Doubling Duel Slides 12th Grade Financial Math
The Doubling Duel
Exact Logarithms vs. The Rule of 72: When is a "Shortcut" Good Enough?
Logarithmic Math
vs.
Rule of 72
Warm-up: Mental Math
05:00
The Rule of 72 is a shortcut to estimate how long it takes an investment to double.
Years ≈ 72 ÷ Rate (%)
*Note: Use the whole number (9) not the decimal (0.09) for this rule!
Quick Estimates:
1 How long at 8%?
2 How long at 12%?
3 How long at 6%?
Video Analysis
4:45 - 5:32
Embedded media
Watch for:
How does the narrator compare the "exact" logarithmic answer to the Rule of 72 estimation?
Key Formula
For continuously compounded interest, doubling time is:
t = ln(2) / r
Where r is the decimal rate.
The Investigation
25:00
Your Mission:
Create a comparison table (on paper or in a spreadsheet) for interest rates ranging from 1% to 25%.
Calculate:
The Shortcut: 72 / Interest RateThe Exact: ln(2) / decimal rateThe Variance: Shortcut - Exact
Rate Rule of 72 Exact (ln 2/r) Difference 1% 72.0 yrs 69.3 yrs +2.7 8% 9.0 yrs 8.7 yrs +0.3 ... ... ... ... 25% 2.88 yrs 2.77 yrs +0.11
When is an estimate
"good enough" in finance?
If you are helping a client plan their retirement over 30 years, is a 0.5-year difference significant? What if you are a high-frequency trader?
Rule of 69.3
The "True" constant doubling factor
Rule of 70
Used for population/economics
Rule of 72
The "Divisibility King" for mental math
Extension: Why 72?
Advanced Concept
The Rule of 72 is actually an approximation of the Taylor Series for natural logarithms.
\[ \ln(2) \approx 0.6931 \] \[ \ln(1+r) \approx r \]
For continuously compounded interest, the exact numerator is 69.3.
Discussion Challenge:
If 69.3 is more "accurate," why did the financial world settle on 72?
Divisible by 2, 3, 4, 6, 8, 9, 12, 18, 24... Easier for Mental Math Annual vs. Continuous Offset
Doubling Duel Worksheet The Doubling Duel
Financial Math: Rule of 72 vs. Logarithmic Accuracy
Student:
Date:
Part 1: Video Analysis
1. Calculate the doubling time for a 9% interest rate using the "exact" formula shown in the video:
Formula hint: \( t = \frac{\ln(2)}{r} \)
2. Now use the "Rule of 72" shortcut for the same 9% rate:
Formula hint: \( t \approx \frac{72}{R} \)
Part 2: Comparison Investigation
Complete the table below to compare the shortcut to the exact logarithmic calculation for continuous compounding. For the exact calculation, use \(\ln(2) \approx 0.693\).
Annual Rate (\(R\)) Shortcut: \( \frac{72}{R} \) Exact: \( \frac{0.693}{r} \) Difference (years) % Error 1% 4% 8% 12% 18% 25%
*Note: \( r \) is the decimal form (e.g., \( 0.08 \) for 8%), while \( R \) is the whole number (8) for the shortcut.
Part 3: Critical Analysis
Looking at your data, where is the Rule of 72 most accurate? Where does it begin to fail significantly?
In the financial world, Rule of 72 is the standard shortcut. Why do you think advisors use 72 instead of 69 or 70, which are mathematically "closer" to \( \ln(2) \)? (Hint: Look at the numbers you can divide by 72).
The Taylor Series Extension
For advanced students: The Rule of 72 is an approximation of the logarithmic growth formula \( t = \frac{\ln(2)}{\ln(1+r)} \). For small values of \( r \), \( \ln(1+r) \) is approximately equal to \( r \). Using calculus, explain why the "Shortcut" is always slightly higher than the exact continuous calculation.
Doubling Duel Anchor Chart The Rule of 72
Estimating the Power of Doubling
THE QUICK SHORTCUT
Years ≈ 72 ÷ Rate
Use Whole Numbers (e.g. 8) No Logarithms Needed!
The Exact Math
For Continuous Compounding , we use the natural log of 2 (\(\ln 2 \approx 0.693\)):
\( t = \frac{\ln(2)}{r} \)
Where \(r\) is a decimal (e.g. 0.08)
The Difference
Rule of 72 is best for mental math (72 has many factors!)Exact Formula is required for high-stakes modeling.
Accuracy Snapshot
Low Rate (3%)
Diff: ~1.0 Year
Sweet Spot (8%)
Diff: ~0.3 Years
High Rate (18%)
Diff: ~0.1 Year
PRO TIP: Why 72?
Advisors use 72 because it's divisible by 2, 3, 4, 6, 8, 9, and 12. Try doing 69.3 ÷ 8 in your head during a meeting!
Doubling Duel Teacher Key Teacher Guide & Key
The Doubling Duel
Grade 12: Financial Math
Lesson Pacing
Warm-up
5 Min
Video
10 Min
Investigation
25 Min
Discussion
5 Min
Extension
Ongoing
Answer Key: Part 2 Table
Rate (R) Shortcut (72/R) Exact (0.693/r) Diff (Years) % Error 1% 72.00 69.31 +2.69 3.8% 4% 18.00 17.33 +0.67 3.8% 8% 9.00 8.66 +0.34 3.9% 12% 6.00 5.78 +0.22 3.8% 18% 4.00 3.85 +0.15 3.8% 25% 2.88 2.77 +0.11 3.9%
*Note for teachers: The % error for continuous compounding remains constant because the Rule of 72 is essentially approximating 69.3 with 72 (\( 72/69.3 \approx 1.038 \)).
Critical Analysis Guidance
Question: Where is it most accurate? continuous compounding, the "error" (difference in years) shrinks as interest rates rise. However, for annual compounding (common in bank savings), the Rule of 72 is most accurate around 8%. This is a great point for discussion.
Question: Why 72? Factors of 72 : 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36. Mental math is significantly easier. Using the "exact" 69.3 requires a calculator, defeating the purpose of a rule of thumb.
The Taylor Series Connection
The derivation comes from \( 2P = Pe^{rt} \), simplifying to \( \ln(2) = rt \). Since \( \ln(2) \approx 0.6931 \), the exact relationship is \( t = 0.6931/r \).
Rule of 72 is essentially: \( t \approx 72 / (100r) \), which simplifies to \( t \approx 0.72 / r \). Students should notice that \( 0.72 > 0.693 \), explaining why the Rule of 72 consistently slightly overestimates the time needed for continuous doubling.
