A salesperson...

Question

A salesperson drove the first 105 miles of a trip in one hour more than it took to drive the last 90 miles. If the average rate during tha last 90 miles was 10mph faster than the average rate during the first 105 miles, find the average rate for each portion of the trip.

Answer 1

Hi Hann, this problem is an application of the D(istance) = R(age) * T(ime) formula.

 

Given:

 

Distance on first part of trip: 105 miles.

 

Distance on second part of trip: 90 miles.

 

Let x be the time it took to drive 90 miles, or the second part of the trip.

 

Then x + 1 is the time of the first part of the trip.

 

Let R be the rate on the first part of the trip.

 

Then R + 10 will be the rate on the second part of the trip.

 

Set up the equations:

 

D_f = R_f * T_f - first part of the trip.

 

Substitute and solve for T_f:

 

105 = R * (x + 1)

 

x + 1 = 105 / R

 

x = 105 / R - 1, time for the first part of the trip in terms of x.

 

------------------------------------------

 

D_s = R_s * T_s - second part of the trip.

 

Substitute and solve for T_s

 

90 = (R + 10) * x

 

x = 90 / (R + 10), time for the second part of the trip in terms of x.

 

Equate the x's:

 

105 / R - 1 = 90 / (R + 10)

 

Solve for R:

 

Multiply all terms by R * (R + 10), the common denominator:

 

105 * (R + 10) - R * (R + 10) = 90 * R

 

Clear parenthesis and move all terms to the left side of the equal sign:

 

105R + 1050 - R² - 10R - 90R = 0

 

Combine terms:

 

-R² + 5R + 1050 = 0

 

Reverse the signs:

 

R² - 5R - 1050 = 0

 

Solve for R:

 

(R - 35) * (R + 30) = 0, Note the factors of 1050 are 30 * 35, without the sign)

 

Using the zero product rule:

 

R - 35 = 0

 

R = 35

 

And 

 

R + 30 = 0

 

R = -30, this solution can be discarded since the rate cannot be negative.

 

Average rate of speed for each part of the trip:

 

The rate of speed for the first part or the 105 mile part of the trip was 35 miles per hour.

 

The rate of speed for the second part or the 90 miles part of the trip was 45 miles per hour.

 

Questions?