The constant of proportionality and proportional tables
Cheese costs the same for every kilogram, so one number turns any weight into its price. What is that number, and how can a table of prices show whether such a number exists at all?
Where you are 4 in Proportional relationships
Why this exists
At a market, a cheese stall sells one cheese by weight at 18 dollars per kilogram. Customers ask for half a kilogram, 1.5 kilograms, 0.3 kilograms. The stallholder does not keep a separate price for every weight: one multiplication does them all, the weight times 18. A garden center pays its staff the same way, the hours worked times the hourly wage.
In each case one number turns every amount of the first quantity into the matching amount of the second, and once you have it every price and every pay packet is one step away.
Not every price list works like this. A farm that charges an entry fee, or a shop that cuts the price for bulk buyers, has no single number that does the job. So there are two things to learn: what the number is and what it measures, and how to tell from a table or a description whether there is such a number at all.
Definition
Two quantities x and y are in a proportional relationship when there is one number k such that every value of y is k times the matching value of x:
We say that y is proportional to x, and k is the constant of proportionality.
Dividing both sides by x, when x \ne 0, gives
so the constant is the quotient of y by x, and it is the same quotient for every pair. It is the unit rate of y with respect to x: the amount of y that goes with exactly one unit of x. Because of that it carries a unit, the unit of y per unit of x. For the cheese, x is the weight in kilograms, y is the price in dollars, and k = 18 dollars per kilogram.
Which quantity is called x matters. With the roles swapped, \displaystyle x = \frac{1}{k}\,y, so x is proportional to y with constant \displaystyle \frac{1}{k}, measured in the unit of x per unit of y. For the cheese that is \displaystyle \frac{1}{18} kilogram per dollar.
A table of values is proportional when it could come from a proportional relationship: every pair in it with x \ne 0 gives the same quotient \displaystyle \frac{y}{x}, and a pair with x = 0, if there is one, has y = 0.
Intuition
Part 1: one number, used on every pair
Pears cost 4 dollars per kilogram. Draw a line of weights and, under it, a line of prices, with each weight directly above what it costs. A double number line like this shows the whole relationship at once.
Each kilogram adds the same 4 dollars, so 2 kg sits above 8 dollars and 3 kg above 12. Halfway between 2 and 3 kilograms, 2.5 kg sits above 10 dollars, halfway between 8 and 12. Both lines start together at 0: no pears, no cost. And the price under 1 kilogram is the constant itself, 4.
A table of the same prices holds two different kinds of multiplier, and it is worth keeping them apart.
Going across, from 1 kg to 2 kg is a step of \times 2, from 2 kg to 3 kg a step of \times 1.5, and from 3 kg to 6 kg a step of \times 2. These are the scale factors of Equivalent ratios and ratio tables and What a proportion says. They change from step to step, and each one acts on both rows alike. Going down, from a weight to its price, the multiplier is 4 in every column. That is the constant of proportionality: the one multiplier that never changes.
Since every pair obeys y = kx, any single pair is enough to find k, and once k is known every missing entry follows. Multiply an x by k to get its y; divide a y by k to get its x.
Part 2: tables that are not proportional
A pick-your-own strawberry farm charges 5 dollars to enter and then 6 dollars for each kilogram picked. Its prices are 11 dollars for 1 kg, 17 dollars for 2 kg and 23 dollars for 3 kg. The price rises by the same 6 dollars for every extra kilogram, which looks steady, but the quotients are
and they keep falling, because the 5 dollar entry is shared out over more and more kilograms. No single number multiplies every weight to its price. Run the pattern back to 0 kg and the table says 5 dollars for no fruit, while a proportional table must pair 0 with 0.
A bulk price breaks it another way: the quotients agree for small orders, then drop.
So the test of a table is the definition itself. Find \displaystyle \frac{y}{x} for every pair. If all the quotients agree, and any pair with x = 0 has y = 0, the table is proportional and the shared quotient is k. One quotient out of line is enough to say no. For a description in words, the same test asks: does each unit of x carry the same amount of y, every time, with nothing added on once?
Results
- The constant from any pair. In a proportional relationship, \displaystyle k = \frac{y}{x} for every pair with x \ne 0.
- Two pairs that always fit. The pair (1, k) fits: k is the value of y when x = 1. So does the pair (0, 0).
- Units. k is measured in the unit of y per unit of x. Changing a unit changes the number but not the price: 0.02 dollars per gram is 20 dollars per kilogram.
