1. **State the problem:** We have an investment with simple interest. After 18 months, the amount is 10,900. After an additional 24 months (total 42 months), the amount is 12,100. We need to find the original principal $P$ and the simple interest rate $r$ per year. 2. **Recall the formula for simple interest:** $$ A = P(1 + rt) $$ where $A$ is the amount, $P$ is the principal, $r$ is the annual interest rate (in decimal), and $t$ is the time in years. 3. **Convert months to years:** - 18 months = $\frac{18}{12} = 1.5$ years - 42 months = $\frac{42}{12} = 3.5$ years 4. **Set up equations using the given amounts:** - After 1.5 years: $$ 10,900 = P(1 + r \times 1.5) $$ - After 3.5 years: $$ 12,100 = P(1 + r \times 3.5) $$ 5. **Express both equations:** $$ 10,900 = P + 1.5Pr $$ $$ 12,100 = P + 3.5Pr $$ 6. **Subtract the first equation from the second:** $$ 12,100 - 10,900 = (P + 3.5Pr) - (P + 1.5Pr) $$ $$ 1,200 = 2Pr $$ 7. **Solve for $Pr$:** $$ Pr = \frac{1,200}{2} = 600 $$ 8. **Use the first equation to find $P$:** $$ 10,900 = P + 1.5 \times 600 = P + 900 $$ $$ P = 10,900 - 900 = 10,000 $$ 9. **Find $r$ using $Pr = 600$:** $$ r = \frac{600}{P} = \frac{600}{10,000} = 0.06 $$ 10. **Convert $r$ to percentage:** $$ r = 0.06 \times 100 = 6\% $$ **Final answer:** - Original principal $P = 10,000$ - Simple interest rate $r = 6\%$ per annum