1. **Stating the problem:** Chip and Kelsey have different amounts of money now, and they discuss who will have more money in the future if no more money is added to their accounts. 2. **Understanding the problem:** Chip claims that even though he has less money now, his amount will surpass Kelsey's after a few years without additional deposits. Kelsey claims Chip will always have less. 3. **Key concept:** If both accounts earn interest at the same rate and no more money is added, the account with the larger initial amount will always have more money because compound interest grows proportionally to the principal. 4. **Formula for compound interest:** $$ A = P(1 + r)^t $$ where $A$ is the amount after $t$ years, $P$ is the principal (initial amount), and $r$ is the annual interest rate. 5. **Applying the formula:** - Let $P_C$ be Chip's initial amount, $P_K$ be Kelsey's initial amount, with $P_C < P_K$. - After $t$ years, Chip's amount: $$ A_C = P_C(1 + r)^t $$ - Kelsey's amount: $$ A_K = P_K(1 + r)^t $$ 6. **Comparing amounts:** Since $P_C < P_K$ and both grow by the same factor $(1 + r)^t$, it follows that: $$ A_C = P_C(1 + r)^t < P_K(1 + r)^t = A_K $$ for all $t > 0$. 7. **Conclusion:** Kelsey is correct. Chip will always have less money if no additional deposits are made and both earn the same interest rate. **Final answer:** Kelsey is right; Chip will always have less money under these conditions.