How to make sense of graphing relationships?

Question

Let x be an apple (x = apple).

Let y be an orange (y = orange).

let 3x = 2y be a proportional relationship. That is, having 3 apples is the equivalent as having 2 oranges.

Now lets graph our relationship:

Y
9           1
8
7
6       1
5
4
3   1
2
1
0 1 2 3 4 5 6 7 8 9 X

The question is, if I am saying that x + x + x = y + y, then why the graph is showing 2 apples in the apple dimension and 3 oranges in the orange dimension. It sounds like the opposite of what I am trying to say.

Instead, I was expecting 3 apples in the apple dimension and 2 oranges in the orange dimension:

Y
9
8
7
6                 1
5
4           1
3
2     1
1
0 1 2 3 4 5 6 7 8 9 X

Why this looks counter intuitive?

Here's your confusion: $x$ is "apple" or $x$ is the amount of apples? — jjagmath, Aug 17 '22 at 15:19
@jjagmath, the context is presented as x being a single unit of apple. Note that I am not assigning a value/price to a single unit of apple. — YGranja, Aug 17 '22 at 15:26
"having 3 apples is the equivalent as having 2 oranges" - this does not define a direct proportion. You should say that the ratio of apples to oranges (or vice versa) is constant. — Vasili, Aug 17 '22 at 15:27
@YGranja You're using $x$ and $y$ to label the axes, which means that you're using $x$ as "amount of apples" and $y$ as "amount of oranges" — jjagmath, Aug 17 '22 at 15:30
@Vasili, When I say 3 dollars = 2 euros or 3 dollars is the equivalent as 2 euros, isn't that a direct proportion? — YGranja, Aug 17 '22 at 15:36
@YGranja: No, this is only one data point which does not define a relationship. But if you say "for every three dollars, a bank gives two euros", this defines many data points. This also means that for $x$ dollars you will get $\frac{2}{3}x$ euros as the ratio of euros to dollars is $2:3$. — Vasili, Aug 17 '22 at 15:43
You appear to be confusing "apples" as units with "number of apples". Let's do inches and feet instead. There are, of course, $12$ inches in a foot. Thus $12$ inches = $1$ foot as units. however it is not the case that $12$ times the number of inches equals the number of feet. Indeed, phrased this way, you need $\frac 1{12}$ in place of $12$. — lulu, Aug 17 '22 at 16:02
Take a look at "Ratio" and "proportion" confusions and What does being proportional mean?. — ryang, Aug 17 '22 at 17:43

Ethan Bolker · Answer 1 · 2022-08-17T16:04:39.990

If

3 dollars = 2 euros

then the proportionality constant is $$ \frac{3}{2}\frac{\text{dollars}}{\text{euro}} $$ which is just another way to write the number $1$.

Then the relationship is

$$ \text{quantity of dollars} = \frac{3}{2} \frac{\text{dollars}}{\text{euro}} \times \text{quantity of euros}. $$ If you draw the graph with quantity of euros on the $x$-axis and quantity of dollars on the $y$-axis the line will have slope $3/2$ and go through the point $(2,3)$.

score 1 · Answer 2 · answered Aug 17 '22 at 15:37

1

Your equation 3x=2y is wrong, if x is the number of apples, put in x=3 and you know y=2 so you would have $3*3=2*2$ and you see it is wrong. the relation of number of apples to number of oranges x/y=3/2 and so you have 2x=3y your second graph saying "let x be an apple" is misleading , say x is the number of apples ans always try you equation with numbers you know.

answered Aug 17 '22 at 15:37

trula

1,472

When I say 3 dollars = 2 euros, am I misleading the ratio between dollars to euros? – YGranja Aug 17 '22 at 15:45
If you say 3 dollars = 2 euros, it is not misleading, if you use it right, which is convenient, when traveling, since you know if you spend 200€ you have spend 300$ so you multiply each side with the same number, here 100. but if x is the number of $ and y the number of € and you write 3x=2y you will get wrong results y=300 will give you x=400/3$=133$ and you think this is cheap and buy it! so you need the exchange rate $/€=3/2 and will have x/y=3/2, 2x=3y – trula Aug 17 '22 at 16:29

score -1 · Answer 3 · answered Aug 17 '22 at 15:26

-1

You are misinterpreting what $3x=2y$ means. If x=(number of apples) and y=(number of oranges), then $3x=2y$ does not means that 3 apples are equivalent to 2 oranges. This tells you that 3 times the number of apples is equal to 2 times the number of oranges. So, if you have 3 apples, you have $3\cdot3=9$ which is 2 times the number of oranges, so the oranges are $\frac{9}{2}$, so you have 4 and a half oranges.

In order to visualize it graphically, it is better to isolate the $y$ dividing both sides by its coefficient, so the equation becomes: $$y=\frac{3}{2}x \ , $$ which means that for each apple (x) you have $\frac{3}{2}$ oranges, i.e. one orange and a half.

answered Aug 17 '22 at 15:26

SilvioM

1,298

So you are telling me that 3x = 2y does not means that 3 dollars are equivalent to 2 euros, assuming that x and y are acting as dollars and euros? – YGranja Aug 17 '22 at 15:50
Exactly, 3x=2y mathematically means that 3 times the dollars is equal to 2 times the euros. You are confusing variables (x and y which can stay for quantitative of dollars and quantitative of euros) with unit measurements. If you write 3dollars=2€ it means what you are saying, because dollars and € are just unit measurements, in fact you can't change € and dollars like you change x and y which are variables. – SilvioM Aug 17 '22 at 15:55
Ps: I don't understand why I got two undervotes, did I say something wrong? Please correct me if it is the case. – SilvioM Aug 17 '22 at 16:07
1

I don't see why that is the case, you seem to present a direct relationship very similar to other answers. I will cast a vote up sice I don't see anything wrong here. – YGranja Aug 17 '22 at 16:13
@SilvioM More than one year later, but I can guess why this answer got downvotes. The OP's confusion appears to come mostly from confusing the two statements "x is the value of an apple" and "x is the number of apples" (which is a legitimate confusion, and this error creeps in the work of even experienced people occasionally; it's a bit like making a sign error or confusing left and right). In fact the OP's post begins with "Let x be an apple." And your post starts with "x=(number of apples)". So I can guess that the downvoters thought this answer was adding to the OP's confusion. – Stef Dec 08 '23 at 21:20

How to make sense of graphing relationships?

3 Answers3