- Filling a table. y = kx, and when k \ne 0, \displaystyle x = \frac{y}{k}.
- Swapping the roles. If y is proportional to x with constant k \ne 0, then x is proportional to y with constant \displaystyle \frac{1}{k}.
- New pairs from old. If (x_1, y_1) and (x_2, y_2) fit a proportional relationship, so do (x_1 + x_2,\ y_1 + y_2) and (x_1 - x_2,\ y_1 - y_2), and so does (c x_1,\ c y_1) for any number c.
- The table test. A table is proportional exactly when every pair with x \ne 0 has the same quotient \displaystyle \frac{y}{x} and every pair with x = 0 has y = 0. Equal steps in y for equal steps in x are not enough on their own.
Why it is true
The constant from any pair. Every pair obeys y = kx. When x \ne 0, divide both sides by x to get \displaystyle \frac{y}{x} = k. Putting x = 1 into y = kx gives y = k, and putting x = 0 gives y = 0. The unit follows from the division: 54 dollars shared over 3 kilograms is 18 dollars per kilogram.
Swapping the roles. If k \ne 0, divide both sides of y = kx by k: \displaystyle x = \frac{1}{k}\,y. One number, \displaystyle \frac{1}{k}, now multiplies every y to give its x.
New pairs from old. By the distributive property,
so the sums form a pair that fits, and the same steps with minus signs handle the differences. For a multiple, c y_1 = c(k x_1) = k(c x_1), regrouping the product by the associative and commutative properties.
The table test. If every pair with x \ne 0 has \displaystyle \frac{y}{x} = k, multiplying by x gives y = kx for those pairs, and a pair (0, 0) obeys y = kx too, since k \cdot 0 = 0. So one number k fits every pair. The other direction is the first result above.
Equal steps are not enough. Suppose y goes up by the same amount m for each unit of x, and y = b when x = 0. Then y = mx + b, and for x \ne 0,
If two different values x_1 and x_2 gave the same quotient, then \displaystyle \frac{b}{x_1} = \frac{b}{x_2}, which forces b = 0. So the quotient stays the same only when b = 0, that is, only when x = 0 goes with y = 0. With b = 0 the relationship is y = mx, proportional with k = m.
Worked examples
Examples 1 to 3 belong to Part 1, the constant and its units. Examples 4 to 6 belong to Part 2: telling proportional tables and descriptions apart, and solving problems with the constant.
Reading the constant from a table of prices
A cheese stall prices one cheese by weight.
| Weight (kg) | 0.5 | 1.5 | 2 | 3 |
|---|---|---|---|---|
| Price (dollars) | 9 | 27 | 36 | 54 |
(a) Show that price is proportional to weight, and find the constant of proportionality with its unit.
(b) What does the constant say about the cheese?
Solution(a) Divide each price by its weight:
Every quotient is 18, so every price is 18 times its weight. Price is proportional to weight, with k = 18 dollars per kilogram. The unit comes from the division: dollars divided by kilograms.
(b) 18 is the price of exactly one kilogram, the unit rate. It is the entry the table would show under 1 kg, and every other entry is that amount taken x times: 1.5 kg costs 1.5 \times 18 = 27 dollars.
Finding the constant from one week’s pay
Amara is paid the same amount for every hour she works at a garden center. One week she worked 8 hours and was paid 132 dollars. Complete the table.
| Hours worked | 3 | 7.5 | ? | ? |
|---|---|---|---|---|
| Pay (dollars) | ? | ? | 198 | 74.25 |
Every hour carries the same pay, so pay is proportional to hours worked. The one full pair we have gives the constant:
That is 16.50 dollars per hour, her hourly wage. Each missing pay is 16.5 times its hours: 3 \times 16.5 = 49.5, so 49.50 dollars, and 7.5 \times 16.5 = 123.75 dollars. Each missing number of hours is the pay divided by 16.5: 198 \div 16.5 = 12 hours and 74.25 \div 16.5 = 4.5 hours. Check one: 4.5 \times 16.5 = 74.25.
| Hours worked | 3 | 7.5 | 12 | 4.5 |
|---|---|---|---|---|
| Pay (dollars) | 49.50 | 123.75 | 198 | 74.25 |
The columns need not be in order. Each one is a pair, and every pair has the same quotient.
One price, three constants
A stall sells coffee beans loose, at the same price for every gram. A 250 g scoop costs 5 dollars.
(a) Find the constant of price with respect to weight, in dollars per gram and in dollars per kilogram.
