Find the requested value and tell what the other numbers represent.
(a)
Find r: 500 = 1,000(r)(8)
    (Enter the rate as a decimal.)
r =
 
Identify the following.
interest	$ 
principal	$ 
time	
(b)
Find P: 9,520 = P + P(0.06)(6)
P =
 $ 
Identify the following. (Enter the rate as a decimal.)
future value	$ 
rate	
time	


Let's solve each part step-by-step.

### Part (a)
Given equation:
\[
500 = 1000 \cdot r \cdot 8
\]

1. Isolate $ r $ by dividing both sides by $ 1000 \cdot 8 $:
   \[
   r = \frac{500}{1000 \cdot 8}
   \]

2. Simplify the calculation:
   \[
   r = \frac{500}{8000} = 0.0625
   \]

So, the rate $ r $ as a decimal is:
\[
r = 0.0625
\]

#### Identification of Values:
- **Interest**: $500 (this is the amount accrued as interest).
- **Principal**: $1,000 (the initial amount of money, or the principal).
- **Time**: 8 (time period over which the interest accrues).

---

### Part (b)
Given equation:
\[
9520 = P + P \cdot 0.06 \cdot 6
\]

1. Factor $ P $ out on the right side:
   \[
   9520 = P(1 + 0.06 \cdot 6)
   \]

2. Calculate inside the parentheses:
   \[
   9520 = P(1 + 0.36)
   \]
   \[
   9520 = P \cdot 1.36
   \]

3. Isolate $ P $ by dividing both sides by $ 1.36 $:
   \[
   P = \frac{9520}{1.36} = 7000
   \]

So, the principal $ P $ is:
\[
P = 7000
\]

#### Identification of Values:
- **Future Value**: $9,520 (the total amount after interest is added).
- **Rate**: 0.06 (interest rate as a decimal).
- **Time**: 6 (time period over which the interest accrues).

---

Would you like further clarification on any of these steps?

### Relative Questions:
1. How do you convert a percentage interest rate to a decimal for calculations?
2. What are the basic components of the simple interest formula?
3. How would the solutions change if the time period was in months instead of years?
4. How can you use the formula $ A = P(1 + rt) $ for compound interest calculations?
5. What’s the difference between principal and future value in interest calculations?

### Tip:
To find rates as decimals, divide the percentage by 100.

Find the requested value and tell what the other numbers represent.
(a) Find r: 500 = 1,000(r)(8) (Enter the rate as a decimal.)
r =
Identify the following.
interest $
principal $
time
(b) Find P: 9,520 = P + P(0.06)(6)
P = $
Identify the following. (Enter the rate as a decimal.)
future value $
rate
time

Learn how to solve for the rate and principal in simple interest problems using equations. This step-by-step guide covers calculating the rate in '500 = 1,000(r)(8)' and solving for the principal in '9,520 = P + P(0.06)(6)'. Ideal for students in Grades 9-10.

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