Answer :
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]
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]