(b) Find the constant of weight with respect to price, in grams per dollar.
(c) What do 1.2 kg of beans cost, and how many grams does 12 dollars buy?
Solution(a) Price divided by weight is \displaystyle \frac{5}{250} = 0.02, so the constant is 0.02 dollars per gram. A kilogram is 1000 grams, so a kilogram costs 1000 \times 0.02 = 20 dollars, and the constant is 20 dollars per kilogram. The price has not changed; only the unit for x has, and the number changed with it. Converting a rate to other units does this in general.
(b) Weight divided by price is \displaystyle \frac{250}{5} = 50 grams per dollar, the reciprocal of the first constant: \displaystyle \frac{1}{0.02} = 50.
(c) The known amount is a weight in kilograms, so use the constant per kilogram: 1.2 \times 20 = 24 dollars. For the weight, the known amount is a price, so use grams per dollar: 12 \times 50 = 600 grams. Pick the constant whose x is the quantity you already know, and each question is one multiplication.
Which tables are proportional?
Two stalls post price tables. Decide which are proportional, and give the constant of any that is.
(a) Potatoes:
| Weight (kg) | 2 | 5 | 8 | 10 |
|---|---|---|---|---|
| Price (dollars) | 3 | 7.50 | 12 | 15 |
(b) Olives, sold in tubs:
| Tubs | 1 | 2 | 3 | 6 |
|---|---|---|---|---|
| Price (dollars) | 4 | 8 | 11 | 22 |
(a) The quotients are \displaystyle \frac{3}{2} = 1.5, \displaystyle \frac{7.5}{5} = 1.5, \displaystyle \frac{12}{8} = 1.5 and \displaystyle \frac{15}{10} = 1.5. All agree, so the table is proportional with k = 1.50 dollars per kilogram.
(b) The quotients are \displaystyle \frac{4}{1} = 4, \displaystyle \frac{8}{2} = 4, \displaystyle \frac{11}{3} \approx 3.67 and \displaystyle \frac{22}{6} \approx 3.67. Not proportional: from 3 tubs on, the price per tub drops, a bulk offer. The first two pairs agree with each other and so do the last two, but no single number fits all four. At 4 dollars a tub, 3 tubs would cost 12 dollars.
Proportional or not, from a description
Decide whether pay is proportional to hours worked. If it is, give the constant.
(a) A cleaner is paid 18 dollars for every hour worked.
(b) An electrician charges a 45 dollar call-out fee plus 60 dollars per hour.
(c) A warehouse worker is paid 15 dollars an hour for the first 40 hours in a week, and time and a half, 1.5 times the usual rate, for every hour after that.
Solution(a) Yes. Every hour carries the same 18 dollars and no hours earn nothing, so pay is 18 times the hours. The constant is 18 dollars per hour.
(b) No. One hour costs 45 + 60 = 105 dollars and two hours cost 45 + 120 = 165 dollars. The quotients are \displaystyle \frac{105}{1} = 105 and \displaystyle \frac{165}{2} = 82.5, which differ. Doubling the hours does not double the charge, because the fee is paid once.
(c) Not over all weeks. A 20 hour week pays 20 \times 15 = 300 dollars and a 40 hour week 600 dollars, both with quotient 15. A 50 hour week has 10 hours at 1.5 \times 15 = 22.50 dollars, so it pays
and \displaystyle \frac{825}{50} = 16.5. Weeks of at most 40 hours are proportional with k = 15 dollars per hour, but beyond 40 hours the quotient climbs. A table built only from short weeks would look proportional, which is why the rule in words, not a handful of pairs, settles the question.
From hours of work to kilograms of apples
Jonas earns 21 dollars an hour, and apples at his market cost 3.50 dollars per kilogram.
(a) How many kilograms of apples does one hour’s pay buy?
(b) Show that the weight of apples his pay buys is proportional to the hours he works, and give the constant with its unit.
(c) How long must he work to pay for 15 kg, and how many minutes of work does each kilogram take?
Solution(a) One hour pays 21 dollars, and each kilogram costs 3.50 dollars, so one hour buys 21 \div 3.5 = 6 kg.
(b) For h hours his pay is 21h dollars, and that buys 21h \div 3.5 kilograms. Since \displaystyle \frac{21h}{3.5} = 6h, one number, 6, multiplies every number of hours to give the kilograms. So the weight of apples is proportional to hours worked, with k = 6 kilograms per hour. The unit works out the same way: dollars per hour divided by dollars per kilogram leaves kilograms per hour.
