Money And Consumer Math Worksheets pdf | Math Champions - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. We will use the compound interest formula:

$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$

Where:
* $A$ = The final amount (Principal + Interest)
* $P$ = The principal (starting money)
* $r$ = The annual interest rate (as a decimal)
* $n$ = The number of times interest is compounded per year (Quarterly means $n=4$)
* $t$ = The time in years

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Problem 1

Question: You put $\$791$ into a savings account with an interest rate of $8\%$ compounded quarterly which earns $\$481.27$ over a period of time. How long was the period of time?

Step 1: Identify the values.
* Principal ($P$) = $\$791$
* Interest Earned = $\$481.27$
* Final Amount ($A$) = Principal + Interest = $\$791 + \$481.27 = \$1,272.27$
* Rate ($r$) = $8\% = 0.08$
* Compounded Quarterly ($n$) = $4$
* Time ($t$) = ?

Step 2: Set up the equation.
$$1272.27 = 791 \left(1 + \frac{0.08}{4}\right)^{4t}$$

Step 3: Solve for $t$.
First, simplify the part inside the parentheses: $1 + (0.08 / 4) = 1 + 0.02 = 1.02$.
$$1272.27 = 791 (1.02)^{4t}$$

Divide both sides by $791$:
$$\frac{1272.27}{791} = (1.02)^{4t}$$
$$1.60843... = (1.02)^{4t}$$

To solve for the exponent, we can check powers of $1.02$ or use logarithms. Let's estimate.
If $t = 6$, then $4t = 24$.
$1.02^{24} \approx 1.6084$. This matches our number perfectly.

So, $4t = 24$.
$t = 6$.

Final Answer: 6 years

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Problem 2

Question: If you put $\$626$ into a savings account that earns $10\%$ compounded quarterly, how much interest will you receive at the end of eight years?

Step 1: Identify the values.
* $P = \$626$
* $r = 0.10$
* $n = 4$
* $t = 8$

Step 2: Calculate the Total Amount ($A$).
$$A = 626 \left(1 + \frac{0.10}{4}\right)^{4 \times 8}$$
$$A = 626 (1 + 0.025)^{32}$$
$$A = 626 (1.025)^{32}$$

Calculate $(1.025)^{32} \approx 2.203757$
$$A = 626 \times 2.203757 \approx 1379.55$$

Step 3: Calculate the Interest.
Interest = Total Amount - Principal
Interest = $\$1,379.55 - \$626 = \$753.55$

Final Answer: $\$753.55$

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Problem 3

Question: You put $\$747$ into an investment at $4\%$ compounded quarterly for five years. What will the balance be at the end of five years?

Step 1: Identify the values.
* $P = \$747$
* $r = 0.04$
* $n = 4$
* $t = 5$

Step 2: Calculate the Balance ($A$).
$$A = 747 \left(1 + \frac{0.04}{4}\right)^{4 \times 5}$$
$$A = 747 (1 + 0.01)^{20}$$
$$A = 747 (1.01)^{20}$$

Calculate $(1.01)^{20} \approx 1.22019$
$$A = 747 \times 1.22019 \approx 911.48$$

Final Answer: $\$911.48$

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Problem 4

Question: How much interest does a $\$500$ investment earn at $7\%$ compounded quarterly over eight years?

Step 1: Identify the values.
* $P = \$500$
* $r = 0.07$
* $n = 4$
* $t = 8$

Step 2: Calculate the Total Amount ($A$).
$$A = 500 \left(1 + \frac{0.07}{4}\right)^{4 \times 8}$$
$$A = 500 (1 + 0.0175)^{32}$$
$$A = 500 (1.0175)^{32}$$

Calculate $(1.0175)^{32} \approx 1.74742$
$$A = 500 \times 1.74742 \approx 873.71$$

Step 3: Calculate the Interest.
Interest = $\$873.71 - \$500 = \$373.71$

Final Answer: $\$373.71$

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Problem 5

Question: If you borrow $\$683$ for nine years at an interest rate of $3\%$ compounded quarterly, how much interest will you pay?

