Is it proportional?
Tonight's idea
Two quantities are proportional when one is always the same number times the other. Apples cost $2 a kilogram: 1 kg costs $2, 3 kg costs $6, 5 kg costs $10. Cost ÷ weight is always 2. That fixed number is the constant of proportionality, written \(k\). The rule is \(y = kx\), here \(y = 2x\).
A taxi charges $3 to get in, then $2 a kilometre: 1 km costs $5, 2 km costs $7. Double the distance and the cost does not double. Not proportional.
Three tests, and they always agree:
- Table: is \(y \div x\) the same in every row?
- Graph: is it a straight line through \((0, 0)\)?
- Story: does nothing cost nothing, and does double give double?
Straight is not enough. The taxi graph is straight, but it misses \((0, 0)\) because of the $3 you pay before moving.
Watch one: test a table
| \(x\) | 2 | 4 | 6 | 10 |
|---|---|---|---|---|
| \(y\) | 5 | 10 | 15 | 25 |
Divide each \(y\) by its \(x\): \(5 \div 2 = 2.5\), \(10 \div 4 = 2.5\), \(15 \div 6 = 2.5\), \(25 \div 10 = 2.5\). The same every time, so \(y\) is proportional to \(x\), with \(k = 2.5\) and \(y = 2.5x\).
The rule fills the gaps: when \(x = 7\), \(y = 2.5 \times 7 = 17.5\).
Now change the last \(y\) to 26. Then \(26 \div 10 = 2.6\). One bad row is enough: not proportional.
Type in your own table and watch the points land on the line, or miss it: mathdetours.com/demos/proportion.html.
Your turn
- Each list gives pairs \((x, y)\). Which list is proportional? Give its \(k\). List A: \((3, 12)\), \((5, 20)\), \((8, 32)\). List B: \((2, 7)\), \((4, 11)\), \((6, 15)\).
- The cost of notebooks is proportional to how many you buy. 4 notebooks cost $6. (a) Find \(k\) and say what it means. (b) What do 10 notebooks cost? (c) How many can you buy with $27?
- Proportional or not? Give a reason each time. (a) The side of a square and its perimeter. (b) A child's age and their height. (c) A film service that costs $5 a month plus $2 for each film. (d) Time spent cycling at a steady speed and distance covered.
- A straight-line graph passes through \((2, 3)\) and \((6, 7)\). (a) Is \(y\) proportional to \(x\)? (b) Keep \((2, 3)\). Where would the line have to be when \(x = 6\) for the answer to be yes?
Stretch: nothing gives nothing, but…
A square of side 0 has area 0. Is the area of a square proportional to its side? Test sides 1, 2 and 3. Then say what is wrong with "it starts at zero, so it is proportional".
Answers
- List A: \(12 \div 3 = 20 \div 5 = 32 \div 8 = 4\), so it is proportional with \(k = 4\). List B: \(7 \div 2 = 3.5\) but \(11 \div 4 = 2.75\), so it is not.
- (a) \(k = 6 \div 4 = 1.5\): each notebook costs $1.50. (b) \(1.5 \times 10 = \$15\). (c) \(27 \div 1.5 = 18\) notebooks.
- (a) Yes: perimeter is always 4 times the side. (b) No: a baby of age 0 is not 0 cm tall, and a 12-year-old is not twice as tall as a 6-year-old. (c) No: zero films still cost $5. (d) Yes: no time, no distance, and twice the time gives twice the distance.
- (a) No: \(3 \div 2 = 1.5\) but \(7 \div 6\) is about 1.17. (b) \(k\) would be 1.5, so \(y = 1.5 \times 6 = 9\). The line would pass through \((6, 9)\), and through \((0, 0)\).
Stretch. Areas are 1, 4 and 9, so area ÷ side is 1, 2, 3. Not the same, so not proportional. Doubling the side from 1 to 2 makes the area four times as big. The graph does start at \((0, 0)\) but it curves upward. Starting at zero is only half the test: the graph must also be straight.