Starting at home, person(1) traveled uphill to the store(1) store
for TIME_UP minutes at just RATE_UP mph.
He then traveled back home along the same path downhill at a speed of RATE_DOWN mph.
Starting at home, person(1) traveled uphill to the store(1) store
for TIME_UP minutes at just RATE_UP mph.
She then traveled back home along the same path downhill at a speed of RATE_DOWN mph.
What is his average speed for the entire trip from home to the store( 1 ) store and back?
What is her average speed for the entire trip from home to the store( 1 ) store and back?
RATE_AVG mph
The average speed is not just the average of RATE_UP mph and RATE_DOWN mph.
He traveled for a longer time uphill (since he was going slower),
so we can estimate that the average speed is closer to
RATE_UP mph than RATE_DOWN mph.
She traveled for a longer time uphill (since she was going slower),
so we can estimate that the average speed is closer to
RATE_UP mph than RATE_DOWN mph.
To calculate the average speed, we will make use of the following:
\text{AVERAGE_SPEED_TEXT} =
\dfrac{\blue{\text{TOTAL_DISTANCE_TEXT}}}{\pink{\text{TOTAL_TIME_TEXT}}}
\text{DISTANCE_UPHILL} = \text{DISTANCE_DOWNHILL}
What was the total distance traveled?
\blue{\begin{align*}
\text{TOTAL_DISTANCE_TEXT}
&= \text{DISTANCE_UPHILL} + \text{DISTANCE_DOWNHILL} \\
&= 2 \times \text{DISTANCE_UPHILL}
\end{align*}}
\begin{align*}
\text{DISTANCE_UPHILL}
&= \text{SPEED_UPHILL} \times \text{TIME_UPHILL_TEXT} \\
&= RATE_UP\text{ MPH_TEXT} \times TIME_UP\text{ MINUTES_TEXT}
\times \dfrac{1 \text{ HOUR_TEXT}}{60 \text{ MINUTES_TEXT}}\\
&= DISTANCE\text{ MILES_TEXT}
\end{align*}
Substituting to find the total distance:
\blue{\text{TOTAL_DISTANCE_TEXT} = 2 * DISTANCE\text{ MILES_TEXT}}
What was the total time spent traveling?
\pink{\text{TOTAL_TIME_TEXT} = \text{TIME_UPHILL_TEXT} + \text{TIME_DOWNHILL_TEXT}}
\begin{align*}
\text{TIME_DOWNHILL_TEXT}
&= \dfrac{\text{DISTANCE_DOWNHILL}}{\text{SPEED_DOWNHILL_TEXT}} \\
&= \dfrac{DISTANCE\text{ MILES_TEXT}}{RATE_DOWN\text{ MPH_TEXT}}
\times \dfrac{60 \text{ MINUTES_TEXT}}{1 \text{ HOUR_TEXT}}\\
&= TIME_DOWN\text{ MINUTES_TEXT}
\end{align*}
\pink{\begin{align*}
\text{TOTAL_TIME_TEXT}
&= TIME_UP\text{ MINUTES_TEXT} + TIME_DOWN\text{ MINUTES_TEXT} \\
&= TIME_UP + TIME_DOWN\text{ MINUTES_TEXT}
\end{align*}}
Now that we know both the total distance and total time, we can find the average speed.
\begin{align*}
\text{AVERAGE_SPEED_TEXT}
&= \dfrac{\blue{\text{TOTAL_DISTANCE_TEXT}}}{\pink{\text{TOTAL_TIME_TEXT}}} \\
&= \dfrac{\blue{2 * DISTANCE\text{ MILES_TEXT}}}
{\pink{TIME_UP + TIME_DOWN\text{ MINUTES_TEXT}}}
\times \dfrac{60 \text{ MINUTES_TEXT}}{1 \text{ HOUR_TEXT}} \\
&= RATE_AVG\text{ MPH_TEXT}
\end{align*}
The average speed is RATE_AVG mph, and which is closer to
RATE_UP mph than RATE_DOWN mph as we expected.
It takes TIME_INIT minutes for PEOPLE_INIT
people to paint WALL_INIT walls.
How many minutes does it take PEOPLE_FINAL people to paint WALL_FINAL walls?
TIME_FINAL minutes
Imagine that each person is assigned one wall, and all PEOPLE_INIT people begin painting at the same time.
Since everyone will finish painting their assigned wall after TIME_INIT minutes,
it takes one person TIME_INIT minutes to paint one wall.
If we have PEOPLE_FINAL people and WALL_FINAL walls,
we can again assign one wall to each person.
Everyone will take TIME_FINAL minutes to paint their assigned wall.
In other words, it takes TIME_FINAL minutes for
PEOPLE_FINAL people to paint WALL_FINAL walls.
It takes TIME_INIT minutes for PEOPLE_INIT
people to paint WALL_INIT walls.
How many minutes does it take PEOPLE_FINAL people to paint WALL_FINAL walls?
TIME_FINAL minutes
First calculate how long it will take one person to paint one wall.
It will take \dfrac{1}{WALL_INIT} of the time to paint one wall.
So it will take fractionReduce(TIME_INIT, WALL_INIT) minutes for PEOPLE_INIT people to paint one wall.
For only one person, it will take PEOPLE_INIT times as long,
so it will take fractionReduce(TIME_INIT, WALL_INIT) \cdot minutes for one person to paint one wall.
PEOPLE_INIT
= fractionReduce(TIME_INIT * PEOPLE_INIT, WALL_INIT)
So, the number of minutes it will take for one person to paint WALL_FINAL walls is
fractionReduce(TIME_INIT * PEOPLE_INIT, WALL_INIT) \cdot .
WALL_FINAL
= fractionReduce(TIME_INIT * PEOPLE_INIT * WALL_FINAL, WALL_INIT)
The number of minutes it will take for PEOPLE_FINAL people to paint WALL_FINAL walls is
fractionReduce(TIME_INIT * PEOPLE_INIT * WALL_FINAL, WALL_INIT) \div .
PEOPLE_FINAL
= TIME_FINAL