Calculate inverse ratios, split a total amount by inverse proportion, or solve A1×B1 = A2×B2 for unknown values in work-rate problems.
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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):
Split Quantity (share a total in inverse ratio):
Inverse Proportion (solve for an unknown):
- 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 clears the reciprocal denominators and divides by their greatest common divisor to show a simplified whole-number ratio. Entered finite decimal parts are treated as exact decimal fractions; displayed quantity shares are rounded. 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.
