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Leela invests [tex]\$500[/tex] at [tex]4.5\%[/tex] interest according to the equation [tex]V_l = 500(1.045)^t[/tex], where [tex]V_l[/tex] is the value of the account after [tex]t[/tex] years. Adele invests the same amount of money at the same interest rate but begins investing two years earlier according to the equation [tex]V_a = 500(1.045)^{t+2}[/tex]. The total value of Adele's account is approximately what percent of the total value of Leela's account at any time [tex]t[/tex]?

A. [tex]101.3\%[/tex]
B. [tex]104.5\%[/tex]
C. [tex]109.0\%[/tex]
D. [tex]109.2\%[/tex]


Sagot :

To solve the problem, we need to find the ratio of the total value of Adele's account to Leela's account and express this ratio as a percentage.

1. Determine the value of Leela's account after \( t \) years:

The formula for the value of Leela's account is given by:
[tex]\[ V_l = 500 (1.045)^t \][/tex]

2. Determine the value of Adele's account after \( t \) years:

Since Adele begins investing two years earlier, the formula for the value of Adele's account after \( t \) years is:
[tex]\[ V_e = 500 (1.045)^{t+2} \][/tex]

3. Calculate the ratio of \( V_e \) to \( V_l \):

To find how much Adele's account is compared to Leela's, we take the ratio of \( V_e \) to \( V_l \):
[tex]\[ \text{Ratio} = \frac{V_e}{V_l} = \frac{500 (1.045)^{t+2}}{500 (1.045)^t} \][/tex]

4. Simplify the ratio:

Notice that the \( 500 \) terms cancel out:
[tex]\[ \text{Ratio} = \frac{(1.045)^{t+2}}{(1.045)^t} \][/tex]

Using the properties of exponents, specifically \((a^{m+n} = a^m \cdot a^n)\),
[tex]\[ \text{Ratio} = \frac{(1.045)^t \cdot (1.045)^2}{(1.045)^t} = (1.045)^2 \][/tex]

5. Calculate \( (1.045)^2 \):

[tex]\[ (1.045)^2 = 1.045 \times 1.045 = 1.092025 \][/tex]

6. Convert the ratio to a percentage:

To express the ratio as a percentage, multiply by 100:
[tex]\[ \text{Percentage} = 1.092025 \times 100 \approx 109.2\% \][/tex]

So, the total value of Adele's account is approximately \( 109.2\% \) of the total value of Leela's account at any time \( t \).

The correct answer is:
[tex]\( \boxed{109.2\%} \)[/tex]
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