Questions: Leela invests 500 at 4.5% interest according to the equation (V1=500(1.045)^t), where (Vl) is the value of the account after (t) years. Adele invests the same amount of money at the same interest rate, but begins investing two years earlier according to the equation (Va=500(1.045)^t+2). The total value of Adele's account is approximately what percent of the total value of Leela's account at any time, t? 101.3% 104.5% 109.0% 109.2%

Leela invests 500 at 4.5% interest according to the equation (V1=500(1.045)^t), where (Vl) is the value of the account after (t) years. Adele invests the same amount of money at the same interest rate, but begins investing two years earlier according to the equation (Va=500(1.045)^t+2). The total value of Adele's account is approximately what percent of the total value of Leela's account at any time, t?
101.3%
104.5%
109.0%
109.2%

Transcript text: Leela invests $\$ 500$ at $4.5 \%$ interest according to the equation $V_{1}=500(1.045)^{t}$, where $V_{l}$ is the value of the account after $t$ years. Adele invests the same amount of money at the same interest rate, but begins investing two years earlier according to the equation $V_{a}=500(1,045)^{t+2}$. The total value of Adele's account is approximately what percent of the total value of Leela's account at any time, t? $101.3 \%$ $104.5 \%$ $109.0 \%$ $109.2 \%$

Solution

Solution Steps

To determine the total value of Adele's account as a percentage of the total value of Leela's account at any time $ t $, we need to compare the two given equations. Specifically, we need to find the ratio $ \frac{V_a}{V_l} $ and then convert this ratio to a percentage.

Write the equations for $ V_l $ and $ V_a $.
Compute the ratio $ \frac{V_a}{V_l} $.
Convert the ratio to a percentage.

Step 1: Define the Equations for $ V_l $ and $ V_a $

Leela's investment value after $ t $ years is given by: \[ V_l = 500 \times (1.045)^t \]

Adele's investment value after $ t $ years, starting two years earlier, is given by: \[ V_a = 500 \times (1.045)^{t+2} \]

Step 2: Compute the Ratio $ \frac{V_a}{V_l} $

To find the ratio of Adele's account value to Leela's account value, we compute: \[ \frac{V_a}{V_l} = \frac{500 \times (1.045)^{t+2}}{500 \times (1.045)^t} \]

Simplifying the ratio: \[ \frac{V_a}{V_l} = \frac{(1.045)^{t+2}}{(1.045)^t} = (1.045)^2 \]

Step 3: Convert the Ratio to a Percentage

Next, we convert the ratio to a percentage: \[ (1.045)^2 \approx 1.0920 \]

Thus, the percentage is: \[ 1.0920 \times 100\% = 109.2\% \]