The table below shows the price to mail a first class letter for various weights. Is the cost to mail a letter proportional to its weight? Well a one-ounce letter costs thirty-nine cents; a two-ounce letter costs fifty-nine cents; a three-ounce letter costs seventy-nine cents; and a four-ounce letter costs one dollar and thirty-six cents.
The first thing we need to think about is this phrase here: is the cost to mail a letter proportional to its weight? What do we mean when we say something is proportional to something? Well two things are directly proportional if we can express the relationship between them as one being a simple multiple of the other.
So for example, so long as we keep that units the same, is the cost always equal to some number — let’s just call it 𝑘 cause we don’t know what that number is for now — times the weight? Now for example, we know that a one-ounce letter costs thirty-nine cents or nought point three nine dollars.
So nought point three nine equals 𝑘 times one. In other words, 𝑘 would be equal to nought point three nine. So if the cost to mail a letter was proportional to its weight, we could work out the value of 𝑘 and then we could apply that formula to any weight in the table, and it will be true. So let’s try another one.
Now I think the easiest number to work with is gonna be two. So let’s just check, does the cost to mail a two-ounce letter follow this formula? Well if it does then the cost would be nought point three nine times two, which would be seventy-eight cents or nought point seven eight dollars. But no, according to our table, the cost for a two-ounce letter is fifty-nine cents.
So straight away, we found a value which breaks the proportional relationship, so this is not a proportional relationship. So our answer is no, the cost to mail a letter is not proportional to its weight.
Now just before we go, another way to tackle that problem would’ve been to say that if they were proportional, that if I had twice the weight, then I would have twice the cost. And that’s clearly not the case: two times nought point three nine is nought point seven eight.
But the actual cost of mailing a two-ounce letter is only nought point five nine dollars. So it’s not twice the cost for twice the weight, so it’s not proportional.