(c) Hours are kilograms divided by the constant: 15 \div 6 = 2.5 hours. Check: 2.5 hours pay 2.5 \times 21 = 52.50 dollars, and 15 kg cost 15 \times 3.50 = 52.50 dollars. The constant the other way round is \displaystyle \frac{1}{6} hour per kilogram, which is 60 \div 6 = 10 minutes of work per kilogram.
Boundaries
- The quotient needs x \ne 0. \displaystyle k = \frac{y}{x} cannot be found from a pair with x = 0. Such a pair is tested by asking whether y = 0 instead.
- A table shows only some pairs. A table can prove a relationship is not proportional (one bad quotient is enough), but a table that passes only shows that the listed pairs fit. Whether the whole situation is proportional depends on its rule: the overtime week in Example 5 passes for every week of at most 40 hours and fails after that. A table with only one pair, and that pair’s x not 0, always fits the constant \displaystyle k = \frac{y}{x}, so on its own it says nothing.
- The number depends on the units and the order. 0.02 dollars per gram, 20 dollars per kilogram and 50 grams per dollar all describe the same beans. A constant quoted without its unit, or without saying which quantity is x, is incomplete.
- Rounding. Prices are rounded to the cent: 0.333 kg at 18 dollars per kilogram is 5.994 dollars, charged as 5.99. So printed quotients can differ slightly even when the relationship is proportional.
- The sign of k. Every constant in this lesson is positive, because prices, weights, hours and pay are. The definition y = kx makes sense for a negative k too, and Direct variation uses it that way. With k = 0 every y is 0 and the constant cannot be reversed, so that case is left out: the constant of a proportional relationship is never 0 (Recognizing a proportional relationship in an equation).
Where this goes next
Plot the pairs of a proportional relationship and they land on one straight line through the point (0, 0), passing through (1, k): recognizing a proportional relationship in a graph starts there, and the constant becomes how steep the line is, the idea behind slope. Written as an equation, the relationship is y = kx with nothing added on, which recognizing a proportional relationship in an equation takes up. In algebra the same idea, with k allowed to be negative, is direct variation.
A fixed fee, like the call-out fee in Example 5, gives a straight line y = mx + b that is not proportional; modeling with linear relationships handles those. When one missing value is wanted and k is not, a proportion can be solved directly. Comparing two stalls by their constants is comparing two ratios. When two quantities multiply to the same number every time, the relationship is inversely proportional. And the constant between a drawing and the real object is a scale factor.
Exercises
Part 1: the constant and its units
Each table in Exercises 1 to 3 is proportional. Find the constant of proportionality with its unit, and say what it means.
- Tomatoes:
| Weight (kg) | 2 | 3 | 5 | 7 |
|---|---|---|---|---|
| Price (dollars) | 6.80 | 10.20 | 17 | 23.80 |
- A shop assistant’s pay:
| Hours worked | 5 | 8 | 10 |
|---|---|---|---|
| Pay (dollars) | 92.50 | 148 | 185 |
- Eggs in cartons. Give also the constant of cartons with respect to eggs.
| Cartons | 2 | 5 | 9 |
|---|---|---|---|
| Eggs | 24 | 60 | 108 |
- Grapes cost 4.80 dollars per kilogram. Complete the table.
| Weight (kg) | 0.5 | 2 | 3.5 | ? | ? |
|---|---|---|---|---|---|
| Price (dollars) | ? | ? | ? | 12 | 21.60 |
- A lifeguard is paid the same for every hour and earned 86.40 dollars for a 6 hour shift. Find the hourly wage, and complete the table.
| Hours | 2.5 | 9 | ? |
|---|---|---|---|
| Pay (dollars) | ? | ? | 115.20 |
- Ground pepper is sold loose at the same price for every gram, and 20 g costs 3 dollars. Find the constant of price with respect to weight (a) in dollars per gram and (b) in dollars per kilogram, and (c) the constant of weight with respect to price, in grams per dollar.
- Length in inches is proportional to length in feet. Find the constant, the two missing entries, and the constant of feet with respect to inches.
| Feet | 2.5 | 4 | ? | 0.75 |
|---|---|---|---|---|
| Inches | 30 | 48 | 78 | ? |
- At a market stall, 2.5 kg of cherries cost 27 dollars, and price is proportional to weight. (a) Find the constant in dollars per kilogram. (b) Find the constant of weight with respect to price, in kilograms per dollar as a fraction, and in grams per dollar to the nearest gram. (c) What do 4 kg cost? (d) How many kilograms does 64.80 dollars buy?
