Present value (with changing years). When they are first born, Grandma gives each of her grandchildren a $$ \$ 5,000 $$ savings bond that matures in 18 years. For each of the following grandchildren, what is the present value of each savings bonds if the current discount rate is $$ 6 \% $$ ? a. Seth turned fifteen years old today. b. Shawn turned thirteen years old today. c. Sherry turned eight years old today. d. Sheila turned four years old today. e. Shane was just born. a. Seth just turned fifteen years old today and the current discount rate is $$ 6 \% $$. What is the present value of his savings bond? $$ \$ \square $$ (Round to the nearest cent.)

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To find the present value of the savings bond for each grandchild, we can use the formula for present value:

$$ PV = \frac{FV}{(1 + r)^n} $$

where:
- $$ PV $$ is the present value,
- $$ FV $$ is the future value (the amount the bond will be worth in 18 years),
- $$ r $$ is the discount rate (6% in this case),
- $$ n $$ is the number of years until the bond matures (18 years).

Given that the future value of the savings bond is $5,000 and the discount rate is 6%, we can calculate the present value for each grandchild.

a. Seth turned fifteen years old today:
Since Seth is 15 years old, the number of years until the bond matures is 18 - 15 = 3 years.

b. Shawn turned thirteen years old today:
Since Shawn is 13 years old, the number of years until the bond matures is 18 - 13 = 5 years.

c. Sherry turned eight years old today:
Since Sherry is 8 years old, the number of years until the bond matures is 18 - 8 = 10 years.

d. Sheila turned four years old today:
Since Sheila is 4 years old, the number of years until the bond matures is 18 - 4 = 14 years.

e. Shane was just born:
Since Shane is 0 years old, the number of years until the bond matures is 18 years.

Now, we can calculate the present value for each grandchild using the formula:

$$ PV = \frac{5000}{(1 + 0.06)^n} $$

Let's calculate the present value for each grandchild.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{5000}{\left(1+0.06\right)^{3}}$$
- step1: Add the numbers:
  $$\frac{5000}{1.06^{3}}$$
- step2: Convert the expressions:
  $$\frac{5000}{\left(\frac{53}{50}\right)^{3}}$$
- step3: Simplify the expression:
  $$\frac{5000}{\frac{53^{3}}{50^{3}}}$$
- step4: Rewrite the expression:
  $$\frac{5000\times 50^{3}}{53^{3}}$$
The present value of Seth's savings bond is approximately $4,198.10.

Now, let's calculate the present value for Shawn.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{5000}{\left(1+0.06\right)^{5}}$$
- step1: Add the numbers:
  $$\frac{5000}{1.06^{5}}$$
- step2: Convert the expressions:
  $$\frac{5000}{\left(\frac{53}{50}\right)^{5}}$$
- step3: Simplify the expression:
  $$\frac{5000}{\frac{53^{5}}{50^{5}}}$$
- step4: Rewrite the expression:
  $$\frac{5000\times 50^{5}}{53^{5}}$$
The present value of Shawn's savings bond is approximately $3,736.29.

Next, let's calculate the present value for Sherry.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{5000}{\left(1+0.06\right)^{10}}$$
- step1: Add the numbers:
  $$\frac{5000}{1.06^{10}}$$
- step2: Convert the expressions:
  $$\frac{5000}{\left(\frac{53}{50}\right)^{10}}$$
- step3: Simplify the expression:
  $$\frac{5000}{\frac{53^{10}}{50^{10}}}$$
- step4: Rewrite the expression:
  $$\frac{5000\times 50^{10}}{53^{10}}$$
The present value of Sherry's savings bond is approximately $2,791.97.

Now, let's calculate the present value for Sheila.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{5000}{\left(1+0.06\right)^{14}}$$
- step1: Add the numbers:
  $$\frac{5000}{1.06^{14}}$$
- step2: Convert the expressions:
  $$\frac{5000}{\left(\frac{53}{50}\right)^{14}}$$
- step3: Simplify the expression:
  $$\frac{5000}{\frac{53^{14}}{50^{14}}}$$
- step4: Rewrite the expression:
  $$\frac{5000\times 50^{14}}{53^{14}}$$
The present value of Sheila's savings bond is approximately $2,211.50.

Finally, let's calculate the present value for Shane.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{5000}{\left(1+0.06\right)^{18}}$$
- step1: Add the numbers:
  $$\frac{5000}{1.06^{18}}$$
- step2: Convert the expressions:
  $$\frac{5000}{\left(\frac{53}{50}\right)^{18}}$$
- step3: Simplify the expression:
  $$\frac{5000}{\frac{53^{18}}{50^{18}}}$$
- step4: Rewrite the expression:
  $$\frac{5000\times 50^{18}}{53^{18}}$$
The present value of Shane's savings bond is approximately $1,751.72.

Therefore, the present values of the savings bonds for each grandchild are:
- Seth: $4,198.10
- Shawn: $3,736.29
- Sherry: $2,791.97
- Sheila: $2,211.50
- Shane: $1,751.72
The present values of the savings bonds are: - Seth: \$4,198.10 - Shawn: \$3,736.29 - Sherry: \$2,791.97 - Sheila: \$2,211.50 - Shane: \$1,751.72

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