Math Problem Speed/Distance/Time

Question

Betty and Carl live in the same apartment building, work at the same office, and set off for work in the morning, at the same time. They must each travel 65km, and they arrive at work at the same time. Betty travels by car at 45km/h, parks at the car park, and then walks at 5km/h the rest of the way to work. Carl takes the bus, which travels at 40km/h, to the same car park as Betty uses, and then rides his bike the rest of the way at 6km/h. If they both leave home at 6.a.m., what time do they arrive at work?

Answer 1

45 * tCar + 5 * tWalk = 65 <--- officially the first equation

40 * tBus + 6 * tBike = 65 <--- officially the 2nd equation

tCar + tWalk = tBus + tBike <--- officially the third equation: travel times are the same

45 * tCar = 40 * tBus <--- officially the fourth equation: distance from apartment to parking lot

First, solves the fourth equation for tCar, it is tCar = (8/9)*tBus

Then substitutes it into equation #1 and equation #3

45( 8/9) tBus + 5 * tWalk = 65

40 * tBus + 6 * tBike = 65

(8/9) tBus + tWalk = tBus + tBike

Writes the system in standard form:

40 * tBus + 5 * tWalk = 65 <--- equation ALPHA

40*tBus + 6 * tBike = 65 <--- equation BETA

(1/9)*tBus + tBike - tWalk = 0 <--- equation DELTA

Solves the 3 x 3 by eliminating tBus....

First Subtracts equation BETA - ALPHA:

6 * tBike - 5 * tWalk = 0

tBike = (5/6)* tWalk <--- equation OMEGA

Next, does -360 * DELTA + BETA:

-354* tBike + 360 * tWalk = 65 <--- equation EPSILON

Substitutes OMEGA into EPSILON:

-354 ( 5/6 * tWalk ) + 360 * tWalk = 65

-1770 * tWalk + 2160 * tWalk = 390 <--- multiplies both sides by 6

390 * tWalk = 390

tWalk = 1

per OMEGA, tBike = 5/6

Per original first equation:

45 * tCar + 5 * tWalk = 65

45 * tCar + 5 * 1 = 65

45 * tCar + 5 = 65

45 * tCar = 60

tCar = 60/45 = 4/3

Per original second equation:

40 * tBus + 6 * tBike = 65

40 * tBus + 6(5/6) = 65

40 * tBus + 5 = 65

40 * tBus = 60

tBus = 60/40 = 3/2

tCar + tWalk = 4/3 + 1 = 4/3 + 3/3 = 7/3

tBus + tBike = 3/2 + 5/6 = 9/6 + 5/6 = 14/6 = 7/3

So it is (4/3)*45 = 60 km from the apartment to the parking lot

=(3/2)*40

The total travel time is 7/3 hours = 2 and 1/3 hours = 2 hours and 20 minutes