Yusuf invested $180 in an account paying an interest rate of 2, one eigth281​% compounded monthly. Malik invested $180 in an account paying an interest rate of 2, start fraction, 3, divided by, 8, end fraction283​% compounded quarterly. To the nearest *hundredth of a year*, how much longer would it take for Yusuf's money to triple than for Malik's money to triple?

To determine how much longer it will take for Yusuf's money to triple compared to Malik's money, we need to calculate the time required for each investment to triple, given their respective interest rates and compounding periods. 

### Step-by-Step Solution

1. **Determine the interest rate and compounding periods:**
   - Yusuf's interest rate: $ 2 \frac{1}{8} \% = 2.125\% $ compounded monthly.
   - Malik's interest rate: $ 2 \frac{3}{8} \% = 2.375\% $ compounded quarterly.

2. **Convert the annual interest rates to decimal form:**
   - Yusuf: $ r = 0.02125 $
   - Malik: $ r = 0.02375 $

3. **Use the compound interest formula:**
   \[
   A = P \left(1 + \frac{r}{n}\right)^{nt}
   \]
   Where:
   - $ A $ is the final amount.
   - $ P $ is the principal amount.
   - $ r $ is the annual interest rate.
   - $ n $ is the number of compounding periods per year.
   - $ t $ is the number of years.

4. **Set up the equations for tripling the investment ( $ A = 3P $ ):**
   - For Yusuf (monthly compounding): $ 180 \left(1 + \frac{0.02125}{12}\right)^{12t} = 540 $
   - For Malik (quarterly compounding): $ 180 \left(1 + \frac{0.02375}{4}\right)^{4t} = 540 $

5. **Solve for $ t $ in both equations:**

   - For Yusuf:
     \[
     \left(1 + \frac{0.02125}{12}\right)^{12t} = 3
     \]
     \[
     \left(1 + \frac{0.00177083}{12}\right)^{12t} = 3
     \]
     Taking the natural logarithm on both sides:
     \[
     12t \ln\left(1 + \frac{0.02125}{12}\right) = \ln 3
     \]
     \[
     12t \ln(1.00177083) = \ln 3
     \]
     \[
     t = \frac{\ln 3}{12 \cdot \ln(1.00177083)}
     \]
     Using $\ln(1.00177083) \approx 0.00176907$:
     \[
     t = \frac{\ln 3}{12 \cdot 0.00176907} \approx \frac{1.0986}{0.02122884} \approx 51.75 \text{ years}
     \]

   - For Malik:
     \[
     \left(1 + \frac{0.02375}{4}\right)^{4t} = 3
     \]
     \[
     \left(1 + \frac{0.0059375}{4}\right)^{4t} = 3
     \]
     Taking the natural logarithm on both sides:
     \[
     4t \ln\left(1 + \frac{0.02375}{4}\right) = \ln 3
     \]
     \[
     4t \ln(1.0059375) = \ln 3
     \]
     \[
     t = \frac{\ln 3}{4 \cdot \ln(1.0059375)}
     \]
     Using $\ln(1.0059375) \approx 0.00592068$:
     \[
     t = \frac{\ln 3}{4 \cdot 0.00592068} \approx \frac{1.0986}{0.02368272} \approx 46.39 \text{ years}
     \]

6. **Calculate the difference in time:**
   \[
   51.75 - 46.39 = 5.36 \text{ years}
   \]

Therefore, it will take approximately **5.36 years longer** for Yusuf's money to triple than for Malik's money.

Would you like more details or have any questions about this solution?

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### Related Questions:
1. How would the calculations change if the interest rates were compounded annually instead of monthly and quarterly?
2. What is the impact of a higher interest rate on the time required for an investment to triple?
3. How can the compound interest formula be adjusted for continuous compounding?
4. What would be the effect of different initial investments on the time to triple at the same interest rates?
5. How does the frequency of compounding affect the total amount accrued over a fixed period?
6. What are the key differences between simple interest and compound interest?
7. How do you derive the natural logarithm values used in these calculations?
8. How does inflation affect the real value of money in the context of compound interest?

### Tip:
Always compare interest rates and compounding periods when evaluating different investment options to understand their long-term growth potential accurately.

Explore how long it takes for investments to triple under different compound interest rates and periods. Use our step-by-step solution to understand the calculations involved for Yusuf and Malik's investments. Suitable for advanced high school students and anyone interested in finance.

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