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Recognize proportional relationships in tables, graphs, equations and percent stories.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn to decide whether two quantities are proportional. You will test a table by dividing in every row, test a graph by looking for a straight line through the origin, and test an equation by writing it as $y = kx$. You will see that a percent is a constant of proportionality, and that a fixed fee or a starting amount stops a relationship from being proportional, even when its graph is straight.
You can write a ratio, find equivalent ratios and read a table of values. You can plot points on a coordinate plane. You know that the constant of proportionality $k$ is the number that $y$ is always multiplied by in $y = kx$, and you can turn a percent into a decimal and use it as a multiplier. This lesson asks one question of all those tools: is this relationship proportional at all? You will learn to decide from a table, from a graph, from an equation and from a story, including stories about percents.
| Term | What it means |
|---|---|
| Proportional relationship | Two quantities where one is always the same multiple of the other: $y = kx$. |
| Constant of proportionality | The multiple $k$ in $y = kx$; it equals $y \div x$ for every pair. |
| Origin | The point $(0, 0)$ on a graph. Every proportional graph passes through it. |
| Linear | Having a straight-line graph. Every proportional relationship is linear, but not every linear one is proportional. |
| Fixed fee | An amount charged once, whatever the quantity, such as a delivery charge. It stops a relationship from being proportional. |
| Percent | A rate per hundred: 8% means 8 for every 100, or 0.08 as a multiplier. |
| Commission | Pay that is a percent of the value of what a person sells. |
Two quantities are proportional when one is always the same multiple of the other. That single idea shows up in three ways, and each gives you a test.
| Where you see it | The test |
|---|---|
| a table | $y \div x$ gives the same number in every row |
| a graph | the points lie on a straight line through the origin |
| an equation | it can be written $y = kx$, with nothing added |
The tests always agree, because they are the same fact seen three ways. If every row has $y \div x = 3$, then $y = 3x$, and the graph of $y = 3x$ is a straight line through $(0, 0)$.
Percents fit the same pattern. Taking 8% of a price means multiplying by 0.08, so the tax on a price is proportional to the price, with $k = 0.08$. But as soon as a fixed amount is added, such as a shipping fee, the relationship stops being proportional, even though its graph is still a straight line.
Another way: picture
Picture two straight lines on the same grid. One starts at the corner, the origin, and rises steadily: proportional. The other is exactly as straight and just as steep, but it starts 5 units up the $y$-axis: not proportional. The only difference you can see is where each line crosses the $y$-axis.
Another way: story
At a farm stand, apples cost 50 cents each. No apples cost nothing, two apples cost a dollar, and every apple adds the same 50 cents: that is proportional. If the stand also charged 2 dollars to park, then buying no apples would still cost 2 dollars, and the cost would no longer be proportional to the number of apples.
To test a table, divide each $y$ value by the $x$ value in the same row. If every quotient is the same, the table is proportional and that quotient is $k$. If even one quotient is different, it is not.
Check every row, not just the first two. A table with $x$ of 2, 4, 6 and $y$ of 5, 10, 16 passes for two rows, $5 \div 2 = 2.5$ and $10 \div 4 = 2.5$, and fails on the third, $16 \div 6 \approx 2.67$.
You can also look across rows. In a proportional table, multiplying $x$ by some number multiplies $y$ by the same number: when $x$ doubles, $y$ doubles; when $x$ is tripled, so is $y$. And if the table contains $x = 0$, then $y$ must be 0 too.
A proportional relationship always graphs as a straight line, because each step of 1 in $x$ adds the same amount $k$ to $y$. It also always passes through the origin, because $y = k \times 0 = 0$ when $x = 0$.
So check two things. Is the graph a straight line? If it curves, it is not proportional. Does it pass through $(0, 0)$? If it crosses the $y$-axis anywhere else, it is not proportional. Both answers must be yes.
Two points on a proportional graph are especially useful. The origin $(0, 0)$ says that none of one quantity goes with none of the other. The point $(1, k)$ shows the constant itself: the $y$ value when $x$ is 1, the unit rate. Any other point $(x, y)$ on the line means that $x$ units go with $y$ units, and $y \div x = k$.
An equation is proportional when it can be written as $y = kx$, a number times $x$ with nothing added or subtracted. So $y = 4.5x$, $d = 60t$ and $c = \tfrac{3}{4}n$ are proportional. So is $y = \tfrac{x}{5}$, because dividing by 5 is multiplying by $\tfrac{1}{5}$.
But $y = 4.5x + 2$ is not, because of the $+2$, and $y = x^2$ is not, because $y \div x$ changes as $x$ changes. An equation like $y = 3(x + 2)$ is not proportional either: multiplying out gives $3x + 6$. Look at the equation after it is simplified, not before.
