Question

A computer takes tokens for 10 minutes and for 30 minutes, 30% of the tokens used are for ten minutes. The tokens cost £1 for 10 minutes and £2 for 30 minutes. What is the mean for the amount of time per token?

The answer is 24 but I have no idea how to get there.

Answer 1

The average amount of time per token is `24` minutes.

Explanation:

We are asked to find the average time of a used token. What is the definition of average? It's the total of all the values, divided by how many there are.

In this case, that means adding up all the used tokens' times, then dividing by the number of tokens. But we don't know how many there are—just that 30% of them are 10-minute tokens. How can we use this?

Let's imagine that there's a computer with a number of tokens in it; call this number of tokens `n.` We don't know what `n` is, but right now, it doesn't matter. With the given information, we should be able to look inside the "token-box" for this computer, and of the 100% of the `n` tokens inside, we'd find 30% of them to be 10-minute ones and (therefore) the remaining 70% to be 30-minute ones.

The number of 10-minute tokens is 30% of `n`, which can be written as `0.3n`. Likewise, the number of 30-minute tokens is `0.7n`.

To find the average token value, we first add all the tokens' time values together. Since we have `0.3n` tokens worth 10 minutes each, the total time of these tokens is their quantity multiplied by the time per token—in other words,

`color(white)[color(black)((0.3n),("tokens"))] xx color(white)[color(black)((10),("min./token"))]=color(white)[color(black)((3n),("minutes"))]`

Similarly, the total time value of all the 30-minute tokens is `0.7n xx 30=21n " minutes".`

So that means the sum of all the time values of the tokens is:

`3n " minutes" +21n" minutes"=24n" minutes"`

Now, we just need to divide this result of `24n " minutes"` by the number of tokens. But wait—we let `n` be the number of tokens! So the average token time value is:

`(24n " minutes")/(n " tokens")" "=" "24" ""min."/"token"`

In other words, we showed that the average time value per token doesn't depend on the number of tokens `n`. All we needed were the percentages of each kind of token. Thus, you can calculate the average token time using:

`color(white)= 30%xx10" "+" "70%xx30`
`=3+21`
`=24.`