Answer :
To solve this problem, let's break down the information given and analyze the different premiums for each company. Aaron needs a life insurance policy worth [tex]$250,000, and the premiums are calculated based on a $[/tex]2000 face value.
First, we need to find the number of units of [tex]$2000 in $[/tex]250,000:
[tex]\[ \text{Number of units} = \frac{\text{Policy amount}}{\text{Face value}} = \frac{250,000}{2,000} = 125 \][/tex]
Now, let's calculate the premiums for each company, considering the additional smoker charges:
For Company A:
- Base premium per unit: [tex]$2.56$[/tex]
- Additional smoker surcharge: 20%
- Total base premium for all units: [tex]\(2.56 \times 125 = 320 \)[/tex]
- Additional smoker charge: [tex]\( 0.20 \times 320 = 64 \)[/tex]
- Total premium: [tex]\( 320 + 64 = 384 \)[/tex]
For Company B:
- Base premium per unit: [tex]$1.86$[/tex]
- Additional smoker surcharge: 18%
- Total base premium for all units: [tex]\(1.86 \times 125 = 232.5 \)[/tex]
- Additional smoker charge: [tex]\( 0.18 \times 232.5 = 41.85 \)[/tex]
- Total premium: [tex]\( 232.5 + 41.85 = 274.35 \)[/tex]
For Company C:
- Base premium per unit: [tex]$2.07$[/tex]
- Additional smoker surcharge: 25%
- Total base premium for all units: [tex]\(2.07 \times 125 = 258.75 \)[/tex]
- Additional smoker charge: [tex]\( 0.25 \times 258.75 = 64.6875 \)[/tex]
- Total premium: [tex]\( 258.75 + 64.6875 = 323.4375 \)[/tex]
Next, let's compare the premiums and calculate the differences:
- Difference between Company A and Company B: [tex]\( 384 - 274.35 = 109.65 \)[/tex]
- Difference between Company C and Company B: [tex]\( 323.4375 - 274.35 = 49.0875 \)[/tex]
Let's now evaluate the statements provided to determine which is true:
a. Aaron is paying the smallest premium possible with Company B.
- This statement is incorrect because Company B does not provide the smallest premium when comparing to the others; we see that Company B actually has one of the lower premiums.
b. Aaron is paying [tex]$93 more in premiums than he would with Company A. - This statement is incorrect. The premium difference between Company B and Company A is actually $[/tex]109.65.
c. Aaron is paying [tex]$129 more in premiums than he would with Company C. - This statement is incorrect. The premium difference between Company B and Company C is $[/tex]49.0875.
d. Aaron could have chosen any company because the premiums are all the same.
- This statement is clearly incorrect because the premiums vary between the companies.
Since none of the statements provided are correct, Aaron may need to reconsider his options. However, None of the given statements a, b, c, or d is correct, so based on the numerical findings:
The correct response should be:
None of the provided statements (a, b, c, d) are accurate.
First, we need to find the number of units of [tex]$2000 in $[/tex]250,000:
[tex]\[ \text{Number of units} = \frac{\text{Policy amount}}{\text{Face value}} = \frac{250,000}{2,000} = 125 \][/tex]
Now, let's calculate the premiums for each company, considering the additional smoker charges:
For Company A:
- Base premium per unit: [tex]$2.56$[/tex]
- Additional smoker surcharge: 20%
- Total base premium for all units: [tex]\(2.56 \times 125 = 320 \)[/tex]
- Additional smoker charge: [tex]\( 0.20 \times 320 = 64 \)[/tex]
- Total premium: [tex]\( 320 + 64 = 384 \)[/tex]
For Company B:
- Base premium per unit: [tex]$1.86$[/tex]
- Additional smoker surcharge: 18%
- Total base premium for all units: [tex]\(1.86 \times 125 = 232.5 \)[/tex]
- Additional smoker charge: [tex]\( 0.18 \times 232.5 = 41.85 \)[/tex]
- Total premium: [tex]\( 232.5 + 41.85 = 274.35 \)[/tex]
For Company C:
- Base premium per unit: [tex]$2.07$[/tex]
- Additional smoker surcharge: 25%
- Total base premium for all units: [tex]\(2.07 \times 125 = 258.75 \)[/tex]
- Additional smoker charge: [tex]\( 0.25 \times 258.75 = 64.6875 \)[/tex]
- Total premium: [tex]\( 258.75 + 64.6875 = 323.4375 \)[/tex]
Next, let's compare the premiums and calculate the differences:
- Difference between Company A and Company B: [tex]\( 384 - 274.35 = 109.65 \)[/tex]
- Difference between Company C and Company B: [tex]\( 323.4375 - 274.35 = 49.0875 \)[/tex]
Let's now evaluate the statements provided to determine which is true:
a. Aaron is paying the smallest premium possible with Company B.
- This statement is incorrect because Company B does not provide the smallest premium when comparing to the others; we see that Company B actually has one of the lower premiums.
b. Aaron is paying [tex]$93 more in premiums than he would with Company A. - This statement is incorrect. The premium difference between Company B and Company A is actually $[/tex]109.65.
c. Aaron is paying [tex]$129 more in premiums than he would with Company C. - This statement is incorrect. The premium difference between Company B and Company C is $[/tex]49.0875.
d. Aaron could have chosen any company because the premiums are all the same.
- This statement is clearly incorrect because the premiums vary between the companies.
Since none of the statements provided are correct, Aaron may need to reconsider his options. However, None of the given statements a, b, c, or d is correct, so based on the numerical findings:
The correct response should be:
None of the provided statements (a, b, c, d) are accurate.
