C. Discrete Functions · Financial Applications: Interest, Present Value and AnnuitiesFinding the Number of Compounding Periods
C3.4About 70 minutes
Learning goals- I can find the number of compounding periods, n, using guess and check with a scientific calculator.
- I can use a graph or a TVM Solver to find n.
- I can use the power of a power rule and the rule of 72 to reason about doubling, tripling and other growth.
- I can interpret n in context (for example, round up to a whole number of periods).
Words to know
number of compounding periods (n)
How many times interest is compounded during the term.
guess and check (systematic trial)
Trying values of n, checking the result, and adjusting until the answer is found.
TVM Solver
A time value of money tool on graphing calculators and in apps or websites. Enter any four of N, I%, PV, PMT and FV (plus the periods per year) and it solves for the fifth.
doubling time
The time it takes an investment to double in value.
rule of 72
A quick estimate: at an annual rate of r% compounded annually, money doubles in about 72 ÷ r years.
How long will it take?
You have $2 500 in savings and your goal is $4 000. At 6% compounded annually, how many years must you wait? Now the unknown is the exponent: 2500(1.06)ⁿ = 4000.
You cannot solve this by taking a root, because n is the exponent, not the base. In Grade 12 (Advanced Functions, MHF4U) you will learn logarithms, which solve this kind of equation directly. In this course you will use three strategies that always work: guess and check, graphs, and technology such as a TVM Solver. You will also use exponent laws to reason quickly.
Strategy 1: guess and check
Simplify first: 2500(1.06)ⁿ = 4000 gives (1.06)ⁿ = 1.6. Now try values of n on your calculator:
- n = 7: 1.06⁷ ≈ 1.5036 (too small)
- n = 8: 1.06⁸ ≈ 1.5938 (still too small, just barely)
- n = 9: 1.06⁹ ≈ 1.6895 (bigger than 1.6)
So the answer is between 8 and 9 years. Interest is only added at the end of each year, so after 8 years you have $3 984.62 (not enough) and after 9 years you have $4 223.70. You reach the goal after 9 years.
Be careful about rounding: in a money question, if n is not a whole number, you usually need the next whole period.
Worked exampleExample 1: Doubling with monthly compounding
How long will it take money to double at 9% compounded monthly?
- i = 0.09 ÷ 12 = 0.0075. The principal does not matter for doubling: P(1.0075)ⁿ = 2P gives 1.0075ⁿ = 2.
- Estimate first with the rule of 72: 72 ÷ 9 = 8 years, about 96 months.
- Try n = 96: 1.0075⁹⁶ ≈ 2.0489 (a little too big). Try n = 92: 1.9886. Try n = 93: 2.0035.
- So 92 months is not quite enough and 93 months is enough.
- 93 months = 7 years 9 months.
93 months (7 years and 9 months).
Strategy 2: graphs
Graph y = 2500(1.06)ˣ and y = 4000 on a graphing calculator or a free graphing app, and find the intersection. It is at x ≈ 8.07. As in the guess-and-check method, the money must stay in for a whole number of compounding periods, so the answer is 9 years.
Strategy 3: a TVM Solver
A TVM Solver (on a graphing calculator, or in a free online financial calculator) has these fields:
- N = number of periods; I% = annual interest rate as a percent; PV = present value; PMT = regular payment (0 for a single deposit); FV = future value; P/Y = payments per year; C/Y = compounding periods per year.
Money you pay out is entered as negative, and money you receive as positive. Your deposit of $2 500 leaves your pocket, so PV = −2500. The $4 000 comes back to you, so FV = 4000.
Worked exampleExample 2: The power of a power rule
At 8% compounded annually, 1.08⁹ ≈ 1.999, so money doubles in about 9 years. About how long does it take to become 4 times and 8 times as large?
- 4 = 2², so (1.08⁹)² = 1.08¹⁸ ≈ 2² = 4 by the power of a power rule. Money quadruples in about 18 years.