Many everyday percents are proportional relationships in disguise. A tip of 20% is $t = 0.2b$, where $b$ is the bill. A commission of 5% is $c = 0.05s$. A discount of 25% leaves 75% to pay, so the sale price is $s = 0.75p$, with $k = 0.75$. In each case the percent, written as a decimal, is the constant of proportionality, and each graph is a line through the origin.
Two percent situations are not proportional. The first adds a fixed amount: a price plus 6% tax plus a 4-dollar delivery fee. The second takes percents of different things: the interest on a savings account that is compounded year after year grows faster than any single $k$ can describe. When you meet a percent, ask whether it is simply a multiplier. If it is, the relationship is proportional.
Practice the question on situations you know.
When you are not sure, make a small table of two or three pairs and divide. The numbers settle the question.
Many relationships go up by the same amount each step and are still not proportional. A taxi that charges 3 dollars to start plus 2 dollars a mile gives 5, 7, 9, 11 dollars for 1, 2, 3, 4 miles. The differences are all 2, so the graph is a straight line, but the quotients $5 \div 1 = 5$ and $7 \div 2 = 3.5$ disagree, and 0 miles would cost 3 dollars, not 0.
This is why the test is always a quotient, never a difference. A constant difference only tells you the graph is straight. A constant quotient tells you it is straight and passes through the origin.
Use these steps to decide whether a relationship is proportional.
How to check. Use two tests, not one. If the quotients agree, the graph should pass through the origin and the equation should have nothing added. Test your equation on a pair you did not use to find $k$. And read $k$ back into the story: "8.50 dollars per hour" or "0.38 pounds on Mars for every pound on Earth" should make sense.
The Moon's gravity is weaker than Earth's, and an object's weight there is about 16.5% of its weight on Earth. Weight on the Moon is proportional to weight on Earth, with $k = 0.165$: a 120-pound student would weigh $0.165 \times 120 = 19.8$ pounds, and a 400-pound piece of equipment would weigh 66 pounds. The graph of Moon weight against Earth weight is a straight line through the origin, because an object with no weight on Earth has none on the Moon either. This is why the Apollo astronauts, with their heavy space suits and backpacks, could bound across the surface in long hops.
Real estate agents are often paid only a commission, a percent of the price of each home they help sell. At 3%, selling a 300,000-dollar house earns $0.03 \times 300{,}000 = 9{,}000$ dollars, and selling a house for twice the price earns twice as much: proportional, with $k = 0.03$. Many car salespeople are paid differently: a base salary, say 2,000 dollars a month, plus 2% of their sales. Then pay is $P = 0.02s + 2{,}000$, which is not proportional. A month with no sales still pays 2,000 dollars, and doubling sales does not double the pay: 100,000 dollars of sales pays 4,000 dollars, while 200,000 pays 6,000.
At a gas pump, the price you pay is proportional to the number of gallons, because each gallon costs the same. Suppose gas costs 3.45 dollars a gallon. Then 10 gallons cost 34.50 dollars and 12.4 gallons cost $3.45 \times 12.4 = 42.78$ dollars, and the pump's display is really just showing $C = 3.45g$ as the gallons flow. A car wash that charges 8 dollars plus the gas is a different story: now 10 gallons cost 42.50 dollars, and $42.50 \div 10 = 4.25$ is not the price of a gallon.
The most common mistake is thinking every straight line is proportional. A straight line that does not pass through the origin, such as the graph of a fare with a starting charge, is linear but not proportional.
The second is testing differences instead of quotients. The $y$ values 4, 7, 10 go up by 3 each time, but $4 \div 1$ and $7 \div 2$ differ, so the table $x = 1, 2, 3$ is not proportional.
The third is checking only one or two rows. One disagreeing row is enough to fail the test, so every row must be checked.
The last is treating every percent story as proportional. A percent is a multiplier, but a fixed fee added to it, or a percent of a changing amount, breaks the pattern.
Divide $y$ by $x$ in the first row.
$14 \div 4 = 3.5$
The quotient in one row is only a candidate for $k$.
Divide in the second row.
$21 \div 6 = 3.5$
The second row agrees with the first.
Divide in the third row.
$35 \div 10 = 3.5$
Every row has to agree, not just two of them.
Decide whether the table is proportional.
$3.5 = 3.5 = 3.5 \Rightarrow \text{proportional}$
All three quotients are the same, so $y$ is always 3.5 times $x$.
Write the equation and check it on a row.
$y = 3.5x, \quad 3.5 \times 6 = 21$
The equation gives back a $y$ value from the table.
Read three points from plan A's line.
$(0, 0), \; (2, 5), \; (4, 10)$
Plan A charges for gigabytes of data, and its line starts at the origin.
Divide $y$ by $x$ at the two points that are not the origin.
$5 \div 2 = 2.5, \quad 10 \div 4 = 2.5$
Both points give the same quotient.
Decide about plan A.
$\text{straight, through } (0, 0), \; k = 2.5 \Rightarrow C = 2.5g$
Plan A costs 2.50 dollars for each gigabyte, and nothing for none.
