If $3 \frac{x}{7} \times 2 \frac{y}{5} = 10$, then the values of $x$ and $y$ respectively, would be

### Concept & Logic This problem requires converting mixed fractions containing unknown variables into improper fractions. The resulting equation can be solved efficiently by testing the given options rather than attempting complex algebraic factorization. $$a \frac{b}{c} = \frac{(a \times c) + b}{c}$$ ### Step-by-Step Solution * **Given:** $3 \frac{x}{7} \times 2 \frac{y}{5} = 10$ * **Convert to improper fractions:** * $\frac{(3 \times 7) + x}{7} \times \frac{(2 \times 5) + y}{5} = 10$ * $\frac{21 + x}{7} \times \frac{10 + y}{5} = 10$ * **Simplify the equation:** * $\frac{(21 + x)(10 + y)}{35} = 10$ * $(21 + x)(10 + y) = 350$ * **Test the given options:** * *(a) 3, 4:* $(21 + 3) \times (10 + 4) = 24 \times 14 = 336 \neq 350$ * *(b) 5, 7:* $(21 + 5) \times (10 + 7) = 26 \times 17 = 442 \neq 350$ * *(c) 4, 3:* $(21 + 4) \times (10 + 3) = 25 \times 13 = 325 \neq 350$ * *(d) 4, 4:* $(21 + 4) \times (10 + 4) = 25 \times 14 = 350$ ### Exam Strategy & Shortcut Option testing (plugging in the values) is the fastest way to solve this. Since $350$ ends in $0$, the product of the unit digits of $(21 + x)$ and $(10 + y)$ must end in $0$. For option (d) where $x=4$ and $y=4$, we get $25 \times 14$. The unit digits are $5 \times 4 = 20$, which gives a final unit digit of $0$, making it a very strong first candidate to test. ### Common Pitfall Attempting to expand the brackets to create a polynomial equation: $210 + 21y + 10x + xy = 350$. This creates an equation with two variables that cannot be easily solved without additional information. Always use the options for Diophantine equations (equations with integer solutions). ### Final Answer **Therefore, the correct answer is 4, 4.**