- A shop sells ribbon by the meter: 3 m cost 4.20 dollars and 5 m cost 7 dollars. Without finding the constant first, use those two pairs to find the price of (a) 8 m, (b) 2 m and (c) 1 m. Then check your answers against the constant.
Part 2: is the table proportional?
Decide whether each table is proportional. If it is, give the constant; if it is not, say what breaks it.
- Mushrooms:
| Weight (kg) | 1 | 2 | 4 | 5 |
|---|---|---|---|---|
| Price (dollars) | 3.50 | 7 | 13 | 17.50 |
- A gardener’s pay:
| Hours | 0 | 3 | 5 | 8 |
|---|---|---|---|---|
| Pay (dollars) | 0 | 51 | 85 | 136 |
- A babysitter’s charges:
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Charge (dollars) | 25 | 40 | 55 | 70 |
Part 2: proportional or not, from a description
Decide whether the second quantity is proportional to the first. If it is, give the constant with its unit.
- A sum of money, and the weight of almonds it buys at 16 dollars per kilogram.
- A number of weeks worked, and the pay for them, for someone paid 16.40 dollars an hour for 35 hours every week.
- Apples cost 3 dollars per kilogram. Bag A holds 6 apples weighing 0.9 kg, and bag B holds 6 apples weighing 1.5 kg. Find the price of each bag. Is price proportional to the weight of the apples? Is it proportional to the number of apples?
Part 2: problems
- Stall P sells 2.5 kg of carrots for 4 dollars and stall Q sells 4 kg for 6 dollars, each charging the same for every kilogram. Which stall is cheaper, and how much is saved on 10 kg by buying there?
- Bilal earns 15.60 dollars an hour. (a) What does he earn for 7.5 hours? (b) How many hours must he work to earn 507 dollars?
- Kiri is paid 19.20 dollars an hour, and olive oil costs 12.80 dollars per liter. (a) How much oil does one hour’s pay buy? (b) How long must she work to pay for 6 liters? (c) Is the amount of oil her pay buys proportional to the hours she works? If it is, give the constant.
Reasoning
- Explain why every proportional relationship includes the pair (0, 0) and the pair (1, k), and why a table containing the pair (0, 5) cannot be proportional.
- Show that if (x_1, y_1) and (x_2, y_2) both fit a proportional relationship, then so does (x_1 + x_2,\ y_1 + y_2). A table contains the pairs (2, 9), (3, 13.5) and (5, 22). Use this fact to show it is not proportional.
- Lior says: “In my table the price goes up by 6 dollars for every extra kilogram, so price is proportional to weight with constant 6.” Give a table for which this reasoning is wrong, and name one extra fact that would make it right.
Answers
- k = 3.40 dollars per kilogram, the price of one kilogram of tomatoes: \displaystyle \frac{6.80}{2} = \frac{10.20}{3} = \frac{17}{5} = \frac{23.80}{7} = 3.40.
- k = 18.50 dollars per hour, the pay for one hour, which is the hourly wage: \displaystyle \frac{92.50}{5} = \frac{148}{8} = \frac{185}{10} = 18.50.
- k = 12 eggs per carton: \displaystyle \frac{24}{2} = \frac{60}{5} = \frac{108}{9} = 12. The other way round the constant is \displaystyle \frac{1}{12} carton per egg: each egg fills a twelfth of a carton.
- The prices are 0.5 \times 4.80 = 2.40, 2 \times 4.80 = 9.60 and 3.5 \times 4.80 = 16.80 dollars. The weights are 12 \div 4.80 = 2.5 kg and 21.60 \div 4.80 = 4.5 kg.
- The wage is \displaystyle \frac{86.40}{6} = 14.40 dollars per hour. Then 2.5 \times 14.40 = 36 dollars, 9 \times 14.40 = 129.60 dollars, and 115.20 \div 14.40 = 8 hours.
- (a) \displaystyle \frac{3}{20} = 0.15 dollars per gram. (b) 1000 \times 0.15 = 150 dollars per kilogram. (c) \displaystyle \frac{20}{3} = 6\tfrac{2}{3} grams per dollar.
- k = 12 inches per foot, since \displaystyle \frac{30}{2.5} = \frac{48}{4} = 12. The missing entries are 78 \div 12 = 6.5 feet and 0.75 \times 12 = 9 inches. Feet with respect to inches has constant \displaystyle \frac{1}{12} foot per inch.
