Library / Foundations: Maths / FMA-53
Foundations: Maths
Direct and inverse proportion
How a quantity changes when another is scaled, if y is proportional to a power of x.
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- FMA-53
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- v1
- Field
- Foundations: Maths
- Result
- y2
If y is proportional to a power of x, y = k xn, then scaling x by a factor multiplies y by that factor raised to n: y2 = y1(x2/x1)n. With n = 1 this is direct proportion; with n = −1, inverse proportion; with n = 2, an area-like square law. The constant k cancels, so it is not needed. Starting values and exponents must be nonzero and the powers real.
How a quantity changes when another is scaled, if y is proportional to a power of x. An exponent of 1 is direct proportion, −1 is inverse proportion and 2 is the square law.
Calculator
- Scaling factor for x, x2/x11.5
- Scaling factor for y, (x2/x1)n0.6667
- Constant of proportionality k = y1/x1n24
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With n = −1, changing x from 2 to 3 (a factor of 1.5) changes y from 12 to 8.
- Doubling x would multiply y by 2−1 = 0.5: direct proportion doubles y, inverse proportion halves it and a square law multiplies it by four.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| y1 Starting value of y | 0.1 to 1,000 | 0.06667 | 666.7 | 666.6 | +10% | +10 % |
| x1 Starting value of x | 0.1 to 100 | 0.4 | 400 | 399.6 | +10% | +10 % |
| x2 New value of x | 0.1 to 100 | 0.24 | 240 | 239.8 | +10% | −9.09 % |
| n Exponent n (1 direct, −1 inverse) | −3 to 3 | 3.556 | 40.5 | 36.94 | +0.12 | +4.99 % |
Sweep of New value of x
| New value of x | New value of y |
|---|---|
| 0.1 | 240 |
| 0.2371 | 101.2 |
| 0.5623 | 42.68 |
| 1.334 | 18 |
| 3.162 | 7.589 |
| 7.499 | 3.2 |
| 17.78 | 1.35 |
| 42.17 | 0.5691 |
| 100 | 0.24 |
Sweep one variable
Variables and how each one contributes
New value of y
What the equation gives you. Everything else feeds it.
Starting value of y
The starting value of y, in its own unit. It is multiplied by the scaling factor. Raises the new value in direct proportion, because y₂ is a multiple of y₁. Raising it by 1 (to 13) moves the result by +8.33 %, from 8 to 8.667.
Starting value of x
The starting value of x, in any unit used consistently with x₂. It must be nonzero. Changes the new value through the ratio x₂/x₁: raising x₁ lowers the ratio, which lowers y₂ for a positive exponent and raises it for a negative one. Raising it by 0.5 (to 2.5) moves the result by +25 %, from 8 to 10.
New value of x
The new value of x, in the same unit as x₁. Changes the new value through the ratio x₂/x₁: raising x₂ raises y₂ for a positive exponent and lowers it for a negative one. Raising it by 0.5 (to 3.5) moves the result by −14.3 %, from 8 to 6.857.
Exponent n (1 direct, −1 inverse)
The exponent n in y = kxn: 1 for direct proportion, −1 for inverse, 2 for a square law. Sharpens the dependence on x: for x₂ larger than x₁ a larger n gives a larger y₂, and for x₂ smaller than x₁ it gives a smaller one. Raising it by 0.25 (to −0.75) moves the result by +10.7 %, from 8 to 8.853.
Worked example with the values above
- Ratio of the new to the old x: x2/x1 = 3 / 2 = 1.5.
- Raise it to the power n = −1: 1.5−1 = 0.6667.
- y2 = 12 × 0.6667 = 8.
Assumptions and limits
- y is exactly proportional to x raised to a constant power, with no added constant.
- Values nonzero and the power defined (a negative x with a fractional power is not allowed).
Common mistakes
- Using a straight proportion when the relationship is a square or inverse law.
- Dividing the wrong way: for inverse proportion the new y is y₁ × (x₁/x₂), which is the n = −1 case.
Statistics and uncertainty
For a power law the relative error of y is |n| times the relative error of x: with n = 2 a 1 % error in x gives about 2 %, and with n = −1 about 1 % in the opposite sense. A larger exponent therefore makes the result more sensitive to the input.
Where it is used
- Scaling a result when one quantity changes: how time changes with speed, how force changes with area, how stress changes with diameter.
- Comparing similar designs at different sizes.
Maths you need
- Ratio: one quantity divided by another
- Powers: repeated multiplication