Rule of 72 worksheet

1. Rate needed, rule of 72: 7.2%

1. Use the quick rule in reverse: divide 72 by the years to get the rate needed

   $\frac{72}{10}$$=\mathrm{7.2\%}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{10}$$=\mathrm{7\%}$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{10}$$=\mathrm{6.93\%}$

4. Now find the exact rate, with growth added once a year. Solve (1 + r)^N = 2, where N is the years

   $2^{\frac{1}{10}}-1$$=0.07177346$$=\mathrm{7.177346\%}$

5. See how far off the rule of 72 is: its estimate minus the exact rate, in percentage points

   $\mathrm{7.2\%}-\mathrm{7.177346\%}$$=0.022654$

2. Doubling time, rule of 72: 8 years

1. Use the quick rule: divide 72 by the rate in percent to get the years to double

   $\frac{72}{9}$$=8\text{\hspace{0.17em}years}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{9}$$=7.777778$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{9}$$=7.7$

4. Now find the exact time, with growth added once a year. Solve (1 + r)^t = 2, where r is the rate as a decimal

   $t$$=\frac{\text{ln 2}}{\text{ln}\left(1+0.09\right)}$

5. Divide the two logarithms (ln is the natural log on a calculator) to get the years

   $\frac{0.69314718}{0.0861777}$$=8.043232\text{\hspace{0.17em}years}$

6. See how far off the rule of 72 is: its estimate minus the exact answer

   $8-8.043232$$=\mathrm{-0.043232}\text{\hspace{0.17em}years}$

3. Doubling time, rule of 72: 10.29 years

1. Use the quick rule: divide 72 by the rate in percent to get the years to double

   $\frac{72}{7}$$=10.285714\text{\hspace{0.17em}years}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{7}$$=10$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{7}$$=9.9$

4. Now find the exact time, with growth added once a year. Solve (1 + r)^t = 2, where r is the rate as a decimal

   $t$$=\frac{\text{ln 2}}{\text{ln}\left(1+0.07\right)}$

5. Divide the two logarithms (ln is the natural log on a calculator) to get the years

   $\frac{0.69314718}{0.06765865}$$=10.244768\text{\hspace{0.17em}years}$

6. See how far off the rule of 72 is: its estimate minus the exact answer

   $10.285714-10.244768$$=0.040946\text{\hspace{0.17em}years}$

4. Doubling time, rule of 72: 12 years

1. Use the quick rule: divide 72 by the rate in percent to get the years to double

   $\frac{72}{6}$$=12\text{\hspace{0.17em}years}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{6}$$=11.666667$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{6}$$=11.55$

4. Now find the exact time, with growth added once a year. Solve (1 + r)^t = 2, where r is the rate as a decimal

   $t$$=\frac{\text{ln 2}}{\text{ln}\left(1+0.06\right)}$

5. Divide the two logarithms (ln is the natural log on a calculator) to get the years

   $\frac{0.69314718}{0.05826891}$$=11.895661\text{\hspace{0.17em}years}$

6. See how far off the rule of 72 is: its estimate minus the exact answer

   $12-11.895661$$=0.104339\text{\hspace{0.17em}years}$

5. Rate needed, rule of 72: 8%

1. Use the quick rule in reverse: divide 72 by the years to get the rate needed

   $\frac{72}{9}$$=\mathrm{8\%}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{9}$$=\mathrm{7.777778\%}$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{9}$$=\mathrm{7.7\%}$

4. Now find the exact rate, with growth added once a year. Solve (1 + r)^N = 2, where N is the years

   $2^{\frac{1}{9}}-1$$=0.08005974$$=\mathrm{8.005974\%}$

5. See how far off the rule of 72 is: its estimate minus the exact rate, in percentage points

   $\mathrm{8\%}-\mathrm{8.005974\%}$$=\mathrm{-0.005974}$

6. Doubling time, rule of 72: 7.2 years

1. Use the quick rule: divide 72 by the rate in percent to get the years to double

   $\frac{72}{10}$$=7.2\text{\hspace{0.17em}years}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{10}$$=7$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{10}$$=6.93$

4. Now find the exact time, with growth added once a year. Solve (1 + r)^t = 2, where r is the rate as a decimal

   $t$$=\frac{\text{ln 2}}{\text{ln}\left(1+0.1\right)}$

5. Divide the two logarithms (ln is the natural log on a calculator) to get the years

   $\frac{0.69314718}{0.09531018}$$=7.272541\text{\hspace{0.17em}years}$

6. See how far off the rule of 72 is: its estimate minus the exact answer

   $7.2-7.272541$$=\mathrm{-0.072541}\text{\hspace{0.17em}years}$

7. Rate needed, rule of 72: 5.14%

1. Use the quick rule in reverse: divide 72 by the years to get the rate needed

   $\frac{72}{14}$$=\mathrm{5.142857\%}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{14}$$=\mathrm{5\%}$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{14}$$=\mathrm{4.95\%}$

4. Now find the exact rate, with growth added once a year. Solve (1 + r)^N = 2, where N is the years

   $2^{\frac{1}{14}}-1$$=0.05075664$$=\mathrm{5.075664\%}$

5. See how far off the rule of 72 is: its estimate minus the exact rate, in percentage points

   $\mathrm{5.142857\%}-\mathrm{5.075664\%}$$=0.067193$

8. Doubling time, rule of 72: 6.55 years

1. Use the quick rule: divide 72 by the rate in percent to get the years to double

   $\frac{72}{11}$$=6.545455\text{\hspace{0.17em}years}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{11}$$=6.363636$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{11}$$=6.3$

4. Now find the exact time, with growth added once a year. Solve (1 + r)^t = 2, where r is the rate as a decimal

   $t$$=\frac{\text{ln 2}}{\text{ln}\left(1+0.11\right)}$

5. Divide the two logarithms (ln is the natural log on a calculator) to get the years

   $\frac{0.69314718}{0.10436002}$$=6.641885\text{\hspace{0.17em}years}$

6. See how far off the rule of 72 is: its estimate minus the exact answer

   $6.545455-6.641885$$=\mathrm{-0.09643}\text{\hspace{0.17em}years}$

9. Rate needed, rule of 72: 9%

1. Use the quick rule in reverse: divide 72 by the years to get the rate needed

   $\frac{72}{8}$$=\mathrm{9\%}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{8}$$=\mathrm{8.75\%}$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{8}$$=\mathrm{8.6625\%}$

4. Now find the exact rate, with growth added once a year. Solve (1 + r)^N = 2, where N is the years

   $2^{\frac{1}{8}}-1$$=0.09050773$$=\mathrm{9.050773\%}$

5. See how far off the rule of 72 is: its estimate minus the exact rate, in percentage points

   $\mathrm{9\%}-\mathrm{9.050773\%}$$=\mathrm{-0.050773}$

10. Rate needed, rule of 72: 12%

1. Use the quick rule in reverse: divide 72 by the years to get the rate needed

   $\frac{72}{6}$$=\mathrm{12\%}$

2. Try the same shortcut with 70 instead of 72

   $\frac{70}{6}$$=\mathrm{11.666667\%}$

3. And with 69.3, the version for nonstop growth

   $\frac{69.3}{6}$$=\mathrm{11.55\%}$

4. Now find the exact rate, with growth added once a year. Solve (1 + r)^N = 2, where N is the years

   $2^{\frac{1}{6}}-1$$=0.12246205$$=\mathrm{12.246205\%}$

5. See how far off the rule of 72 is: its estimate minus the exact rate, in percentage points

   $\mathrm{12\%}-\mathrm{12.246205\%}$$=\mathrm{-0.246205}$