1. **State the problem:** Find values of $c$ and $d$ such that $$5 \times 10^c + 5 \times 10^d = 50500.$$\n\n2. **Rewrite the equation:** Factor out 5:\n$$5(10^c + 10^d) = 50500.$$\n\n3. **Divide both sides by 5:**\n$$\cancel{5}(10^c + 10^d) = \frac{50500}{\cancel{5}}$$\n$$10^c + 10^d = 10100.$$\n\n4. **Analyze the sum:** We want two powers of 10 that add to 10100.\n\n5. **Try values:**\n- If $c=4$, then $10^4=10000$.\n- Then $10^d = 10100 - 10000 = 100$.\n- Since $10^2=100$, $d=2$.\n\n6. **Check the solution:**\n$$5 \times 10^4 + 5 \times 10^2 = 5 \times 10000 + 5 \times 100 = 50000 + 500 = 50500,$$ which matches the original equation.\n\n**Final answer:** $c=4$, $d=2$.