11 + rand(9) 41 + rand(9) 31 + rand(9) 1 + rand(9) AVG * TIME (SPEED1 * SPEED2 * TIME - DIST * SPEED2) / (SPEED1 - SPEED2) DIST - DIST2

person(1) traveled by vehicle(1) at an average speed of SPEED1 miles per hour.

Then, he(1) traveled by vehicle(2) at an average speed of SPEED2 miles per hour.

In total, he(1) traveled DIST miles for plural(TIME, "hour").

How many miles did person(1) travel by vehicle(1)? (Round to the nearest mile.)

round(DIST1)

`d_vehicleVar(1) = round(DIST1)` miles

How many miles did person(1) travel by vehicle(2)? (Round to the nearest mile.)

round(DIST2)

`d_vehicleVar(2) = round(DIST2)` miles

How many minutes did person(1) travel by vehicle(1)? (Round to the nearest minute.)

round(DIST1 / SPEED1 * 60)

`t_vehicleVar(1) = round(DIST1 / SPEED1 * 60)` minutes

How many minutes did person(1) travel by vehicle(2)? (Round to the nearest minute.)

round(DIST2 / SPEED2 * 60)

`t_vehicleVar(2) = round(DIST2 / SPEED2 * 60)` minutes

Remember that `d = r * t`, or written another way, `t = d / r`.

`d_vehicleVar(1) =` distance that person(1) traveled by vehicle(1)

`d_vehicleVar(2) =` distance that person(1) traveled by vehicle(2)

Total distance: `d_vehicleVar(1) + d_vehicleVar(2) = DIST`

Total time: `t_vehicleVar(1) + t_vehicleVar(2) = TIME`

`t_vehicleVar(1) = (d_vehicleVar(1) / SPEED1)` and `t_vehicleVar(2) = (d_vehicleVar(2) / SPEED2)`

Substitute the blue equations for: `(d_vehicleVar(1) / SPEED1) + (d_vehicleVar(2) / SPEED2) = TIME`

Multiply the above equation by -1 * SPEED1: `-d_vehicleVar(1) - (SPEED1 / SPEED2) * d_vehicleVar(2) = -SPEED1 * TIME`

Add the two orange equations for: `(SPEED2 - SPEED1 / SPEED2) * d_vehicleVar(2) = DIST - (SPEED1 * TIME)`

Calculating all variables (and then rounding to the nearest integer):
`d_vehicleVar(2) =` roundTo( 4 , DIST2 ) `(round(DIST2))` miles
`d_vehicleVar(1) =` roundTo( 4 , DIST1 ) `(round(DIST1))` miles
`t_vehicleVar(1) =` roundTo( 4 , DIST1 / SPEED1 * 60 ) `(round(DIST1 / SPEED1 * 60))` minutes
`t_vehicleVar(2) =` roundTo( 4 , DIST2 / SPEED2 * 60 ) `(round(DIST2 / SPEED2 * 60))` minutes