This question may be redundant and I apologize in advance but I am really having a hard time to digest the notion of proportional to in mathematics. Kindly, can someone simplify the idea of it and when we can say proportional to something.
Thank you for your help.
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We say that $y$ is proportional to $x$ if there is a constant $c$ for which $y=cx$. Intuitively, this means that
... and indeed, in general, if we scale $x$ multiplicatively by a certain amount, then we scale $y$ multiplicatively by that same amount. The earliest examples of this that we tend to come across are in converting units:
When we start talking more abstractly about proportionality, we should go back to this scaling concept - any 2 squares are proportional, for example, because doubling the length of all sides of a square leads you to another square. However, this is not true of all shapes - to get from a triangle with sidelengths $\{3,4,5\}$ to a triangle with sidelengths $\{5,12,13\}$, you'd necessarily have to scale one side more than one of the others.
One way of showing this is to note that the first triangle has two sides in the ratio $4:3$, and this ought to be preserved by multiplicative scalings - upon a scaling of scale factor $f$, the sides would become $\{3f,4f,5f\}$, and $4f:3f=4:3$ for all nonzero $f$. However, none of the pairs of sides in the second triangle are in this ratio.
On
Two variables $x$ and $y$ are said to be proportional provided that: if you multiply $x$ by $2$, then $y$ will also be multiplied by $2$; if you multiply $x$ by $3$, then $y$ will also be multiplied by $3$; in general, if you multiply $x$ by any constant $k\neq 0$, then $y$ will also be multiplied by $k$. Mathematically, this is expressed by saying that there is a constant $k$ such that $y=kx$.
Examples:
(MILK) Assume that the price is $\$3.00$ per $1$L. The total price is proportional to the amount of liters: if we buy $3$ liters, we will pay $\$9.00$; if we buy $6$ liters (which is $2$ times $3$), we will pay $\$18.00$ (which is $2$ times $9$); in general, if we buy $L$ liters, we will pay $3L$. So, there is a constant $k$ (which is $3$) such that total price $=k$ amount of liters.
(TAXI) Assume that the fare is $\$3.00$ (initial charge) plus $50$ cents per 1/5 mile. The taxi fare isn't proportional to the traveled distance: if we travel $3$ mile, we will pay $\$10.50$ but if we travel $6$ miles (which is $2$ times $3$) we will pay $\$18.00$ (which is not $2$ times $10,50$).
A quantity is proportional to another if you can multiply that quantity with a number $a$ to get the other quantity. That means, both quantities have the same ratio.
For example. The number of human legs is proportional to the number of people, because every person has two legs. Thus, the ratio of people to legs is 1 : 2. There are twice as many legs as people. (I do not mean to discriminate people with leg amputations).