The sleep bank

Think of sleep as a bank. Every night you put hours in, and every day you spend them. The numbers here are only an example: how much sleep a person needs depends on the person, so ask a grown-up or a doctor about yours.

Sleep as a bank: in at night, out by day. Example numbers only; needs differ from person to person.

Sleep: night $\uparrow$, day $\downarrow$ (example numbers).

Say a child needs nine hours a night. On five school nights she sleeps only seven. Each night she is two hours short, so after the week she owes ten hours.

Need: $n=9$ h a night. On school nights she sleeps $s=7$ h, short by $n-s=2$ h, so after $5$ nights the debt is $D=5\times 2=10$ h.

$n=9,s=7$ (h a night); $D=5(n-s)=10$.

At the weekend she sleeps two hours more than the usual nine, on both nights. That pays back four hours, not ten.

At the weekend, $2$ h above the usual $9$, on $2$ nights: $2\times 2=4$ h paid back.

Weekend: $2\times 2=4$ paid back.

So on Monday she still owes six hours. The surprise: the debt does not vanish on Sunday night. It rolls into the next week, and one long morning in bed cannot clear it. A few earlier bedtimes are a gentler way to pay it off.

So $D=10-4=6$ h, and $D>0$: on Monday she still owes. The debt rolls into the next week.

$D=10-4=6$, so $D>0$: it rolls on.