When one doubles, the other halves
Tonight's idea
A journey is 120 km. At 20 km/h it takes 6 hours. At 40 km/h it takes 3. At 60 km/h it takes 2. Double the speed and the time halves. Triple it and the time drops to a third.
What stays the same is the product: speed × time is 120 every time. Two quantities tied together like this are in inverse variation (also called inverse proportion).
\(xy = k\), or the same thing, \(y = \dfrac{k}{x}\). The number \(k\) is the constant product.
Compare direct proportion, \(y = kx\). There the ratio \(y/x\) is constant and the graph is a line through the origin. Here the product is constant and the graph is a curve.
Why does the curve never touch the axes? If it touched the \(x\)-axis, \(y\) would be 0 there, and \(xy\) would be 0, not \(k\). The same goes for \(x = 0\). As \(x\) grows, \(y\) shrinks towards zero without reaching it. No speed gets you there in no time.
Watch one: 6 workers, 10 days
Six workers paint a school in 10 days. How long would 4 workers take, each working at the same pace?
The job is a fixed amount of work: \(6 \times 10 = 60\) worker-days. That is \(k\). Fewer workers means more days, with the product still 60.
With 4 workers: \(d = \dfrac{60}{4} = 15\) days. Check another way: the workers were multiplied by \(\tfrac23\), so the days are multiplied by \(\tfrac32\), and \(10 \times \tfrac32 = 15\).
To finish in 5 days you need \(\dfrac{60}{5} = 12\) workers.
Your turn
- \(y\) varies inversely with \(x\), and \(y = 12\) when \(x = 5\). (a) Find \(k\). (b) Find \(y\) when \(x = 20\). (c) Find \(x\) when \(y = 4\).
- Which tables show inverse variation? Test the products. (a) \(x\): 2, 3, 6 with \(y\): 18, 12, 6. (b) \(x\): 1, 2, 4 with \(y\): 10, 8, 4. (c) \(x\): 4, 10, 25 with \(y\): 5, 2, 0.8.
- For a gas held at a steady temperature, pressure × volume is constant. A sealed syringe holds 60 mL of air at a pressure of 100 kPa. (a) You squeeze it to 24 mL. What is the pressure now? (b) What volume gives 150 kPa?
- You drive a fixed distance. (a) You raise your speed by 25%. By what percent does the journey time fall? It is not 25%. (b) By what percent must you raise your speed to cut the time by half?
Stretch: the best rectangle
Every rectangle with one corner at the origin and the opposite corner on \(xy = 12\) has area 12. Which has the smallest perimeter, and what is it? Use \((x + y)^2 = (x - y)^2 + 4xy\).
Answers
- (a) \(k = 5 \times 12 = 60\). (b) \(y = 60/20 = 3\). (c) \(x = 60/4 = 15\).
- (a) Products 36, 36, 36: yes, \(y = 36/x\). (b) Products 10, 16, 16: no. (It is linear: \(y\) falls by 2 for each 1 in \(x\).) (c) Products 20, 20, 20: yes, \(y = 20/x\).
- \(k = 100 \times 60 = 6000\). (a) \(6000/24 = 250\) kPa. (b) \(6000/150 = 40\) mL.
- (a) Speed is multiplied by 1.25, so time is multiplied by \(1/1.25 = 0.8\): a fall of 20%. (b) Time multiplied by \(\tfrac12\) means speed multiplied by 2: a rise of 100%. A percent change in one is never the same percent in the other.
Stretch. \(4xy = 48\) is fixed, so \((x + y)^2 = (x - y)^2 + 48\) is smallest when \(x - y = 0\). The best rectangle is the square with side \(\sqrt{12} = 2\sqrt3\). Then \(x + y = \sqrt{48} = 4\sqrt3\) and the perimeter is \(8\sqrt3 \approx 13.9\).