> ## Documentation Index > Fetch the complete documentation index at: https://docs.iotools.cloud/llms.txt > Use this file to discover all available pages before exploring further. # Rule of 72 Calculator > Estimate how long it takes an investment to double using the Rule of 72 (years ≈ 72 / rate%), and see the exact logarithmic answer beside it. Compares the Rule of 72, 70 and 69.3 approximations, shows each rule's error, supports any compounding frequency, and works in reverse to find the rate needed to double in a target number of years. [Try Rule of 72 Calculator in your browser →](https://iotools.cloud/tool/rule-of-72-calculator/) ## OpenAPI ````yaml https://api.iotools.cloud/v1/openapi post /v1/tool/rule-of-72-calculator openapi: 3.1.0 info: title: iotools.cloud API version: 1.0.0 description: >- Run any iotools.cloud tool over HTTP. Authenticate with `Authorization: Bearer iot_live_…`. **Only `POST /v1/tool/{slug}` costs credits.** Every GET here — the catalog, a tool's schema, your balance — is free. A tool call is charged its own weight or your plan's per-call minimum, whichever is larger; `x-iotools-credit-cost` on each operation is quoted at the free-tier minimum, and `GET /v1/tools/list` returns the exact figure for your key. `GET /v1/me/credits` reports what you have left, and `GET /v1/me/usage` reports what it went on. Failures are RFC 9457 problem documents — branch on `code`. servers: - url: https://api.iotools.cloud description: Production security: - bearerAuth: [] tags: - name: Catalog description: Find a tool and read its contract. Free. - name: Converters description: >- Convert between formats, encodings, and units — Base64, CSV and JSON, timestamps, and more. Fast, free, and processed right in your browser. - name: Formatters description: >- Format, beautify, minify, and validate code and data — JSON, HTML, CSS, SQL, and regex. Clean up messy input in one click, with nothing to upload. - name: Generators description: >- Generate exactly what you need and on demand — passwords, UUIDs, QR codes, hashes, random numbers, and more. Secure, instant, and free. - name: Calculators description: >- Crunch the numbers fast — from everyday math to specialized conversions and unit work. Free online calculators that run entirely in your browser. - name: Editors description: >- Edit and transform text, code, and images with quick, focused editors that run entirely in your browser — nothing to install, and no sign-up needed. - name: Utilities description: >- Everyday developer and web utilities — DNS and IP lookups, redirect and certificate checkers, and other quick diagnostics. Free and privacy-friendly. - name: Account description: Your key's allowance, limits and usage history. paths: /v1/tool/rule-of-72-calculator: post: tags: - Calculators summary: Rule of 72 Calculator description: >- Estimate how long it takes an investment to double using the Rule of 72 (years ≈ 72 / rate%), and see the exact logarithmic answer beside it. Compares the Rule of 72, 70 and 69.3 approximations, shows each rule's error, supports any compounding frequency, and works in reverse to find the rate needed to double in a target number of years. [Try Rule of 72 Calculator in your browser →](https://iotools.cloud/tool/rule-of-72-calculator/) operationId: run_rule_of_72_calculator requestBody: required: false content: application/json: schema: type: object properties: mode: enum: - rate_to_years - years_to_rate description: Direction rate: type: number description: Annual Interest Rate (%) minimum: 0.01 maximum: 1000 years: type: number description: Years to Double minimum: 0.01 maximum: 1000 frequency: enum: - '1' - '2' - '4' - '12' - '365' - continuous description: Compounding required: [] examples: 8_annual_return_doubling_time: summary: 8% annual return — doubling time value: mode: rate_to_years rate: '8' frequency: '1' 2_low_rate_where_the_rule_drifts: summary: 2% low rate — where the rule drifts value: mode: rate_to_years rate: '2' frequency: '1' double_in_10_years_required_rate: summary: Double in 10 years — required rate value: mode: years_to_rate years: '10' frequency: '1' 6_compounded_monthly: summary: 6% compounded monthly value: mode: rate_to_years rate: '6' frequency: '12' responses: '200': description: Tool output content: application/json: examples: 8_annual_return_doubling_time: summary: 8% annual return — doubling time value: tool: rule-of-72-calculator tool_version: 1.0.1 outputs: summary: |- === Rule of 72 — Doubling Time === At 8% per year, your money doubles in about 9.00 years (Rule of 72). The exact doubling time is 9.01 years — the rule is off by -0.07%. Annual Interest Rate: 8% Compounding: Annually (1x/year) Rule of 72: 9.00 years Rule of 70: 8.75 years Rule of 69.3: 8.66 years Exact (logarithmic): 9.01 years === Formula === Approximation: years ≈ 72 / rate Exact: years = ln(2) / ln(1 + rate/100) table: - method: Rule of 72 result: 9.00 years error-vs-exact: '-0.07%' best-for: Most popular; most accurate for rates around 6-10% - method: Rule of 70 result: 8.75 years error-vs-exact: '-2.85%' best-for: >- Easier mental math; slight under-estimate at low rates - method: Rule of 69.3 result: 8.66 years error-vs-exact: '-3.82%' best-for: Closest to the true constant (69.3 ≈ 100·ln 2) - method: Exact (logarithmic) result: 9.01 years error-vs-exact: — best-for: True value from the compound-interest formula credits_used: 3 credits_remaining: null 2_low_rate_where_the_rule_drifts: summary: 2% low rate — where the rule drifts value: tool: rule-of-72-calculator tool_version: 1.0.1 outputs: summary: |- === Rule