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Direct and inverse variation: find k first, then answer the question

"y varies directly with x" is an equation, y = kx, with one unknown constant. The points get lost when inverse variation is set up as a proportion, when "the square of" is dropped, and when "double the distance" is assumed to halve the answer.

Why it works

Direct variation is a line through the origin with slope k: double x and y doubles. Inverse variation is the opposite relationship: double x and y halves, because x · y has to stay equal to k. The constant k is what makes the problem specific. The sentence "varies directly" gives you the shape of the equation, and the one data pair you're given pins down k. After that, the equation answers any question you like.

Finding k is solving a formula for a letter: from y = kx, k = y/x; from y = k/x, k = xy. Different equations, different formulas for k, which is exactly why you can't skip writing the equation.

Example 1: direct variation

y varies directly with x, and y = 18 when x = 4. Find y when x = 10.

1. y = kxWrite the equation first.
2. 18 = k · 4, so k = 4.5Plug in the given pair and solve for k. The ratio y/x is 18/4 = 9/2.
3. y = 4.5xThe equation with k filled in. This is the actual relationship.
4. y = 4.5(10) = 45Check the ratio: 45/10 = 4.5, the same as 18/4.

Example 2: inverse variation

The time a trip takes varies inversely with speed. At 60 mph a trip takes 4 hours. How long does it take at 75 mph?

1. t = ksInverse: faster means less time. The k is on top.
2. 4 = k60, so k = 240Multiply both sides by 60. Here k is the distance: 240 miles, and speed × time always equals it.
3. t = 24075 = 3.2 hoursCheck the product: 75 × 3.2 = 240. Faster speed, less time, as expected.

Example 3: direct variation with a square

The distance a dropped object falls varies directly with the square of the time. It falls 64 feet in 2 seconds. How long does it take to fall 400 feet?

1. d = kt²"The square of the time" means t² in the equation.
2. 64 = k · 2² = 4k, so k = 16Square the 2 before dividing.
3. 400 = 16t², so t² = 25This time the unknown is t, so plug in d and solve.
4. t = 5 secondsOnly the positive root makes sense for a time. Check: 16 · 25 = 400.

Example 4: joint and combined variation

z varies jointly with x and y and inversely with w. When x = 3, y = 8, and w = 4, z = 12. Find z when x = 5, y = 2, and w = 10.

1. z = kxywJointly: the variables multiply, on top. Inversely: w goes on the bottom. One k for the whole thing.
2. 12 = k · 3 · 84 = 6k, so k = 224/4 = 6.
3. z = 2 · 5 · 210 = 220/10 = 2.

The most common combined case is the inverse square: light intensity varies inversely with the square of the distance, I = k/d². If I = 90 at d = 2, then k = 90 · 4 = 360, and at d = 6 the intensity is 360/36 = 10: three times the distance, one ninth the intensity.

Common mistakes, and the exact line they happen on

1. Setting up a proportion for inverse variation

y varies inversely with x, and y = 6 when x = 8. Find x when y = 16.

1. 68 = 16x, so x = 643This is the mistake. A proportion says the ratio y/x is constant, which is direct variation. In inverse variation the product is constant. This answer has y going up and x going up with it, the opposite of inverse.
2. k = xy = 8 · 6 = 48Inverse: find the constant product.
3. 16x = 48, so x = 3y went up from 6 to 16, so x went down from 8 to 3. Check: 3 · 16 = 48.

Proportions are fine for direct variation and only for direct variation. Writing the equation with k works for every kind, so it's the safer habit.

2. Adding an intercept

y varies directly with x, and y = 18 when x = 4 (Example 1).

1. y = x + 14This is the mistake. It fits the one pair, since 4 + 14 = 18, but it isn't direct variation. Direct variation goes through the origin: when x = 0, y = 0. This line gives y = 14 at x = 0, and it gives y = 24 at x = 10 instead of 45.

"Varies directly" means multiplied by a constant, nothing added. If a problem has a fixed fee plus a per-unit charge, that is a line with an intercept, not a variation problem.

3. Dropping the square

Find the time to fall 400 feet (Example 3).

1. 64 = 2k, so k = 32This is the mistake. The equation is d = kt², so the 2 has to be squared before dividing: 64 = 4k. With k = 32, the next line is 400 = 32t and t = 12.5 seconds, more than twice the right answer.
2. 64 = k(2)² = 4k, so k = 16Write the exponent in the equation before you plug anything in.

4. Halving the intensity when the distance doubles

I = 360/d². Find I at d = 4, given I = 90 at d = 2.

1. Distance doubled, so I = 45This is the mistake. That's the shortcut for plain inverse variation. With the square, doubling d multiplies d² by 4, so I is divided by 4.
2. I = 3604² = 36016 = 22.5One quarter of 90. Shortcuts about doubling and halving only apply to the exact form of the variation; plugging into the equation always works.

Practice

1. y varies directly with x, and y = 21 when x = 6. Find y when x = 14.

   Show answer

   k = 21/6 = 3.5, so y = 3.5(14) = 49.

2. y varies inversely with x, and y = 5 when x = 12. Find y when x = 20.

   Show answer

   k = 5 · 12 = 60, so y = 60/20 = 3.

3. The pressure of a gas varies inversely with its volume. At 12 liters the pressure is 40 psi. What is the pressure at 16 liters?

   Show answer

   k = 40 · 12 = 480, so P = 480/16 = 30 psi. Bigger volume, lower pressure.

4. y varies directly with the cube of x, and y = 54 when x = 3. Find y when x = 5.

   Show answer

   54 = k · 27, so k = 2 and y = 2 · 125 = 250.

5. z varies jointly with x and the square of y, and z = 72 when x = 2 and y = 3. Find z when x = 5 and y = 2.

   Show answer

   z = kxy²; 72 = k · 2 · 9 = 18k, so k = 4. Then z = 4 · 5 · 4 = 80.