docabuzar wrote:
A certain car traveled twice as many miles from Town A to Town B as it did from Town B to Town C. From Town A to Town B, the car averaged 12 miles per gallon, and from Town B to Town C, the car averaged 18 miles per gallon. What is the average miles per gallon that the car achieved on its trip from Town A through Town B to Town C?
A. 13
B. 13.5
C. 14
D. 14.5
E. 15
MGMAT 1 Q. 11
The explanation given is unclear to me, why weightage ave is not right to apply here? I want to understand conceptually. Can someone help!
If the concept you do not understand is why (12*2 + 18*1)/3 doesn't work, here you go:
'Distances traveled' (i.e. ratio of 2:1) cannot be the weights here to find the
average mileage. The weights have to be 'number of gallons'.
If we change the question and make it:
A certain car used [highlight]twice as many gallons[/highlight] from Town A to Town B as it did from Town B to Town C. From Town A to Town B, the car averaged 12
miles per
gallon, and from Town B to Town C, the car averaged 18
miles per
gallon. What is the
average miles per
gallon that the car achieved on its trip from Town A through Town B to Town C?
Now you can use (12*2 + 18 *1)/3
Why?
Whenever you are confused what the weights should be (e.g. here should the weights be the distance traveled or should they be gallons of fuel used...), look at the units.
Average required is \(\frac{
miles}{
gallon}\). So you are trying to find the weighted
average of two quantities whose units must be \(\frac{
miles}{
gallon}\).
\(C_{avg} = \frac{C_1*W_1 + C_2*W_2}{{W_1 + W_2}}\)
\(C_{avg}, C_1, C_2 - \frac{
miles}{
gallon}\)
So \(W_1\) and \(W_2\) should be in
gallon to get:
\(\frac{
miles}{
gallon} = (\frac{
miles}{
gallon}*
gallon + \frac{
miles}{
gallon}*
gallon)/(
gallon +
gallon)\)
Only if weights are in gallons, do we get 'Total
miles' in the numerator and 'Total gallons' in the denominator.
We know that
Average miles/
gallon = Total
miles/Total gallons
[highlight]Takeaway: The weights have to be the denominator units of the
average.[/highlight]
So what do we do in this question?
What is the distance travelled (i.e.total
miles)?
"traveled twice as many
miles from Town A to Town B as it did from Town B to Town C"
We know that the ratio of the two distances must be 2:1 or we can say the distances must be 2d and d. Total distance must be (2d + d)
What is the total gallons used?
Fuel used to go from A to B = 2d/12
Fuel used to go from B to C = d/18
So
Average miles/
gallon = (2d + d)/(2d/12 + d/18)
You need to know what the weights are going to be. Here, weights have to be the amount of fuel used i.e. in gallons because you are looking for
average miles per
gallon.
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