# Lesson 17

Annually, Quarterly, or Monthly?

• Let’s use different time intervals to solve problems.

### 17.1: Finding Equal Expressions

1. Find pairs of expressions that are equal. Be prepared to explain how you know.

$$(3^5)^2$$

$$(3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3) \boldcdot (3 \boldcdot 3)$$

$$3 \boldcdot 3 \boldcdot 9 \boldcdot 9 \boldcdot 9$$

$$3^6$$

$$(3^2)^4$$

$$3^7$$

$$3^{10}$$

$$3 \boldcdot 9 \boldcdot 27$$

2. Write an expression that is equal to $$(2^{30})^7$$ using a single exponent.
3. Without evaluating the expressions, explain why $$2^{15}$$ is equal to $$8^5$$.

### 17.2: How Many Times Per Year?

1. Complete the table.
If something happens... It happens this many times a year... It happens every $$\underline{\hspace{.5in}}$$ months...
annually
semi-annually
quarterly
monthly
2. A gym membership has an annual fee, billed monthly. How much is each bill, if the annual fee in dollars is . . .?
1. 360
2. 540
3. $$g$$
3. An educational foundation gives an annual scholarship, distributed semi-annually. How much is each distribution, if the annual scholarship amount in dollars is . . .?
1. 1,800
2. 5,000
3. $$s$$
4. A magazine subscription has an annual price, billed quarterly. How much is each bill, if the annual price in dollars is . . .?
1. 48
2. 80
3. $$m$$

### 17.3: Your Problems Are Compounded

Match each item in the first column to a representation in the second column.

1. A worker sets aside \$6,000 per year for their retirement fund by saving the same amount monthly. A. $$6,\!000 \boldcdot 1.21^3$$ 2. A business’s revenue increases by 20% per quarter. This happens for 2 years. Initially, their quarterly profit was \$6,000.

B.

 $$x$$ $$y$$ 0 1 2 3 4 5 6,000 7,200 8,640 10,368 12,442 14,930

3. $$6,\!000 \boldcdot ((1.05)^{4})^x$$

C. $$6\boldcdot(3^4)^2$$

4. A man borrows \$6,000 from his sister. He will reduce the amount he owes in 1 year by paying her back quarterly. D.  $$x$$ $$y$$ 0 1 2 3 4 5 6,000 4,800 3,840 3,072 2,457.6 1,966.1 5. A business’s revenue decreases by 20% semi-annually. This happens for 3 years. Initially, their quarterly revenue was \$6,000.

E. $$6,\!000 \boldcdot 1.2155^x$$

6. The number of subscribers to a website triples quarterly for 2 years. Initially there were 6 subscribers.

F. $$6 \boldcdot 4,\!096^2$$

7. $$6,\!000 \boldcdot ((1.1)^2)^3$$

G.

8. The number of likes on a post was 6, and then for the next 2 years, the number of likes doubled, monthly.

H.