Let's resolve each part of the question step-by-step.
a. At the end of 10 years, how much will Tim have paid in premiums?
Tim pays an annual premium of [tex]\( \$2,000 \)[/tex]. Over a period of 10 years, the total amount paid in premiums can be calculated by multiplying the annual premium by the number of years.
[tex]\[
\text{Total premiums paid} = \text{Annual premium} \times \text{Number of years}
\][/tex]
Substituting the given values:
[tex]\[
\text{Total premiums paid} = \$2,000 \times 10 = \$20,000
\][/tex]
So, at the end of 10 years, Tim will have paid [tex]\( \$20,000 \)[/tex] in premiums.
b. At the end of 10 years, what will the cash value of his policy be?
According to the table provided, the cash value per unit after 10 years is [tex]\( \$89.00 \)[/tex]. Assuming that "unit" refers to each year worth of the policy, and since Tim has paid for 10 years, the total cash value of the policy at the end of 10 years can be calculated as:
[tex]\[
\text{Cash value after 10 years} = \text{Cash value per unit} \times \text{Number of years}
\][/tex]
Substituting the given values:
[tex]\[
\text{Cash value after 10 years} = \$89.00 \times 10 = \$890.00
\][/tex]
So, at the end of 10 years, the cash value of Tim's policy will be [tex]\( \$890.00 \)[/tex].
c. What will the ratio of cash value to premiums paid be (as a percent)?
To calculate the ratio of cash value to premiums paid as a percentage, we use the formula:
[tex]\[
\text{Ratio percentage} = \left( \frac{\text{Cash value after 10 years}}{\text{Total premiums paid}} \right) \times 100
\][/tex]
Substituting the values from parts (a) and (b):
[tex]\[
\text{Ratio percentage} = \left( \frac{\$890.00}{\$20,000} \right) \times 100 = 4.45\%
\][/tex]
Therefore, the ratio of cash value to premiums paid will be [tex]\( 4.45\% \)[/tex].
To summarize:
- At the end of 10 years, Tim will have paid [tex]\( \$20,000 \)[/tex] in premiums.
- The cash value of his policy at the end of 10 years will be [tex]\( \$890.00 \)[/tex].
- The ratio of cash value to premiums paid will be [tex]\( 4.45\% \)[/tex].
