15% of A = 20% of B = 25% of C and A + B + C = 5640. A is decreased by 58$$\frac{1}{3}$$​%, B is increased by 20% and C is decreased by 30%.

Statement I: The new value of C is 8% more than the new value of A.

Statement II: The new value of B is 10% less than the initial value of A.

Which of the above statements is/are correct?

Correct option is A

Given:
15% of A = 20% of B = 25% of C
A + B + C = 5640
A is decreased by 58$$\frac{1}{3}$$​​
B is increased by 20%
C is decreased by 30%

Solution:

​$$\textbf{Let the common value be } x \\[5pt]15\% \text{ of } A = 20\% \text{ of } B = 25\% \text{ of } C = x \\[5pt]\Rightarrow A = \frac{x}{0.15} = \frac{100x}{15} = \frac{20x}{3} \\\ \\B = \frac{x}{0.20} = \frac{100x}{20} = 5x \\\ \\C = \frac{x}{0.25} = \frac{100x}{25} = 4x \\[10pt]\text{Given: } A + B + C = 5640 \\[5pt]\Rightarrow \frac{20x}{3} + 5x + 4x = 5640 \\\ \\\Rightarrow \frac{20x + 15x + 12x}{3} = 5640 \\\ \\\Rightarrow \frac{47x}{3} = 5640 \\\ \\\Rightarrow x = \frac{5640 \times 3}{47} = 360 \\[10pt]\textbf{Now calculate initial values:} \\A = \frac{20 \times 360}{3} = 2400 \\\ \\B = 5 \times 360 = 1800 \\C = 4 \times 360 = 1440 \\[10pt]\textbf{Apply changes:} \\\text{New A} = 2400 - \left(\frac{7}{12} \times 2400\right) = \frac{5}{12} \times 2400 = 1000 \\\text{New B} = 1800 + \left(0.20 \times 1800\right) = 1.20 \times 1800 = 2160 \\\text{New C} = 1440 - \left(0.30 \times 1440\right) = 0.70 \times 1440 = 1008 \\[10pt]\textbf{Statement I: } \\\text{Is new C 8\% more than new A?} \\\ \\1000 + (0.08 \times 1000) = 1080 \neq 1008 \Rightarrow \text{False} \\[5pt]\textbf{Statement II: } \\\text{Is new B 10\% less than initial A?} \\2400 - (0.10 \times 2400) = 2160 = \text{new B} \Rightarrow \text{True} \\[10pt]\boxed{\text{Correct Answer: (A) II only}}$$​​