Practice · Grade 7 Percentages
Doubling Time Practice Problems
Practice problems in rising difficulty plus a boss question. Each has a hint and full working. Exact method and the rule of 70 covered.
Q1 of 6
0 correct
Find the exact doubling time at 10% growth per period. Round to 2 decimal places.
ln(2) ÷ ln(1.10)
Quick answer
How do I practice doubling time best?
Work several problems with the exact formula t = ln(2) ÷ ln(1 + r) and also estimate each with the rule of 70 (70 ÷ percent rate) as a quick check. Remember to plug r into the exact formula as a decimal (7% = 0.07), but use the percent number directly in the rule of 70. Increase the growth rate problem by problem and notice: a higher rate means a shorter doubling time, and for large rates the rule of thumb drifts further off.
HowTo
A 4-step solving strategy
This order works for any constant growth rate r given in percent — whether you need the exact value or just an estimate.
- 1Step 1 of 4
Read the rate and make it a decimal
Read the rate r in percent from the problem and convert it to a decimal for the exact formula: 7% = 0.07, 3% = 0.03. For the rule of 70 you keep the percent number (7, 3).
- 2Step 2 of 4
Plug into the exact formula
Use t = ln(2) ÷ ln(1 + r). Substitute your decimal, e.g. t = ln(2) ÷ ln(1.03). ln(2) is always a constant ≈ 0.6931.
- 3Step 3 of 4
Evaluate the denominator and divide
Find ln(1 + r) in the denominator and divide: t = 0.6931 ÷ ln(1 + r). Round the result sensibly to two decimal places.
- 4Step 4 of 4
Check with the rule of 70
Estimate 70 ÷ percent rate as a check. For small rates it matches well; if it is far off, the rate is large — then trust the exact value.
Examples
Worked practice problems with full working
Four typical doubling-time problems. Try each yourself first, then compare with the working.
Easy
Doubling time at 7% growth (exact)
t = ln(2) ÷ ln(1 + r), r = 0.07
t = 0.6931 ÷ ln(1.07)
t = 0.6931 ÷ 0.0677
t ≈ 10.24
Check with the rule of 70: 70 ÷ 7 = 10 ≈ 10.24 ✓
Standard case. First make r a decimal, set ln(2) as 0.6931, then divide by ln(1.07).
Easy
Doubling time at 10% growth (exact)
t = ln(2) ÷ ln(1.10)
t = 0.6931 ÷ 0.0953
t ≈ 7.27
Check with the rule of 70: 70 ÷ 10 = 7 ≈ 7.27 ✓
Higher rate than above — so a shorter doubling time. The rule of thumb is close.
Medium
Doubling time at 2% growth (exact)
t = ln(2) ÷ ln(1.02)
t = 0.6931 ÷ 0.0198
t ≈ 35.00
Check with the rule of 70: 70 ÷ 2 = 35 ≈ 35.00 ✓
For small rates the rule of 70 is almost perfect: ln(1 + r) ≈ r.
Hard
Boss: Doubling time at 8% growth (exact)
t = ln(2) ÷ ln(1.08)
t = 0.6931 ÷ 0.0770
t ≈ 9.01
Rule of 70: 70 ÷ 8 = 8.75 — already noticeably off; exact is 9.01.
The larger the rate, the more the rule of thumb drifts. Here the exact formula matters.
Pitfalls
Common mistakes — and how to avoid them
These five traps show up again and again when practicing doubling time.
Rate not converted to a decimal
In ln(1 + r), r must be a decimal: 7% = 0.07, so ln(1.07). Computing ln(1 + 7) = ln(8) gives complete nonsense.
Mixing up the rule of 70 and the exact formula
In the rule of 70 you use the percent number directly (70 ÷ 7); in the exact formula you use the decimal (ln(1.07)). Don't mix them.
Stretching the rule of thumb to large rates
The rule of 70 is only accurate for small rates. At 8% it is already noticeably off (8.75 vs 9.01); at 70% it fails entirely.
Confusing linear with exponential growth
The formula only holds for a constant percentage growth rate. If a fixed amount is added each period, there is no fixed doubling time.
Confusing ln(2) with log(2)
It means the natural log ln(2) ≈ 0.6931. The base-10 log log(2) ≈ 0.3010 only gives the right answer if the denominator also uses log.
Study
Practice with a plan — three quick tips
Always compute both ways
For every problem do the exact calculation and the rule-of-70 estimate. The comparison instantly tells you whether your exact value is plausible.
Raise the rate systematically
Practice with 1%, 2%, 5%, 10% in turn. You will develop a feel for how fast the doubling time drops as the rate grows.
For every wrong answer: why?
Was it the decimal conversion? The logarithm? Note the cause — next time you will recognise the mistake instantly.
FAQ
Frequently asked questions about practice
Glossary
Terms in one sentence
- Doubling time
- The time it takes a quantity to double at a constant growth rate: t = ln(2) ÷ ln(1 + r).
- Growth rate
- The percentage increase per period, e.g. 7% per year.
- Rule of 70
- A rule of thumb for estimating: doubling time ≈ 70 ÷ percent rate.
- Rule of 72
- A variant of the rule of thumb using 72 instead of 70 — easier to divide.
- Exponential growth
- Growth at a constant percentage rate; each period multiplies by (1 + r).
- ln (natural logarithm)
- Logarithm to base e; ln(2) ≈ 0.6931.
- Half-life
- The counterpart for decay — the time it takes to halve.