You invest $$ \$ 800 $$ at an annual rate of $$ 8 \% $$ for 12 years. How much more interest would you earn with compound interest than with simple interest for (a) the whole 12 years; and (b) in just year 7 by itself?

Question

UpStudy AI · Accepted Answer

To determine the difference in interest earned using **compound interest** versus **simple interest**, we'll analyze both scenarios:

- **Principal (P):** \$800
- **Annual Interest Rate (r):** 8% or 0.08
- **Time (t):** 12 years

### (a) Difference Over the Whole 12 Years

1. **Simple Interest (Iₛ):**
   $$
   Iₛ = P \times r \times t = 800 \times 0.08 \times 12 = \$768
   $$

2. **Compound Interest (I𝚌):**
   - **Amount (A):**
     $$
     A = P \times (1 + r)^t = 800 \times (1.08)^{12}
     $$
   - Calculate $$(1.08)^{12}$$:
     $$
     (1.08)^{12} \approx 2.5161
     $$
   - **Total Amount (A):**
     $$
     A \approx 800 \times 2.5161 = \$2,012.88
     $$
   - **Interest Earned (I𝚌):**
     $$
     I𝚌 = A - P = 2,012.88 - 800 = \$1,212.88
     $$

3. **Difference (I𝚌 - Iₛ):**
   $$
   1,212.88 - 768 = \$444.88
   $$
   
   **Rounded to the nearest dollar:** **\$445** more with compound interest over 12 years.

---

### (b) Difference in Just Year 7

1. **Simple Interest for Year 7 (Iₛ₇):**
   $$
   Iₛ₇ = P \times r = 800 \times 0.08 = \$64
   $$

2. **Compound Interest for Year 7 (I𝚌₇):**
   - **Amount at Beginning of Year 7:**
     $$
     A₆ = P \times (1 + r)^6 = 800 \times (1.08)^6 \approx 800 \times 1.5869 = \$1,269.52
     $$
   - **Interest Earned in Year 7:**
     $$
     I𝚌₇ = A₆ \times r = 1,269.52 \times 0.08 \approx \$101.56
     $$

3. **Difference (I𝚌₇ - Iₛ₇):**
   $$
   101.56 - 64 = \$37.56
   $$
   
   **Rounded to the nearest dollar:** **\$38** more with compound interest in year 7.

---

### **Summary**

- **(a)** Over **12 years**, compound interest earns **\$445** more than simple interest.
- **(b)** In **year 7**, compound interest earns **\$38** more than simple interest.
Over 12 years, you earn \$445 more with compound interest. In year 7, you earn \$38 more with compound interest.

You invest at an annual rate of for 12 years. How much more interest would you earn
with compound interest than with simple interest for (a) the whole 12 years; and (b) in just year
7 by itself?

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Arithmetic Questions

You invest at an annual rate of for 12 years. How much more interest would you earn with compound interest than with simple interest for (a) the whole 12 years; and (b) in just year 7 by itself?