Read three points from plan B's line.
$(0, 3), \; (2, 7), \; (4, 11)$
Plan B's line is just as straight, but it starts at 3 on the $y$-axis.
Divide at the two points after the start.
$7 \div 2 = 3.5, \quad 11 \div 4 = 2.75$
The quotients disagree, so there is no single $k$.
Decide about plan B and explain.
$\text{not proportional}: C = 2g + 3$
Plan B has a 3-dollar monthly fee, so zero gigabytes still cost 3 dollars.
Write the total with 7% tax as a multiplication.
$T = 1.07p$
The price plus 7% of the price is 107% of the price.
Work out two totals.
$p = 30: T = 32.10; \quad p = 80: T = 85.60$
Two pairs are needed to compare quotients.
Divide each total by its price.
$32.10 \div 30 = 1.07, \quad 85.60 \div 80 = 1.07$
The quotients agree and equal the multiplier.
Decide about the total with tax.
$\text{proportional}, \; k = 1.07$
Every dollar of price costs 1.07 dollars with tax, whatever the price.
Now add a 6-dollar shipping fee to each order.
$T = 1.07p + 6$
The fee is charged once per order, not per dollar.
Work out the new totals.
$p = 30: T = 38.10; \quad p = 80: T = 91.60$
Each total is 6 dollars more than before.
Divide each new total by its price.
$38.10 \div 30 = 1.27, \quad 91.60 \div 80 = 1.145$
The quotients no longer agree.
Decide, and explain with the origin.
$p = 0: T = 6 \ne 0 \Rightarrow \text{not proportional}$
An empty order would still cost the fee, so the graph misses the origin.
Compare the equation with $y = kx$.
$d = 12t \Rightarrow k = 12$
It is a number times $t$ with nothing added, so it is proportional.
Check the origin.
Interpret the point $(1, 12)$.
In each table, $x$ takes the values $4$, $7$ and $11$. Which list of $y$ values makes a proportional relationship?
A climbing gym charges an entry fee plus the same price for every hour. Two hours cost $28$ dollars and five hours cost $40$ dollars. Complete the worked solution to decide whether the cost is proportional to the time.
Divide the cost of two hours by 2.
$28 \div 2 =$ a
If the cost were proportional, this quotient would be the price of one hour.
Divide the cost of five hours by 5.
$40 \div 5 =$ b
A proportional relationship gives the same quotient for every pair.
Compare the two quotients.
$\text{the quotients differ} \Rightarrow \text{not proportional}$
One pair that disagrees is enough to rule out a single constant $k$.
Find the hourly price from the difference, then the cost of zero hours.
$\text{3 more hours cost } 12, \quad \text{cost at 0 hours} =$ z
Taking two hours' price off the two-hour cost leaves the entry fee, which a proportional graph could not have.
A proportional table has $x$ values $4$, $12$ and $18$ with $y$ values $22$, $66$ and $99$. What is the constant of proportionality, and what is $y$ when $x = 24$?
$k =$ k, and $y =$ y
$y$ is proportional to $x$, and $y = 44$ when $x = 4$. Complete the table.
| $x$ | $y$ |
|---|---|
| 1 | |
| 7 | |
| 14 |
The graph of a proportional relationship is a straight line through the origin and the point $(10, 55)$. Write $y$ in terms of $x$.
Answer:
A salesperson at a furniture store is paid only a commission of $5\%$ of her sales, so her pay $c$ is proportional to her sales $s$. What is the constant of proportionality as a decimal, and what does she earn in a week when she sells $5300$ dollars of furniture?
$k =$ k. Pay: c dollars.
A shoe store takes $10\%$ off every pair of shoes, so the sale price $s$ is proportional to the regular price $p$. What is the constant of proportionality, and what is the sale price of shoes that regularly cost $160$ dollars?
$k =$ k. Sale price: s dollars.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Weight on Mars is proportional to weight on Earth. An astronaut in a space suit who weighs $150$ pounds on Earth would weigh $57$ pounds on Mars. What is the constant of proportionality, and how much would a robot that weighs $200$ pounds on Earth weigh on Mars?
$k =$ k. Robot on Mars: r pounds.
You can decide whether a relationship is proportional and explain why. Without looking: is a table with $x$ of 1, 2, 3 and $y$ of 4, 7, 10 proportional? Is a price plus 6% tax proportional to the price, and what if a 5-dollar fee is added?
19. Your turn: a cyclist's distance is $d = 12t$, with $t$ in hours, step 2
$t = 0: d = 12 \times 0 = 0$
No time riding means no distance, so the graph passes through $(0, 0)$.
19. Your turn: a cyclist's distance is $d = 12t$, with $t$ in hours, step 3
$(1, 12): \; 12 \text{ miles in 1 hour}$
The point where $t = 1$ shows the constant: 12 miles per hour.