ForHosting KIT · Developer Utilities

Inverse variation calculator

The inverse variation calculator solves relationships in the form y = k / x. Enter one known x and y pair to determine the constant of variation, then provide a new x value to predict its corresponding y.

● BetaFree · in your browser

Run — free How it works & API

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The result includes the constant, the completed equation, and clear substitution steps, making it useful for checking homework, exploring reciprocal relationships, or applying a model to a new measurement without rearranging the formula by hand.

Recognize an inverse variation relationship

Inverse variation describes two quantities whose product remains constant. It is written as y = k / x, where k is the constant of variation and x cannot be zero. As the magnitude of x increases, the magnitude of y decreases in a way that preserves x times y. For example, if four workers can complete an idealized fixed task in six hours, the model gives k = 24 worker-hours. Eight workers would then correspond to three hours, provided the assumptions of the model remain reasonable. The calculator does not decide whether a real data set truly follows inverse variation; it applies the model you specify. A quick way to test several observed pairs is to multiply x by y for each one. Products that are equal, apart from ordinary measurement or rounding error, support a common inverse variation model. Products that differ substantially indicate that another relationship or additional factors may be involved. Signs also matter: a positive k gives x and y the same sign, while a negative k gives them opposite signs.

Find the constant and predict the new value

Start with the known pair and rearrange y = k / x by multiplying both sides by x. This gives k = xy. The calculator performs that multiplication and then places the resulting constant into the model. To predict the output for another input, it substitutes the new x and evaluates y = k / new_x. Suppose the known pair is x = 4 and y = 6. Their product is 24, so the specific equation is y = 24 / x. With new_x = 8, division gives y = 3. The returned steps preserve this sequence so you can audit the calculation or show the reasoning in written work. Both the known x and the new x must be non-zero because division by zero is undefined. The known y may be zero; in that case k is zero and the model predicts zero for every permitted x. Decimal and negative inputs are supported, and calculated values are rounded consistently to keep the JSON result stable and readable.

Interpret the answer and the model's limits

A computed answer is only as meaningful as the inverse variation assumption behind it. The formula is appropriate when the product of the two quantities should remain fixed: speed and travel time for a fixed distance, pressure and volume in a simplified constant-temperature gas model, or the number of equal-rate machines and completion time for a fixed amount of work. Real situations may violate those assumptions. Workers can interfere with one another, machines may have setup overhead, and physical measurements have uncertainty. Use the returned equation to distinguish the mathematical result from the practical claim. The constant k carries the combined units of x times y, while predicted y has the same units as the known y. If x doubles, y is halved; if x is multiplied by a factor of five, y is divided by five. Negative values follow the same reciprocal arithmetic, although they may not make sense for quantities such as elapsed time or a count of people. The API price is $0.002 per request, and the browser version can run the same deterministic calculation locally.

What you can do with it

Estimate time for a fixed workload

Use a known number of equal-rate workers or machines and its completion time to estimate the idealized time for a different count.

Check inverse variation homework

Confirm the constant k, the completed equation, and the substitution used to calculate a requested value.

Apply a reciprocal science model

Predict a new measurement when two modeled quantities have a constant product, while keeping the model assumptions visible.

FAQ

What is the inverse variation formula?

The standard formula is y = k / x, equivalently xy = k, where k is constant and x is not zero.

How is the constant of variation calculated?

Multiply the known values: k = x times y. The calculator returns this constant and inserts it into the equation.

Why can x not equal zero?

The formula divides k by x, and division by zero is undefined. This applies to both the known x and the new x.

Can I enter decimal or negative values?

Yes. Finite decimal and negative values are accepted. Consider separately whether a negative value is meaningful in your real-world context.

What does the API request cost?

The API price is $0.002 per request. The same deterministic calculation is also available in the browser.

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/algebra/inverse-variation

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/algebra/inverse-variation \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"x":4,"y":6,"new_x":8}'

const res = await fetch("https://api.kit.forhosting.com/algebra/inverse-variation", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "x": 4,
    "y": 6,
    "new_x": 8
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/algebra/inverse-variation",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "x": 4,
    "y": 6,
    "new_x": 8
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/inverse-variation", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"x":4,"y":6,"new_x":8}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"x":4,"y":6,"new_x":8}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/inverse-variation", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "x": 4,
  "y": 6,
  "new_x": 8
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "algebra.inverse_variation",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Read the full KIT documentation →