of 72 — Doubling Time === At 2% per year, your money doubles in about 36.00 years (Rule of 72). The exact doubling time is 35.00 years — the rule is off by +2.85%. Annual Interest Rate: 2% Compounding: Annually (1x/year) Rule of 72: 36.00 years Rule of 70: 35.00 years Rule of 69.3: 34.65 years Exact (logarithmic): 35.00 years === Formula === Approximation: years ≈ 72 / rate Exact: years = ln(2) / ln(1 + rate/100) table: - method: Rule of 72 result: 36.00 years error-vs-exact: +2.85% best-for: Most popular; most accurate for rates around 6-10% - method: Rule of 70 result: 35.00 years error-vs-exact: '-0.01%' best-for: >- Easier mental math; slight under-estimate at low rates - method: Rule of 69.3 result: 34.65 years error-vs-exact: '-1.01%' best-for: Closest to the true constant (69.3 ≈ 100·ln 2) - method: Exact (logarithmic) result: 35.00 years error-vs-exact: — best-for: True value from the compound-interest formula credits_used: 3 credits_remaining: null double_in_10_years_required_rate: summary: Double in 10 years — required rate value: tool: rule-of-72-calculator tool_version: 1.0.1 outputs: summary: |- === Rule of 72 — Required Rate === To double in 10 years, you need about 7.200% per year (Rule of 72). The exact required rate is 7.177% — the rule is off by +0.32%. Years to double: 10 Compounding: Annually (1x/year) Rule of 72: 7.200% Rule of 70: 7.000% Rule of 69.3: 6.930% Exact (logarithmic): 7.177% === Formula === Approximation: rate ≈ 72 / years Exact: rate = (2^(1/years) − 1) × 100 table: - method: Rule of 72 result: 7.200% error-vs-exact: +0.32% best-for: Most popular; most accurate for rates around 6-10% - method: Rule of 70 result: 7.000% error-vs-exact: '-2.47%' best-for: >- Easier mental math; slight under-estimate at low rates - method: Rule of 69.3 result: 6.930% error-vs-exact: '-3.45%' best-for: Closest to the true constant (69.3 ≈ 100·ln 2) - method: Exact (logarithmic) result: 7.177% error-vs-exact: — best-for: True value from the compound-interest formula credits_used: 3 credits_remaining: null 6_compounded_monthly: summary: 6% compounded monthly value: tool: rule-of-72-calculator tool_version: 1.0.1 outputs: summary: |- === Rule of 72 — Doubling Time === At 6% per year, your money doubles in about 12.00 years (Rule of 72). The exact doubling time is 11.58 years — the rule is off by +3.62%. Annual Interest Rate: 6% Compounding: Monthly (12x/year) Rule of 72: 12.00 years Rule of 70: 11.67 years Rule of 69.3: 11.55 years Exact (logarithmic): 11.58 years === Formula === Approximation: years ≈ 72 / rate Exact: years = ln(2) / (n · ln(1 + rate/(100·n))) table: - method: Rule of 72 result: 12.00 years error-vs-exact: +3.62% best-for: Most popular; most accurate for rates around 6-10% - method: Rule of 70 result: 11.67 years error-vs-exact: +0.74% best-for: >- Easier mental math; slight under-estimate at low rates - method: Rule of 69.3 result: 11.55 years error-vs-exact: '-0.27%' best-for: Closest to the true constant (69.3 ≈ 100·ln 2) - method: Exact (logarithmic) result: 11.58 years error-vs-exact: — best-for: True value from the compound-interest formula credits_used: 3 credits_remaining: null schema: type: object required: - tool - tool_version - outputs - credits_used - credits_remaining properties: outputs: type: object properties: summary: type: string description: Result table: type: array description: Comparison items: type: object properties: method: type: string description: Method result: type: string description: Result error-vs-exact: type: string description: Error vs exact best-for: type: string description: Best for x-iotools-columns: - method - result - error-vs-exact - best-for tool: type: string description: The tool's slug, echoing the {slug} in the request path. tool_version: type: string description: Output-contract version for this tool. credits_used: type: integer description: >- Credits this call consumed, after any settlement refund. 0 when metering is disabled. credits_remaining: type: - integer - 'null' description: >- Credits left in the current monthly allowance, or null when metering is disabled. request_id: type: string description: Correlation id, also sent as x-request-id. '400': $ref: '#/components/responses/ValidationError' '401': $ref: '#/components/responses/Unauthorized' '402': $ref: '#/components/responses/InsufficientCredits' '403': $ref: '#/components/responses/ToolNotAllowed' '404': $ref: '#/components/responses/ToolNotFound' '413': $ref: '#/components/responses/PayloadTooLarge' '429': $ref: '#/components/responses/RateLimited' '500': $ref: '#/components/responses/ToolFailed' '503': $ref: '#/components/responses/ToolDisabled' security: - bearerAuth: [] components: responses: ValidationError: description: Invalid request. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/validation_error title: type: string description: Short human-readable summary of the failure. examples: - Invalid request status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - validation_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. fields: type: - object - string additionalProperties: type: string description: >- What failed: a map of field name → message. Absent when the body itself is malformed; a single string when the failure isn't tied to one field. example: type: https://iotools.cloud/docs/errors/validation_error title: Invalid request status: 400 code: validation_error detail: One or more inputs are invalid — see `fields`. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 fields: inputString: Required Unauthorized: description: Missing or invalid API key. