 7y = 5x + 1 $$ $$ -5x + 7y = 1 \quad \text{--- (Eq 2)} $$ * Solve the system by eliminating $y$. Multiply (Eq 1) by $7$ and (Eq 2) by $4$: $$ 21x - 28y = 21 $$ $$ -20x + 28y = 4 $$ * Add the two new equations together to cancel out $y$: $$ (21x - 20x) = 21 + 4 $$ $$ x = 25 $$ * Substitute $x = 25$ back into (Eq 1) to solve for $y$: $$ 3(25) - 4y = 3 $$ $$ 75 - 3 = 4y $$ $$ 72 = 4y \implies y = 18 $$ ### Exam Strategy & Shortcut For "find the numbers" multiple-choice questions, **Option Elimination** is dramatically faster than solving simultaneous equations. Simply plug the options into the first condition. Testing $25$ and $18$: $3 \times 25 = 75$. Divide $75$ by $18$: $18 \times 4 = 72$, leaving a remainder of $3$. It works perfectly and takes only seconds. ### Common Pitfall Students often set up the division equations backwards by confusing the dividend and the divisor (e.g., writing $y = 4(3x) + 3$ instead of $3x = 4y + 3$). Pay close grammatical attention to what "is divided by" what. ### Final Answer Therefore, the correct answer is 25, 18.