There are 3 bike-sharing companies in the current market. For each ride, $\alpha$-bike charges a certain amount per 5 min block or part thereof, $\beta$-bike charges a certain amount per 10 min block or part thereof and $\mu$-bike charges a certain amount per 15 min block or part thereof. Rebecca rode each of the bike-sharing companies' bikes once in each month. The table below shows the amount of time Rebecca clocked for each ride and her total spending for each month. In celebration of the company's first anniversary, the pricings in February and March 2017 of $\mu$-bikes are a 5% discount off the immediate previous month's pricing.

	January 2017	February 2017	March 2017
$\alpha$-bike	25 min	17 min	36 min
$\beta$-bike	30 min	10 min	39 min
$\mu$-bike	15 min	44 min	33 min
Total spending	$5.70	$5.72	$9.71

Determine which bike-sharing company offers the cheapest rate (without any discount) for a 40-min ride. Justify your answer clearly.

1. Set up variables and count the number of charged blocks for each ride

Let $a, $b, $m be the (January, un-discounted) charge per block for $\alpha$-, $\beta$- and $\mu$-bike respectively. Since each company charges per block 'or part thereof', round each time up to the next whole block:

$\alpha$ (per 5 min): $25\to5$, $17\to4$, $36\to8$ blocks.
$\beta$ (per 10 min): $30\to3$, $10\to1$, $39\to4$ blocks.
$\mu$ (per 15 min): $15\to1$, $44\to3$, $33\to3$ blocks.

2. Form a system of equations including the monthly 5% discount on μ-bike

$\mu$-bike rate: January $m$, February $0.95m$, March $0.95^2m=0.9025m$.

Jan: $5a+3b+m=5.70$
Feb: $4a+b+2.85m=5.72$
Mar: $8a+4b+2.7075m=9.71$

3. Solve the system for the per-block rates

Using a graphing calculator to solve the $3\times3$ system:
$a\approx0.4079,\quad b\approx0.8402,\quad m\approx1.1397.$

4. Cost a 40-min ride on each company and compare

Blocks for 40 min: $\alpha$: $\lceil40/5\rceil=8$; $\beta$: $\lceil40/10\rceil=4$; $\mu$: $\lceil40/15\rceil=3$.

$\alpha$: $8a\approx8(0.4079)=\char36{}3.26$
$\beta$: $4b\approx4(0.8402)=\char36{}3.36$
$\mu$: $3m\approx3(1.1397)=\char36{}3.42$

Since $\char36{}3.26<\char36{}3.36<\char36{}3.42$, $\alpha$-bike is cheapest.