Enter any ratio (X/Y) into the Inverse Ratio Calculator. The calculator will evaluate and display the Inverse Ratio.
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Inverse Ratio Formula
The calculator uses one of three formulas depending on the mode you choose.
Inverse Ratio (flip a ratio):
Inverse of a:b:c = 1/a : 1/b : 1/c
Split Quantity (share a total in inverse ratio):
Share_i = Total × (1/r_i) / Σ(1/r_j)
Inverse Proportion (solve for an unknown):
A₁ × B₁ = A₂ × B₂
- a, b, c — parts of the original ratio
- r_i — the i-th part of the ratio being inverted
- Total — the quantity being divided
- A₁, B₁ — initial pair of inversely related values
- A₂, B₂ — new pair, where one is unknown
All ratio parts must be positive and non-zero. After inverting, the calculator multiplies through by the product of the original parts and simplifies by the GCD so the result is shown in whole numbers when possible. Inverse proportion assumes the two quantities have a constant product (k = A×B), which is the defining property of inverse variation.
Reference Tables
Common ratios and their inverses, shown in simplified form:
| Original Ratio | Inverse Ratio | Check (decimal) |
|---|---|---|
| 1 : 2 | 2 : 1 | 0.5 → 2.0 |
| 2 : 3 | 3 : 2 | 0.667 → 1.5 |
| 3 : 5 | 5 : 3 | 0.6 → 1.667 |
| 4 : 7 | 7 : 4 | 0.571 → 1.75 |
| 2 : 3 : 4 | 6 : 4 : 3 | LCD = 12 |
| 1 : 2 : 5 | 10 : 5 : 2 | LCD = 10 |
Typical situations modeled by inverse proportion (A × B = constant):
| Scenario | A | B |
|---|---|---|
| Workforce and time | Workers | Days to finish |
| Travel | Speed | Time taken |
| Gas at constant temperature | Pressure | Volume |
| Gear pair | Teeth count | RPM |
| Lever / seesaw | Distance from pivot | Force needed |
Worked Examples
Example 1 — Flip a ratio. Invert 4 : 9.
Reciprocals: 1/4 : 1/9. Multiply by 36: 9 : 4. Result: 9 : 4.
Example 2 — Split in inverse ratio. Divide $600 between two people in the inverse ratio of 2 : 3.
Inverse of 2 : 3 is 3 : 2. Sum = 5. Shares: 600 × 3/5 = $360, and 600 × 2/5 = $240.
Example 3 — Inverse proportion. 4 workers finish a job in 12 days. How long will 6 workers take?
k = 4 × 12 = 48. So 6 × B₂ = 48, giving B₂ = 8 days.
FAQ
Is the inverse of a ratio the same as its reciprocal? Yes. Each part is replaced with its reciprocal, then the result is rescaled to whole numbers when possible.
Why does 2 : 3 become 3 : 2 instead of 1/2 : 1/3? Both are correct, but 3 : 2 is the simplified integer form after multiplying both reciprocals by 6.
Can I invert a three-part ratio? Yes. For a : b : c the inverse is bc : ac : ab, then simplified by the GCD.
What if a part is zero? An inverse ratio is undefined when any part is zero, since 1/0 has no value. The calculator rejects zero or negative inputs.
