An initial investment of $1000 grows at an interest rate of 4% per year. How long will it take that investment to reach $1,500? Assume that you make no further deposits or withdrawals from the investment account. Round to the nearest tenth of a year.

Question

12.6 years

9.8 years

10.3 years

5.4 years

Answer 1

To determine how long it will take for an initial investment to grow from $1000 to $1500 at an interest rate of 4% per year, we can use the formula for compound interest:

$$A=P(1+r{)}^t$$

where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial investment).
• $r$ is the annual interest rate (decimal).
• $t$ is the number of years the money is invested or borrowed.

In this case:

• $A=1500$
• $P=1000$
• $r=0.04$

We need to solve for $t$:

$$1500=1000(1+0.04{)}^t$$

This simplifies to:

$$1500=1000(1.04{)}^t$$

Dividing both sides by 1000:

$$1.5=(1.04{)}^t$$

Next, to solve for $t$, we take the logarithm of both sides:

$$\log (1.5)=\log ((1.04{)}^t)$$

Using the power rule of logarithms:

$$\log (1.5)=t\cdot \log (1.04)$$

Now, we can solve for $t$:

$$t=\frac{\log (1.5)}{\log (1.04)}$$

Using a calculator to find the logarithms:

• $\log (1.5)\approx 0.1761$
• $\log (1.04)\approx 0.0170$

Now substitute these values in:

$$t\approx \frac{0.1761}{0.0170}\approx 10.35$$

Rounding to the nearest tenth, we find:

$$t\approx 10.3\text{ years}$$

Thus, it will take approximately 10.3 years for the investment to grow from $1000 to $1500 at an interest rate of 4% per year.