- 8 = 2³, so (1.08⁹)³ = 1.08²⁷ ≈ 2³ = 8. Money becomes 8 times as large in about 27 years.
- Check: 1.08²⁷ ≈ 7.988, just under 8 (because 1.08⁹ is just under 2). So in fact it takes a little more than 27 years. The pattern is a fast estimate: every doubling time adds another factor of 2.
About 18 years to quadruple and about 27 years to grow 8 times (a little more, by checking).
| Annual rate (compounded annually) | Rule of 72 estimate | Actual doubling time |
|---|
| 2% | 36 years | 35.00 years |
| 3% | 24 years | 23.45 years |
| 4% | 18 years | 17.67 years |
| 6% | 12 years | 11.90 years |
| 8% | 9 years | 9.01 years |
| 9% | 8 years | 8.04 years |
| 12% | 6 years | 6.12 years |
Worked exampleExample 3: A quarterly goal
Leah has $1 200. How long will it take to grow to $2 000 at 6% compounded quarterly?
- i = 0.015. (1.015)ⁿ = 2000 ÷ 1200 ≈ 1.6667.
- Try n = 34: 1.015³⁴ ≈ 1.6590 (too small). Try n = 35: 1.015³⁵ ≈ 1.6839 (big enough).
- n = 35 quarters = 8.75 years = 8 years and 9 months.
35 quarters, which is 8 years 9 months.
Just for funThe rule of 72 is more than 500 years old. It appears in a mathematics book by the Italian mathematician Luca Pacioli, printed in 1494, and people still use it to estimate doubling times in their heads.
Hands-on activityTest the rule of 72
You need: a scientific calculator and paper.
- Choose three rates: 3%, 6% and 12%.
- For each rate, start with 1 on your calculator and keep multiplying by the growth factor (1.03, 1.06 or 1.12), counting the presses, until the display passes 2.
- Compare your count with 72 ÷ rate.
- Repeat for tripling (passing 3). Can you find a "rule of ___" for tripling? Try dividing your tripling times into your rates.
What to look for: doubling takes 24 years at 3% (72 ÷ 3 = 24), 12 years at 6% (72 ÷ 6 = 12) and 7 years at 12% (72 ÷ 12 = 6, but 1.12⁶ ≈ 1.97 is just short). Tripling takes about 38, 19 and 10 years. Rate × tripling time is roughly 114 to 120, so a "rule of about 115" works for tripling.
Your turn
1How many years does it take $3 000 to grow to at least $4 500 at 4% compounded annually?
1.04ⁿ = 1.5. 1.04¹⁰ ≈ 1.4802, 1.04¹¹ ≈ 1.5395. 11 years.
2Use the rule of 72 to estimate the doubling time at 6% compounded annually.
3How many months does it take money to double at 9% compounded monthly?
1.0075ⁿ = 2 is reached at n = 93 months (1.0075⁹² ≈ 1.9886, 1.0075⁹³ ≈ 2.0035).
4How long for $1 200 to reach $2 000 at 6% compounded quarterly?
1.015ⁿ ≈ 1.6667: n = 35 quarters (8 years 9 months).
5What should you enter for PV and FV on a TVM Solver if you deposit $500 and want to know when it grows to $800?
PV = −500 (money paid out) and FV = 800 (money received).
6A TVM Solver gives N = 12.3 for a monthly compounding question. How many months until the goal is reached?
Interest is only added at the end of each month, so 13 months.
More practice: warm-up
1Simplify 4000(1.05)ⁿ = 6000 to the form (1.05)ⁿ = a number.
2Which is larger: 1.05⁸ or 1.5?
1.05⁸ ≈ 1.4775, so 1.5 is larger (8 years at 5% is not quite enough to grow by 50%).
3Rule of 72: estimate the doubling time at 4%.
4What does N stand for on a TVM Solver?
The number of periods (payment or compounding periods).
5Name three strategies for finding n without logarithms.
