# Rule of 72 Calculator

> Estimates the doubling time of an amount growing at a fixed annual rate with the Rule of 72, and compares it with the exact result ln 2 / ln(1 + r) for annual compounding and ln 2 / r for continuous compounding.

- Calculator id: `rule-of-72` · Category: Finance (`finance`) · Tool name: `calculate_rule_of_72`
- Canonical page: https://tttkmbb.com/finance/rule-of-72 · This document: https://tttkmbb.com/finance/rule-of-72.md · JSON definition: https://tttkmbb.com/finance/rule-of-72.json

## Purpose

Estimates the doubling time of an amount growing at a fixed annual rate with the Rule of 72, and compares it with the exact result ln 2 / ln(1 + r) for annual compounding and ln 2 / r for continuous compounding.

**Use when:** You want a quick mental-math estimate of how long money (or debt, or prices) takes to double at a given annual growth rate, plus the exact figure.

**Do not use when:** You need the balance after a specific number of years (use compound-interest) or the rate implied by two values (use cagr).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `annual_rate_percent` | number | % | required | Annual interest, return or inflation rate in percent; 8 means 8 %. (> 0, max 100) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `rule_of_72_years` | number | years | 72 / annual_rate_percent. |
| `exact_years` | number | years | ln(2) / ln(1 + rate). |
| `continuous_years` | number | years | ln(2) / rate ≈ 69.3 / annual_rate_percent. |
| `rule_error_percent` | number | % | (rule_of_72_years − exact_years) / exact_years × 100. |
| `years_to_triple_exact` | number | years | ln(3) / ln(1 + rate). |

## Formula

`rule_of_72_years = 72 / annual_rate_percent; exact_years = ln(2) / ln(1 + annual_rate_percent/100); continuous_years = ln(2) / (annual_rate_percent/100)`

The rule is a first-order approximation that is most accurate between about 6 % and 10 %; below that the Rule of 70 or 69.3 is closer, and at high rates the rule overstates the doubling time.

## Data Sources

- Wikipedia – Rule of 72 — https://en.wikipedia.org/wiki/Rule_of_72 (reference, retrieved 2026-09-23)
- Investopedia – The Rule of 72: Definition, Usefulness and How to Use It — https://www.investopedia.com/terms/r/ruleof72.asp (reference, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/rule-of-72?annual_rate_percent=…`
- `POST https://tttkmbb.com/api/v1/calculate/rule-of-72` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/rule-of-72 · OpenAPI operationId `calculate_rule_of_72` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "rule-of-72", "inputs": {…}}`

## Example

- 8 % per year: inputs `{"annual_rate_percent":8}` → `{"rule_of_72_years":9,"exact_years":9.01,"continuous_years":8.66,"rule_error_percent":-0.07}`
- 6 % per year: inputs `{"annual_rate_percent":6}` → `{"rule_of_72_years":12,"exact_years":11.9,"continuous_years":11.55,"years_to_triple_exact":18.85}`

```
GET https://tttkmbb.com/api/v1/calculate/rule-of-72?annual_rate_percent=8
```

## Limitations

You need the balance after a specific number of years (use compound-interest) or the rate implied by two values (use cagr). The rule is a first-order approximation that is most accurate between about 6 % and 10 %; below that the Rule of 70 or 69.3 is closer, and at high rates the rule overstates the doubling time. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Why 72 and not 69.3?**

ln 2 ≈ 0.693 gives the exact continuous-compounding rule, but 72 has many divisors (2, 3, 4, 6, 8, 9, 12) and happens to correct for annual compounding near 8 %.

**Does it work for inflation or debt?**

Yes, any constant growth rate: at 3 % inflation prices double in about 24 years; at 18 % APR an unpaid credit-card balance doubles in about 4 years.

## Related

- [Compound Interest Calculator](https://tttkmbb.com/finance/compound-interest.md) — Exact balance after any number of years.
- [CAGR Calculator](https://tttkmbb.com/finance/cagr.md) — Growth rate implied by a starting and ending value.