Question

A small car company operates machinery for $t$ hours per day that produces $c$ cars per hour, where $c$ and $t$ are positive integers. After upgrading the machinery, the company can now produce $2c+5$ cars per hour. If the machinery now operates for $t-2$ hours per day and yet produces the same number of cars per day, how many cars per day does it produce?

Answer 1

Let daily production be tc before and (t-2)(2c+5) after. So
tc = (t-2)(2c+5).
Expand and rearrange:
ct = 2ct - 4c + 5t - 10
0 = ct - 4c + 5t - 10
c(t-4) + 5(t-2) = 0
Hence c(t-4) = -5(t-2).

Since c>0, examine integer t:
- If t≥4 the left side is ≥0 while the right side ≤0, impossible (and t=4 fails).
- Try t=3: c(−1) = −5 ⇒ c=5, valid.
- t=2 or 1 give nonpositive or nonintegral c.

Thus t=3, c=5 and daily production is tc = 3·5 = 15 cars.