What would be a formula for this?

Question

I'm trying to find a formula for this scenario:

I need \$$302$.

I gain \$$2$ every second.

I gain \$$5$ every $5$ seconds.

When will I reach \$$302$? Both of these rates are working together.

Visually:

$2 $2 $2 $2 ($2 $5) $2 $2 $2 $2 ($2 $5)

I'm able to find the answer by diving by \$$15$ and figuring out how many sequences there were, but I'm looking for an equation.

$302 / 15 = ~20$

$20$ x $5$s + $1$s = $101$s

@barakmanos thought there would be an equation for this like summation of rates or something like that — John Smith, Mar 06 '16 at 12:07
You've already written an equation for your specific values. If you want a general equation, then you need to specify general names for your values (a.k.a variables). — barak manos, Mar 06 '16 at 12:09
On average you get 3 dollars per second. After 100 seconds you get 300 dollars. At the second 101 you get your last 2 dollars. — Crostul, Mar 06 '16 at 12:17

score 2 · Accepted Answer · answered Mar 06 '16 at 12:15

2

After $n$ seconds you have $2n+5\left\lfloor \frac{n}{5} \right\rfloor$ dollars. Now you want to find $n$ for which $$302 \leq 2n+5\left\lfloor \frac{n}{5} \right\rfloor$$ and since $\left\lfloor \frac{n}{5} \right\rfloor \leq \frac{n}{5}$ you have $$302 \leq 2n+5\frac{n}{5}=3n$$ so $n\geq 101$.

Here $\left\lfloor .\right\rfloor$ is the floor function.

answered Mar 06 '16 at 12:15

Sil

16,612

so this is like LCM, only it's relative to 1 second. this seems more like a rule than an equation? – John Smith Mar 06 '16 at 12:27
oh i get it, the floor is what I'm looking for – John Smith Mar 06 '16 at 12:28
You can turn it into general inequality, if you replace $302$ with some variable. – Sil Mar 06 '16 at 12:39
Thanks, I think this is what i'm looking for. – John Smith Mar 06 '16 at 12:48
You are welcome. – Sil Mar 06 '16 at 12:48

What would be a formula for this?

1 Answers1