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).

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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).

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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.