Guess and check, graphing (intersection), and a TVM Solver or other technology. (Exponent laws and the rule of 72 help to estimate.)
More practice: on your own
1How many years for $5 000 to grow to $7 500 at 5% compounded semi-annually?
i = 0.025: 1.025ⁿ = 1.5. 1.025¹⁶ ≈ 1.4845, 1.025¹⁷ ≈ 1.5216. n = 17 half-years = 8.5 years.
2How many months for $800 to grow to $1 000 at 4.2% compounded monthly?
i = 0.0035: 1.0035ⁿ = 1.25. 1.0035⁶³ ≈ 1.2462, 1.0035⁶⁴ ≈ 1.2506. 64 months (5 years 4 months).
3At 3% compounded annually, money doubles in about 23.4 years. Use the power of a power rule to estimate how long it takes to become 4 times as large.
4 = 2², so about 2 × 23.4 ≈ 47 years.
4$10 000 is invested at 4.5% compounded semi-annually. When will it first be worth at least $15 000?
1.0225ⁿ = 1.5. 1.0225¹⁸ ≈ 1.4926, 1.0225¹⁹ ≈ 1.5262. n = 19 half-years = 9.5 years.
5Explain why we round n up in these money questions, even if the decimal part is small, like 8.07.
Interest is added only at the end of each compounding period. After 8 periods the goal has not been reached; it is reached when the 9th period's interest is added.
Challenge
1At 6% compounded annually, how long does it take money to triple? Use guess and check.
1.06ⁿ = 3. 1.06¹⁸ ≈ 2.8543, 1.06¹⁹ ≈ 3.0256. 19 years.
2At 7% compounded annually, 1.07¹⁰ ≈ 1.967. Use the power of a power rule to estimate how long it takes money to grow 8 times, then check.
1.07¹⁰ is about 2, so 8 = 2³ suggests about 30 years. Check: 1.07³⁰ ≈ 7.612, less than 8 because 1.967 is a bit under 2. Guess and check: 1.07³¹ ≈ 8.145, 1.07³² ≈ 8.715. So it takes 31 years.
3Mira has $2 000 at 5% compounded annually. Noah has $2 500 at 3% compounded annually. After how many full years will Mira's account first be worth more than Noah's?
Need 2000(1.05)ⁿ > 2500(1.03)ⁿ, so (1.05/1.03)ⁿ > 1.25. 1.05 ÷ 1.03 ≈ 1.019417. Try n = 11: ≈ 1.2356 (not yet). n = 12: ≈ 1.2593. After 12 years.
Quiz
1. Which equation must be solved to find how long $1 000 takes to reach $1 800 at 5% compounded annually?
2. 1.06⁸ ≈ 1.594 and 1.06⁹ ≈ 1.690. How many years does it take money to grow by 60% at 6% compounded annually?
3. The rule of 72 estimates the doubling time at 8% compounded annually as:
4. Money doubles in about 10 years at a certain rate. About how long to become 4 times as large?
5. On a TVM Solver, you deposit $2 000 and want to know when it will be worth $3 000. How should PV and FV be entered?
6. A TVM Solver gives N = 23.2 for a quarterly compounding question. When is the goal reached?
Big idea: Expectation C3.4 asks students to find the number of compounding periods using technology (a scientific calculator, a TVM Solver or online tools) and to describe strategies such as guess and check, the power of a power rule and graphing. Logarithms are not part of this course; they come in Grade 12 Advanced Functions (MHF4U).
How to help: Guess and check is a real mathematical strategy here, not a shortcut. Encourage your teen to record each guess in a small table so they can see the answer narrowing.
Common difficulties:
- Forgetting to simplify to (1 + i)ⁿ = A ÷ P before guessing.
- Rounding n to the nearest whole number instead of rounding up.
- Giving the answer in periods when the question asks for years (or the other way around).
- TVM sign convention: one of PV and FV must be negative.