12. Simple Interest 5. For a simple interest rate of $$ 6 \% $$ p.a., we want to calculate the number of quarters needed for $$ \$ 2968.00 $$ to be invested to earn $$ \$ 178.08 $$ in interest. 5a First calculate the number of years, $$ T $$, taken to earn the interest. Write your answer correct to two decimal places. Enter each line of working as an equation. ( $$ T=1 $$ 5b Now find the number of quarters taken to earn the interest.

Question

UpStudy AI · Accepted Answer

To solve the problem, we will use the formula for simple interest:

$$
I = P \times r \times T
$$

where:
- $$ I $$ is the interest earned,
- $$ P $$ is the principal amount (the initial investment),
- $$ r $$ is the interest rate (as a decimal),
- $$ T $$ is the time in years.

### Step 1: Calculate the number of years, $$ T $$

Given:
- $$ I = 178.08 $$
- $$ P = 2968.00 $$
- $$ r = 6\% = 0.06 $$

We can rearrange the formula to solve for $$ T $$:

$$
T = \frac{I}{P \times r}
$$

Now, substituting the known values into the equation:

$$
T = \frac{178.08}{2968.00 \times 0.06}
$$

Let's calculate $$ T $$.

### Step 2: Perform the calculation for $$ T $$

Calculating the denominator:

$$
2968.00 \times 0.06 = 178.08
$$

Now substituting back into the equation for $$ T $$:

$$
T = \frac{178.08}{178.08} = 1
$$

Thus, the number of years $$ T $$ is:

$$
T = 1.00 \text{ years}
$$

### Step 3: Find the number of quarters

Since there are 4 quarters in a year, we can find the number of quarters $$ Q $$ by multiplying the number of years $$ T $$ by 4:

$$
Q = T \times 4
$$

Substituting the value of $$ T $$:

$$
Q = 1 \times 4 = 4
$$

### Final Answers

5a. The number of years $$ T $$ taken to earn the interest is:

$$
T = 1.00
$$

5b. The number of quarters taken to earn the interest is:

$$
Q = 4
$$
The number of years is 1.00, and the number of quarters is 4.

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