olve for the unknown quantity in Parts​ (a) through​ (d) that makes the equivalent value of cash outflows equal to the equivalent value of the cash​ inflow, F.
a. If Fequals​$9 comma 500​, Gequals​$550​, and Nequals6​, then iequals​?
b. If Fequals​$9 comma 500​, Gequals​$550​, and iequals5​% per​ period, then Nequals​?
c. If Gequals​$1 comma 000​, Nequals12​, and iequals10​% per​ period, then Fequals​?
d. If Fequals​$7 comma 000​, Nequals6​, and iequals10​% per​ period, then Gequals​?


To solve for the unknown quantities in parts (a) through (d), we are using the **future value of an annuity formula**, which relates a series of equal cash flows $ G $ over $ N $ periods, with a future value $ F $ at an interest rate $ i $ per period. The formula is:

\[
F = G \times \left(\frac{(1 + i)^N - 1}{i}\right)
\]

Where:
- $ F $ is the future value (cash inflow),
- $ G $ is the periodic cash outflow (annuity),
- $ N $ is the number of periods,
- $ i $ is the interest rate per period.

### Part (a) — Solve for $ i $
Given:
- $ F = 9500 $,
- $ G = 550 $,
- $ N = 6 $,

We solve for the interest rate $ i $ using the future value of an annuity formula by rearranging it and applying numerical methods.

### Part (b) — Solve for $ N $
Given:
- $ F = 9500 $,
- $ G = 550 $,
- $ i = 5\% = 0.05 $,

We solve for $ N $ by rearranging the formula:

\[
N = \frac{\log \left( \frac{F \cdot i}{G} + 1 \right)}{\log(1 + i)}
\]

### Part (c) — Solve for $ F $
Given:
- $ G = 1000 $,
- $ N = 12 $,
- $ i = 10\% = 0.10 $,

Simply plug these values into the future value formula to solve for $ F $.

### Part (d) — Solve for $ G $
Given:
- $ F = 7000 $,
- $ N = 6 $,
- $ i = 10\% = 0.10 $,

Rearrange the formula to solve for $ G $:

\[
G = \frac{F \cdot i}{(1 + i)^N - 1}
\]

Now, let me calculate the unknowns for each part.Here are the solutions for each part:

### Part (a) — Solve for $ i $
The interest rate per period is approximately:
\[
i = 42.32\% \text{ (per period)}
\]

### Part (b) — Solve for $ N $
The number of periods is approximately:
\[
N = 12.76 \text{ periods}
\]
(Since $ N $ must be a whole number, this would typically round to 13 periods.)

### Part (c) — Solve for $ F $
The future value $ F $ is approximately:
\[
F = 21,384.28
\]

### Part (d) — Solve for $ G $
The periodic cash outflow $ G $ is approximately:
\[
G = 907.25
\]

Would you like more details or have any other questions?

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Here are 5 relative questions:
1. How does compounding frequency affect the interest rate calculation in part (a)?
2. What rounding strategy should

Solve for the unknown quantity in Parts (a) through (d) that makes the equivalent value of cash outflows equal to the equivalent value of the cash inflow, F. a. If F equals $9,500, G equals $550, and N equals 6, then i equals? b. If F equals $9,500, G equals $550, and i equals 5% per period, then N equals? c. If G equals $1,000, N equals 12, and i equals 10% per period, then F equals? d. If F equals $7,000, N equals 6, and i equals 10% per period, then G equals?

Learn how to solve annuity problems, including calculating interest rates, periods, future values, and periodic payments. Given data like F=$9,500, G=$550, N=6, or i=10%, solve for unknowns using annuity formulas. Suitable for finance students and advanced high school math.

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