Question

On a 400 mile trip, car X travelled half the distance at 40 miles per hour (mph) and the other half at 50 mph. What was the average speed of the car?
(a) \[44\dfrac{4}{9}\] mph
(b) 48 mph
(c) 45 mph
(d) 46 mph

Answer 1

Hint: Here, we need to find the average speed of the car. First, we will find the time the car travels at 40 mph and 50 mph. Then, we will find the total time taken by the car to complete the trip. Finally, we will use the speed distance and time formula to find the average speed using the total time taken and total distance covered.

Formula Used:
 We will use the following formulas:
1.The time taken to cover a distance at a uniform speed is given by \[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\].
2.The speed at which a certain distance is covered in a certain time is given by \[{\rm{Speed}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Time}}}}\].

Complete step-by-step answer:
First, we will find the distance covered at 40 mph, and 50 mph.
The car travels half the distance at 40 mph, and the remaining half at 50 mph.
Dividing 400 miles by 2, we get half the distance as \[\dfrac{{400}}{2} = 200\] miles.
Now, we will use the speed distance time formula to find the time taken to travel 200 miles at 40 mph.
Distance travelled by the car at 40 mph \[ = \] 200 miles
Now substituting values of distance and speed in the formula \[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\], we get
Time taken by the car to travel 200 miles at 40 mph \[ = \dfrac{{200}}{{40}}\] hours
Simplifying the expression, we get
Time taken by the car to travel 200 miles at 40 mph \[ = 5\] hours
Next, we will use the speed distance time formula to find the time taken to travel 200 miles at 50 mph.
Now substituting values of distance and speed in the formula \[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\], we get
Distance travelled by the car at 50 mph \[ = \] 200 miles
Therefore, we get
Time taken by the car to travel 200 miles at 50 mph \[ = \dfrac{{200}}{{50}}\] hours
Simplifying the expression, we get
Time taken by the car to travel 200 miles at 50 mph \[ = 4\] hours
Now, we will add the time taken to travel 200 miles at 40 mph, and the time taken to travel 200 miles at 40 mph to find the time taken to cover the entire distance of 400 miles.
Therefore, we get
Time taken to travel 400 miles \[ = \left( {5 + 4} \right)\] hours
Adding the terms in the expression, we get
Time taken to travel 400 miles \[ = 9\] hours
Thus, we get the time taken by the car to travel 400 miles as 9 hours.
Finally, we will use the speed distance time formula to find the speed of the car if it takes 9 hours to cover 400 miles.
Substituting the distance as 400 miles and time taken as 9 hours in the formula \[{\rm{Speed}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Time}}}}\], we get
\[{\rm{Speed}} = \dfrac{{400}}{9}\] mph
Rewriting the improper fraction as a mixed fraction by converting to the form \[\dfrac{{a \times c + b}}{c}\], we get
\[\begin{array}{l} \Rightarrow {\rm{Speed}} = \dfrac{{396 + 4}}{9}\\ \Rightarrow {\rm{Speed}} = \dfrac{{44 \times 9 + 4}}{9}\end{array}\]
Simplifying the expression, we get
\[\therefore \text{Speed}=44\dfrac{4}{9}\]
Therefore, we get the average speed of the car as \[44\dfrac{4}{9}\] mph.
Thus, the correct option is option (a).

Note: We can make a mistake by finding the average of the two speeds, 40 mph and 50 mph, to get the answer as 45 mph. This is incorrect because the car travels at both speeds for a different number of hours. If the car travelled at both 40 mph and 50 mph for an equal number of hours, then the average speed would be 45 mph, but the distance travelled will be different at both speeds.
We converted an improper fraction to mixed fraction. An improper fraction is a fraction whose numerator is larger than its denominator. A mixed fraction is a fraction in the form \[a\dfrac{b}{c}\]. Every improper fraction of the form \[\dfrac{{a \times c + b}}{c}\] can be converted to an improper fraction \[a\dfrac{b}{c}\].