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. 

\(x\) 0 1 2 3 4 5
\(y\) 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. 

\(x\) 0 1 2 3 4 5
\(y\) 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.

Horizontal axis, time in months. Vertical axis, money in dollars. Point plotted at 0 comma 6,000, 3 comma 4,500, 6 comma 3,000, and 9 comma 1,500.

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

H.

Horizontal axis, time in months. Vertical axis, money in dollars. Points plotted include: origin, 1 comma 500, 2 comma 1,000, 3 comma 1,500, and continue in this pattern of y increasing by 500 as x increases by 1 , last point is: 11 comma 5,500. 

Summary