Inverse Variation Calculator

Inverse variation is a relation y = k/x in which the product x·y stays equal to a constant k: when x doubles, y halves. If y = 24/x, then x = 2 gives y = 12, x = 4 gives y = 6 and x = 8 gives y = 3.

• k is the constant of variation (or constant of proportionality).
• "y varies inversely as x" and "y is inversely proportional to x" mean the same thing.
• x and y can never be 0: y = k/0 is undefined, and no x gives y = 0.

A picture helps: every point (x, y) on y = 24/x is the corner of a rectangle with area 24. A wider rectangle has to be shorter.

To solve an inverse variation problem, find k = x₁·y₁ from the known pair, then compute y₂ = k/x₂. If y varies inversely with x and y = 6 when x = 4, find y when x = 8:

1. Find the constant: k = 4 × 6 = 24.
2. Write the equation: y = 24/x.
3. Substitute x = 8: y = 24/8 = 3.
4. Check: 8 × 3 = 24 = k.

A shortcut skips k: x₁·y₁ = x₂·y₂, so y₂ = x₁·y₁/x₂ = 4 × 6 / 8 = 3.

Pick a mode, type the values, and the calculator finds k, the equation and the missing value as you type. The steps:

1. Choose Find k and y₂ when you know one pair and a new x, y from k when you know the constant, or Check a table to test pairs.
2. Pick the variation type: y = k/x, y = k/x², y = k/√x or y = k/x³.
3. Type the numbers. Fractions such as 3/4 and expressions such as √2 work.
4. Read y, k, the equation, the check and the values at 2x and x/2.
5. Open the graph or the equal-area rectangles, and the worked steps.

The inverse variation formula is y = k/x, with k = x·y; between 2 pairs it reads x₁·y₁ = x₂·y₂.

• Missing y: y₂ = x₁·y₁ / x₂.
• Missing x: x₂ = x₁·y₁ / y₂.
• Inverse square variation: y = k/x², with k = x²·y.
• Inverse square root variation: y = k/√x, with k = √x·y.
• Scaling rule: multiplying x by c divides y by c (or by c² for k/x²).

In direct variation y = kx, so y/x is constant and both rise together; in inverse variation y = k/x, so x·y is constant and one falls as the other rises.

A quick test: divide y by x for every pair. Equal ratios mean direct variation. Multiply instead; equal products mean inverse variation.

A table shows inverse variation when x·y gives the same number for every row; that number is k.

• (2, 12), (3, 8), (4, 6), (6, 4): every product is 24, so y = 24/x.
• (1, 5), (2, 10), (3, 15): the products are 5, 20, 45, so it is not inverse. y/x = 5 every time, so it is direct: y = 5x.
• (1, 10), (2, 6), (4, 3): products 10, 12, 12. One row breaks the rule, so it is neither.

Rounded data, such as measurements, gives products that are close but not equal. Then the data fits inverse variation approximately.

The graph of y = k/x is a hyperbola with 2 branches, asymptotic to both axes. For k > 0 the branches sit in quadrants I and III; for k < 0, in quadrants II and IV.

• Vertical asymptote x = 0 and horizontal asymptote y = 0.
• Symmetric about the line y = x: y = k/x is its own inverse function.
• The slope is −k/x², always negative for k > 0.
• Every point (x, y) on the curve encloses a rectangle of area |k| with the axes.

In inverse square variation, y = k/x², so doubling x divides y by 4 and tripling x divides y by 9. If y = 25 when x = 2, then k = 2² × 25 = 100 and y = 100/5² = 4 when x = 5.

• Light, sound and gravity follow inverse square variation with distance.
• The product x²·y stays constant, not x·y.
• y = k/x² is never negative for k > 0, so both branches sit above the x-axis.

Word problems are inverse variation when one quantity has to fall as the other rises with a fixed total: work, distance, gas volume or gear teeth.

• Workers and days: 6 workers finish a job in 10 days, so k = 60 worker-days. 4 workers need 60/4 = 15 days.
• Speed and time: a 240 km trip takes 4 h at 60 km/h and 3 h at 80 km/h.
• Boyle's law: at fixed temperature P·V is constant. 100 kPa at 4 L becomes 200 kPa at 2 L.
• Gears: a 40-tooth gear at 300 rpm drives a 20-tooth gear at 600 rpm.
• Pressure and area: the same force on half the area gives twice the pressure.

Inverse variation needs only multiplication and division: y₂ = x₁·y₁/x₂.

• Excel and Google Sheets: =A1*B1/C1 with x₁, y₁ and x₂ in A1, B1 and C1.
• Python: y2 = x1 * y1 / x2.
• Constant check in Excel: =A2*B2 filled down a column of pairs shows k for every row.

The table lists y = 60/x for whole-number x: every row multiplies back to k = 60. Click a row to load it into the calculator.

Read it as a work schedule: 60 worker-days of work takes 1 worker 60 days, 5 workers 12 days and 12 workers 5 days.