- (a) \displaystyle \frac{27}{2.5} = 10.80 dollars per kilogram. (b) \displaystyle \frac{2.5}{27} = \frac{5}{54} kilogram per dollar, which is \displaystyle \frac{2500}{27} \approx 93 grams per dollar. (c) 4 \times 10.80 = 43.20 dollars. (d) 64.80 \div 10.80 = 6 kg, or equally \displaystyle 64.80 \times \frac{5}{54} = 6.
- (a) 8 = 3 + 5, so 8 m cost 4.20 + 7 = 11.20 dollars. (b) 2 = 5 - 3, so 2 m cost 7 - 4.20 = 2.80 dollars. (c) 1 m is half of 2 m, so it costs 2.80 \div 2 = 1.40 dollars. The constant is \displaystyle \frac{4.20}{3} = 1.40 dollars per meter, and indeed \displaystyle \frac{7}{5} = 1.40 and 8 \times 1.40 = 11.20.
- Not proportional. Three quotients are 3.50, but \displaystyle \frac{13}{4} = 3.25. The 4 kg pair breaks the pattern; at 3.50 dollars per kilogram it would cost 14 dollars.
- Proportional, with k = 17 dollars per hour: \displaystyle \frac{51}{3} = \frac{85}{5} = \frac{136}{8} = 17, and 0 hours goes with 0 dollars.
- Not proportional. The quotients are 25, 20, \displaystyle \frac{55}{3} \approx 18.33 and 17.5. Each extra hour adds 15 dollars, but the first hour costs 25: a fixed 10 dollar charge plus 15 dollars per hour.
- Proportional. Each dollar buys the same \displaystyle \frac{1}{16} kg, and no money buys no almonds, so \displaystyle k = \frac{1}{16} kilogram per dollar, which is \displaystyle \frac{1000}{16} = 62.5 grams per dollar. For example 40 dollars buy 40 \div 16 = 2.5 kg. Money is the first quantity here, so the constant is the reciprocal of the price per kilogram.
- Proportional. Every week carries the same 35 hours, so the same pay, and no weeks earn nothing: k = 35 \times 16.40 = 574 dollars per week. Three weeks pay 3 \times 574 = 1722 dollars.
- Bag A costs 0.9 \times 3 = 2.70 dollars and bag B costs 1.5 \times 3 = 4.50 dollars. Price is proportional to weight, with k = 3 dollars per kilogram. It is not proportional to the number of apples: both bags hold 6 apples yet cost different amounts, so no single price per apple fits.
- Stall Q is cheaper: \displaystyle \frac{4}{2.5} = 1.60 dollars per kilogram at P and \displaystyle \frac{6}{4} = 1.50 at Q. For 10 kg that is 16 dollars against 15, a saving of 1 dollar.
- (a) 7.5 \times 15.60 = 117 dollars. (b) 507 \div 15.60 = 32.5 hours.
- (a) 19.20 \div 12.80 = 1.5 liters. (b) 6 \div 1.5 = 4 hours; check: 4 \times 19.20 = 76.80 dollars, the price of 6 \times 12.80 = 76.80 dollars of oil. (c) Yes. h hours pay 19.20h dollars, which buy 19.20h \div 12.80 = 1.5h liters, so k = 1.5 liters per hour.
- Every pair obeys y = kx. At x = 0 this gives y = k \cdot 0 = 0, and at x = 1 it gives y = k. A pair (0, 5) would need 5 = k \cdot 0 = 0, which is false for every k, so no constant can produce it.
- By the distributive property y_1 + y_2 = kx_1 + kx_2 = k(x_1 + x_2), so the summed pair obeys the same rule. If the table were proportional, (2, 9) and (3, 13.5) would give the pair (5,\ 9 + 13.5) = (5,\ 22.5), but the table pairs 5 with 22. The quotients show the same thing: \displaystyle \frac{9}{2} = \frac{13.5}{3} = 4.5 but \displaystyle \frac{22}{5} = 4.4.
- One such table: 1 kg costs 10 dollars, 2 kg cost 16 and 3 kg cost 22, a 4 dollar charge plus 6 dollars per kilogram. The price rises by 6 for every kilogram, yet \displaystyle \frac{10}{1} \ne \frac{16}{2}. The missing fact is that 0 kg costs 0 dollars (or, equivalently, that 1 kg costs 6 dollars). With a rise of 6 per kilogram starting from 0, x kilograms cost 6x dollars, which is proportional with k = 6.