Step 1: Identify the values.
* $P = \$683$
* $r = 0.03$
* $n = 4$
* $t = 9$

Step 2: Calculate the Total Amount ($A$).
$$A = 683 \left(1 + \frac{0.03}{4}\right)^{4 \times 9}$$
$$A = 683 (1 + 0.0075)^{36}$$
$$A = 683 (1.0075)^{36}$$

Calculate $(1.0075)^{36} \approx 1.308645$
$$A = 683 \times 1.308645 \approx 893.80$$

Step 3: Calculate the Interest.
Interest = $\$893.80 - \$683 = \$210.80$

Final Answer: $\$210.80$

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Problem 6

Question: You invested $\$986$ and after seven years the total amount of the investment was $\$1,215.45$. What was the interest rate if it was compounded quarterly?

Step 1: Identify the values.
* $P = \$986$
* $A = \$1,215.45$
* $t = 7$
* $n = 4$
* $r = ?$

Step 2: Set up the equation.
$$1215.45 = 986 \left(1 + \frac{r}{4}\right)^{4 \times 7}$$
$$1215.45 = 986 \left(1 + \frac{r}{4}\right)^{28}$$

Step 3: Solve for $r$.
Divide by $986$:
$$\frac{1215.45}{986} = \left(1 + \frac{r}{4}\right)^{28}$$
$$1.2327... = \left(1 + \frac{r}{4}\right)^{28}$$

Take the 28th root of both sides (raise to the power of $1/28$):
$$(1.2327)^{(1/28)} = 1 + \frac{r}{4}$$
$$1.0075 = 1 + \frac{r}{4}$$

Subtract $1$:
$$0.0075 = \frac{r}{4}$$

Multiply by $4$:
$$r = 0.03$$

Convert to percentage: $0.03 = 3\%$

Final Answer: $3\%$

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Problem 7

Question: The ending balance on an investment is $\$357.15$. If the principal was invested at $5\%$ compounded quarterly for eight years, what was the principal?

Step 1: Identify the values.
* $A = \$357.15$
* $r = 0.05$
* $n = 4$
* $t = 8$
* $P = ?$

Step 2: Set up the equation.
$$357.15 = P \left(1 + \frac{0.05}{4}\right)^{4 \times 8}$$
$$357.15 = P (1 + 0.0125)^{32}$$
$$357.15 = P (1.0125)^{32}$$

Step 3: Solve for $P$.
Calculate $(1.0125)^{32} \approx 1.48813$
$$357.15 = P (1.48813)$$

Divide by $1.48813$:
$$P = \frac{357.15}{1.48813} \approx 240.00$$

Final Answer: $\$240$

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Problem 8

Question: If you received $\$325.96$ on $\$578$ invested at a rate of $5\%$ compounded quarterly, for how long did you invest the principal?

*Note: The phrase "received $\$325.96$ on $\$578$" usually implies the interest earned is $\$325.96$, making the total balance higher. However, in many textbook problems, wording can be tricky. Let's look at the numbers. If $\$325.96$ is the interest, Total $A = 578 + 325.96 = 903.96$. If $\$325.96$ is the Total Balance $A$, that would mean losing money, which doesn't happen with positive interest. So, $\$325.96$ is the Interest.*

Step 1: Identify the values.
* $P = \$578$
* Interest = $\$325.96$
* $A = 578 + 325.96 = \$903.96$
* $r = 0.05$
* $n = 4$
* $t = ?$

Step 2: Set up the equation.
$$903.96 = 578 \left(1 + \frac{0.05}{4}\right)^{4t}$$
$$903.96 = 578 (1.0125)^{4t}$$

Step 3: Solve for $t$.
Divide by $578$:
$$\frac{903.96}{578} = (1.0125)^{4t}$$
$$1.56394... = (1.0125)^{4t}$$

Let's test integer years.
If $t = 9$ years, $4t = 36$.
$1.0125^{36} \approx 1.5639$. This matches perfectly.

So, $4t = 36$.
$t = 9$.

Final Answer: 9 years