₹ 600 are divided among $A$, $B$, $C$ so that ₹ 40 more than $\frac{2}{5}$ of $A$'s share, ₹ 20 more than $\frac{2}{7}$ of $B$'s share and ₹ 10 more than $\frac{9}{17}$ of $C$'s share may all be equal. What is $A$'s share?

### Concept & Algebraic Formulation When specific modifications of shares are equal, set them all to a constant variable $k$. Express each share in terms of $k$, sum them, and equate to the total amount. ### Step-by-Step Solution Let the shares of $A, B, C$ be $a, b, c$. Given: $a + b + c = 600$. Let $\frac{2}{5}a + 40 = \frac{2}{7}b + 20 = \frac{9}{17}c + 10 = k$. 1. Express $a, b, c$ in terms of $k$: * $\frac{2}{5}a = k - 40 \Rightarrow a = \frac{5}{2}(k - 40)$ * $\frac{2}{7}b = k - 20 \Rightarrow b = \frac{7}{2}(k - 20)$ * $\frac{9}{17}c = k - 10 \Rightarrow c = \frac{17}{9}(k - 10)$ 2. Substitute into the sum equation: $\frac{5}{2}(k - 40) + \frac{7}{2}(k - 20) + \frac{17}{9}(k - 10) = 600$ 3. Expand and simplify: $\frac{5}{2}k - 100 + \frac{7}{2}k - 70 + \frac{17}{9}k - \frac{170}{9} = 600$ $(\frac{5}{2} + \frac{7}{2} + \frac{17}{9})k - 170 - \frac{170}{9} = 600$ $(\frac{12}{2} + \frac{17}{9})k = 600 + 170 + \frac{170}{9}$ $(6 + \frac{17}{9})k = 770 + \frac{170}{9}$ $(\frac{54 + 17}{9})k = \frac{6930 + 170}{9}$ $\frac{71}{9}k = \frac{7100}{9}$ 4. Solve for $k$: $71k = 7100 \Rightarrow k = 100$ 5. Find $A$'s share ($a$): $a = \frac{5}{2}(100 - 40) = \frac{5}{2}(60) = 5 \times 30 = 150$ ### Exam Strategy & Shortcut This is a standard "equal parts" problem that requires algebra. While you can test options, plugging in $A = 150$ yields $k = \frac{2}{5}(150) + 40 = 60 + 40 = 100$. With $k=100$, check if $B$ and $C$ sum up nicely. $B = \frac{7}{2}(80) = 280$. $C = \frac{17}{9}(90) = 170$. Sum = $150 + 280 + 170 = 600$. This confirms 150 is correct much faster than full algebraic expansion if you guess the right option quickly. ### Common Pitfall Making arithmetic errors during the expansion and collection of terms involving fractions. Back-solving from the options is often safer here. ### Final Answer Therefore, the correct answer is **₹ 150**.