A certain number of small parts need to be produced. 30 parts are scheduled to be produced after each day. After 1/3 of the parts are produced, the rate of production increases by 10% thanks to improvement on efficiency. It takes 4 fewer days to produce all the parts than scheduled. How many parts are in total?
The factory produced 30 per day originally, so 33 after efficiency increase. It takes 4 fewer days, so that is 120 parts rest. I need help with the rest after this.
$\endgroup$1 Answer
$\begingroup$Let the total number of parts be $N$. Let the original production rate per day be $r$.
Since $1/3$ of the parts are produced at the original rate, that takes $\dfrac{N/3}{r}$ days. The remaining $2N/3$ parts are produced at a rate that's 10% higher, that is, $1.1r$, which takes $\dfrac{2N/3}{1.1r}$.
The total time in days to produce these parts is then $\dfrac{N/3}{r} + \dfrac{2N/3}{1.1r}$. Producing all $N$ parts at the original rate would take $\dfrac{N}{r}$ days. You are told that the new production time is 4 days less, that is,
$$ \dfrac{N/3}{r} + \dfrac{2N/3}{1.1r} = \dfrac{N}{r} - 4 $$
Set r = 30 parts per day and rearrange. You have an equation for N
$$ \frac{N}{90} + \frac{2N}{99} = \frac{N}{30} - 4 $$
which gives $N$ = 1980 parts when you solve it.
You can then plug everything back in to check that the original production time is 62 days and the new (reduced) production time is 58 days.
$\endgroup$
