The bath that took too long

News from the bathroom: a bath that should have filled in fifteen minutes took almost nineteen. The bath holds a hundred and fifty litres and the tap gives ten litres every minute, so fifteen minutes was the plan.

Bath news: it should fill in $15$ min, but took almost $19$. The bath holds $150$ L and the tap gives $10$ L/min, so the plan was $150÷10=15$ min.

Volume $V$, tap $r$, leak $\ell$ (litres, minutes): $V=150,r=10,\ell =2$. Plan: $\frac{V}{r}=15$.

The culprit was the plug. It leaked two litres every minute, so the bath gained only eight litres a minute. After fifteen minutes there were just a hundred and twenty litres in it.

The culprit: the plug leaks $2$ L/min, so the net gain is $10-2=8$ L/min. After $15$ min there are $15\times 8=120$ L.

Real: $r-\ell =8$, and $15\times 8=120<V$.

The last thirty litres took a further three minutes and forty-five seconds: eighteen minutes and forty-five seconds in all. Meanwhile thirty-seven and a half litres ran away down the drain.

The last $150-120=30$ L took $30÷8=3.75$ min, so $15+3.75=18.75$ min in all. Lost down the drain: $18.75\times 2=37.5$ L.

$t=\frac{V}{r-\ell }=\frac{150}{8}=18.75$; lost: $\ell t=37.5$.

Plumbers advise pressing the plug in firmly. With no leak, the bath fills in fifteen minutes and not a drop is lost.

Advice: press the plug in firmly. No leak $\Rightarrow$ time $=15$ min and loss $=0$ L.

$\ell =0\Rightarrow t=\frac{V}{r}=15$.