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/invalid_api_key title: type: string description: Short human-readable summary of the failure. examples: - Invalid API key status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - invalid_api_key detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://iotools.cloud/docs/errors/invalid_api_key title: Invalid API key status: 401 code: invalid_api_key detail: 'Provide ''Authorization: Bearer ''.' request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 InsufficientCredits: description: Monthly credit allowance exhausted. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/insufficient_credits title: type: string description: Short human-readable summary of the failure. examples: - Insufficient credits status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - insufficient_credits detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: Always 0 — a refused call charges nothing. credits_remaining: type: integer description: Credits left in the allowance — fewer than this call costs. example: type: https://iotools.cloud/docs/errors/insufficient_credits title: Insufficient credits status: 402 code: insufficient_credits detail: This call costs 1 credit and 0 remain in this month's allowance. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 credits_remaining: 0 ToolNotAllowed: description: Tool exists but has no API surface. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/tool_not_allowed title: type: string description: Short human-readable summary of the failure. examples: - Tool not available over the API status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_allowed detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://iotools.cloud/docs/errors/tool_not_allowed title: Tool not available over the API status: 403 code: tool_not_allowed detail: >- "Background Remover" is available on iotools.cloud but has no API endpoint. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 ToolNotFound: description: No such tool. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/tool_not_found title: type: string description: Short human-readable summary of the failure. examples: - Tool not found status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_found detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://iotools.cloud/docs/errors/tool_not_found title: Tool not found status: 404 code: tool_not_found detail: No tool with that slug. See GET /v1/tools/list. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 PayloadTooLarge: description: Body too large. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/payload_too_large title: type: string description: Short human-readable summary of the failure. examples: - Payload too large status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - payload_too_large detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://iotools.cloud/docs/errors/payload_too_large title: Payload too large status: 413 code: payload_too_large detail: Request body exceeds this tool's size limit. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 RateLimited: description: Per-minute rate limit exceeded. headers: Retry-After: description: Seconds to wait before retrying (RFC 9110 delta-seconds). schema: type: integer content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/rate_limited title: type: string description: Short human-readable summary of the failure. examples: - Rate limit exceeded status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - rate_limited detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. retry_after: type: integer description: >- Seconds until the window resets — the same value as the `Retry-After` header. example: type: https://iotools.cloud/docs/errors/rate_limited title: Rate limit exceeded status: 429 code: rate_limited detail: Too many requests. Retry in 30s. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 retry_after: 30 ToolFailed: description: Tool failed to run. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/processing_error - https://iotools.cloud/docs/errors/internal_error title: type: string description: Short human-readable summary of the failure. examples: - Tool failed to run - Internal error status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - processing_error - internal_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: >- Always 0 on `processing_error` — a failed run is refunded, floor included. credits_remaining: type: integer description: >- Credits left after the refund. Absent when metering is disabled. example: type: https://iotools.cloud/docs/errors/processing_error title: Tool failed to run status: 500 code: processing_error detail: The tool failed to run. Please try again. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 ToolDisabled: description: Tool temporarily disabled. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://iotools.cloud/docs/errors/tool_disabled - https://iotools.cloud/docs/errors/api_unconfigured title: type: string description: Short human-readable summary of the failure. examples: - Tool temporarily disabled - API not configured status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_disabled - api_unconfigured detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://iotools.cloud/docs/errors/tool_disabled title: Tool temporarily disabled status: 503 code: tool_disabled detail: This tool is temporarily unavailable. Try again shortly. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 securitySchemes: bearerAuth: type: http scheme: bearer bearerFormat: iot_live_… ````