Becca deposits $100 into a bank account that earns an annual interest rate of 4%. If she does not make any additional deposits and makes no withdrawals, how long will it take her, in years, to increase the value of her account by at least 60%?

Options

A. 12
B. 15
C. 25
D. 30

Answer

To solve the problem step by step, we will use the formula for compound interest and logarithms.

### Step 1: Understand the Problem
Becca deposits $100 into a bank account with an annual interest rate of 4%. We need to find out how long it will take for her account to increase by at least 60%.

### Step 2: Calculate the Target Amount
We need to find out what 60% of the initial deposit is:
- Initial deposit = $100
- 60% of $100 = \( 0.60 \times 100 = 60 \)
- Therefore, the total amount she needs to reach is \( 100 + 60 = 160 \).

### Step 3: Use the Compound Interest Formula
The formula for compound interest is:
\[ Y = A(1 + r)^t \]
Where:
- \( Y \) is the amount after \( t \) years,
- \( A \) is the initial amount ($100),
- \( r \) is the annual interest rate (4% or 0.04),
- \( t \) is the number of years.

Substituting the known values:
\[ Y = 100(1 + 0.04)^t \]
This simplifies to:
\[ Y = 100(1.04)^t \]

### Step 4: Set Up the Inequality
We want to find \( t \) such that:
\[ 100(1.04)^t \geq 160 \]
Dividing both sides by 100 gives:
\[ (1.04)^t \geq \frac{160}{100} = 1.6 \]

### Step 5: Apply Logarithms
To solve for \( t \), we take the logarithm of both sides:
\[ \log((1.04)^t) \geq \log(1.6) \]
Using the logarithmic identity \( \log(a^b) = b \cdot \log(a) \):
\[ t \cdot \log(1.04) \geq \log(1.6) \]

### Step 6: Solve for \( t \)
Now, we can isolate \( t \):
\[ t \geq \frac{\log(1.6)}{\log(1.04)} \]

### Step 7: Calculate the Values
Using a calculator:
- \( \log(1.6) \approx 0.2041 \)
- \( \log(1.04) \approx 0.0170 \)

Now substituting these values in:
\[ t \geq \frac{0.2041}{0.0170} \approx 12.004 \]

### Step 8: Determine the Smallest Integer
Since \( t \) must be a whole number, we round up to the next whole number:
\[ t = 13 \]

### Conclusion
It will take Becca **13 years** to increase the value of her account by at least 60%.