The sum of radii of two circles is 140 cm and the difference of their circumferences is 88 cm. The diameters of the circles are

### Concept & Logic Set up a system of linear equations using the formulas for sum of radii and difference of circumferences, then solve for the individual radii and double them to find the diameters. $$C_1 - C_2 = 2\pi r_1 - 2\pi r_2 = 2\pi(r_1 - r_2)$$ ### Step-by-Step Solution * Given: Sum of radii, $r_1 + r_2 = 140$. * Given: Difference of circumferences, $2\pi(r_1 - r_2) = 88$. * Solve for $r_1 - r_2$: $2 \times \frac{22}{7} \times (r_1 - r_2) = 88$ $\frac{44}{7} \times (r_1 - r_2) = 88$ $r_1 - r_2 = \frac{88 \times 7}{44} = 14$. * Now, solve the system of equations: $r_1 + r_2 = 140$ $r_1 - r_2 = 14$ * Add the two equations together: $2r_1 = 154 \Rightarrow r_1 = 77$ cm. * Subtract the two equations: $2r_2 = 126 \Rightarrow r_2 = 63$ cm. * The question asks for the diameters ($d = 2r$): $d_1 = 2 \times 77 = 154$ cm. $d_2 = 2 \times 63 = 126$ cm. ### Exam Strategy & Shortcut Work backward from the options! The sum of the radii is 140, so the sum of the diameters must be $2 \times 140 = 280$. Check the sums in the options: (a) $77 + 63 = 140$ (Incorrect) (b) $150 + 120 = 270$ (Incorrect) (c) $154 + 126 = 280$ (Matches!) (d) $160 + 120 = 280$ (Matches!) Between (c) and (d), check the circumference difference. For (c), difference in diameters is $154 - 126 = 28$. Difference in circumference is $\pi \times d = \frac{22}{7} \times 28 = 88$. It matches perfectly. ### Common Pitfall Calculating the correct radii (77 cm and 63 cm) and hastily selecting option (a), completely forgetting that the question specifically asked for the **diameters**. ### Final Answer Therefore, the correct answer is **154 cm, 126 cm**.