Question

On a trip, a motorist drove 150 miles in the morning and 50 miles in the afternoon. His average rate in the morning was twice his average rate in the afternoon. He spent 5 hours driving. Find his average rate on each part of the trip.

Answer 1

The average speed in the morning trip is $50{\text{ mile/h}}$ and the speed in the afternoon trip is $25{\text{ mile/h}}.$

Further explanation:

The relationship between speed, distance and time can be expressed as follows,

$\boxed{{\text{Speed}} = \frac{{{\text{Distance}}}}{{{\text{Dime}}}}}$

Given:

A motorist drove $150{\text{ miles}}$ in the morning and $50\:{\text{miles}}$ in the afternoon.

The time taken to travel is $5{\text{ hours}}.$

Explanation:

The average rate in the morning was twice his average rate in the afternoon.

Consider the speed of the person in afternoon be $x{\text{ miles/hour}}.$

So speed of the person in the morning is $2x{\text{ miles/hour}}.$

The time taken to $150\: {\text{miles}$ in the morning can be calculated as follows,

$\begin{aligned}{\text{speed}}&=\frac{{{\text{distance}}}}{{{\text{time}}}}\\2x&=\frac{{150}}{{{t_1}}}\\{t_1}&= \frac{{150}}{{2x}}\\{t_1}&=\frac{{75}}{x}\\\end{aligned}$

The time taken to $50\:{\text {miles}$ in the afternoon can be calculated as follows,

$\begin{aligned}{\text{speed}}&=\frac{{{\text{distance}}}}{{{\text{time}}}}\\x&= \frac{{50}}{{{t_2}}}\\{t_2}&=\frac{{50}}{x}\\\end{aligned}$

The total time taken by the person is 5 hours.

$\begin{aligned}t&= {t_1} + {t_2}\\5&=\frac{{75}}{x} + \frac{{50}}{x}\\5&= \frac{{75 + 50}}{x}\\x&= \frac{{125}}{5}\\x &= 25\\\end{aligned}$

The speed of the person in the afternoon is $25{\text{ miles/h}}.$

The speed of the person in the morning can be calculated as follows,

$\begin{aligned}{\text{Speed} &= 2\times 25\\&= 50{\text{ miles/h}}\\\end{aligned}$

The average speed in the morning trip is $\boxed{50{\text{ mile/h}}}$ and the speed in the afternoon trip is $\boxed{25{\text{ mile/h}}}.$

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Speed

Keywords: motorist, 150 miles, drove, morning, afternoon, twice, spent, trip, average rate, 5 hours driving, 50 miles.

Answer 2

Answer:

The average speed in the morning trip is  $50\;\rm{miles/hour}$ and the speed in the afternoon trip is $25\;\rm{miles/hour}$.

Step-by-step explanation:

Given: On a trip, a motorist drove $150$ miles in the morning and $50$ miles in the afternoon. His average rate in the morning was twice his average rate in the afternoon. He spent $5$ hours driving.

As per question,

Let the speed of the motorist in afternoon be $x\; \rm{miles\; per\; hour}$.

Then speed of the motorist in the morning is $2x\; \rm{miles\; per\; hour}$.

Now, using the relationship of speed, distance and time can be expressed as follows:

$\rm{Speed}=\frac{\rm{Distance}}{\rm{Time}}$

Here, time taken to $150 \; \rm{miles}$ in the morning can be calculated as follows,

$2\rm{x}=\frac{150}{\rm{t}_{1} }\\\rm{t}_{1}=\frac{150}{2x}\\\\\rm{t}_{1}=\frac{75}{x}$            ............(1)

Now, time taken to $50 \; \rm{miles}$ in the afternoon can be calculated as follows,

$\rm{x}=\frac{50}{\rm{t}_{2} }\\\rm{t}_{2}=\frac{50}{x}\\\\\rm{t}_{2}=\frac{50}{x}$             .............(2)

The total time taken by motorist in $5$ hours is calculated as $\rm{t_{1} \;+\;\rm{t_{2}=5$.

$\frac{75}{x}+ \frac{50}{x}=5$

      $\frac{125}{x}=5$

         $\\x=\frac{125}{5}\\x=25\;\rm{miles/hour}$

The speed of the motorist in the afternoon is $x=25\;\rm{miles/hour}$.

The speed of the motorist in the morning is $2x=2\times25=50\;\rm{miles/hour}$.

Hence, the average speed in the morning trip is  $50\;\rm{miles/hour}$ and the speed in the afternoon trip is $25\;\rm{miles/hour}$.

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