# Question d8878

Sep 18, 2017

$25$ months

#### Explanation:

Suppose that the number of months is $x$.

We can express the two situations as functions of $x$.

If Lauren were to subscribe to the music website, she'd have to pay $35 a month. So over a certain number of months, she'd have to pay $35 times the number of months she downloaded music.

Mathematically, this is expressed as $35 x$.

But with this membership, Lauren would also need to pay an annual fee of $500. As a whole function, we can write this as $f \left(x\right) = 35 x + 500$. If Lauren downloads music without becoming a member, she'd have to pay $55# a month, without any extra fees.

Similarly, this can be expressed mathematically as $55 x$, or in function form as $h \left(x\right) = 55 x$.

Now, we need to find the number of months $x$ that Lauren would have to download music so as both costs are the same.

Basically, we need to find the "meeting point" between these two functions.

So let's set the two functions equal to each other, i.e $f \left(x\right) = h \left(x\right)$:

$R i g h t a r r o w 35 x + 500 = 55 x$

Let's solve for $x$:

$R i g h t a r r o w 500 = 55 x - 35 x$

$R i g h t a r r o w 500 = 20 x$

$R i g h t a r r o w 25 = x$

$\therefore x = 25$

Therefore, Lauren would have to download music for $25$ months for the cost to be the same with and without a membership.