4.5 miles/gallon = 9/2 miles/gallon at 55 mph
3.5 miles/gallon = 7/2 miles/gallon at 60 mph
500/120 = 4.1667 = 4(1/6) = 25/6 miles/gallon average

Note that we are taking the average of miles/gallon. The weights will be gallons, not miles.
Check this post: https://www.veritasprep.com/blog/2014/1 ... -averages/

w1/w2 = (A2 - Aavg)/(Aavg - A1) = (7/2 - 25/6) / (25/6 - 9/2) = (-4/6) / (-1/6) = 2/1

So gallons consumed were in the ratio 2:1 for the two speeds of 55 and 60. Total 120 gallons were consumed so 80 gallons at 55 mph and 40 gallons at 60 mph.
To use 80 gallons, he must have traveled 80*4.5 = 360 miles

Answer (E)
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Let the truck used X gallons of fuel travelling at 55mph, then it used (120-X) gallons of fuel travelling at 60mph.

So, in X gallons of fuel travelling at at an economy of 4.5 miles per gallon it traveled 4.5X miles, and in (120-X) gallons of fuel travelling at an economy of 3.5 miles per gallon it traveled 3.5(120-X) miles.
Since the total Distance traveled is 500 miles, the equation becomes:

4.5X + 3.5(120-X) = 500
Solving for X, it comes to 80.

So, distance traveled at a speed of 55mph = 4.5*80 =>360 miles.

Hence, IMO Option E is the answer.

let distance covered @ 55mph = x
so mileage would be x/4.5
and @ 60 mph ; 500-x ; mileage 500-x/3.5

solve for

x/4.5+500-x/3.5 = 120
x= 360
IMO E

Let the truck travels x miles at 55mph and remaining 500-x miles at 60mph

For 1 mile, at 55mph, number of gallons used = 1/4.5
For x miles, number of gallons used = x/4.5 = 2x/9

For 1 mile, at 60mph, number of gallons used = 1/3.5
For 500-x miles, number of gallons used = (500 - x)/3.5 = 2(500 - x)/7

Given, Total gallons used = 120
--> 2x/9 + (1000 - 2x)/7 = 120
--> (14x + 9000 - 18x)/63 = 120
--> 9000 - 4x = 7560
--> 4x = 1440
--> x = 360

IMO Option E

Hello VeritasKarishma
I hope you are doing well.

Your method has worked so far so well for me but when I try in this question, it seems to not work.
Average miles=500/120= 4.17 milles/ Gallon

Let 55milies case be A and 60 miles be B.

To find the ratio of A and B= (3.5-4.17)/(4.17-4.5)= 2/1
At 55 miles per hour, the trust traveled 333.33 miles.
I am sure I am missing something. please help check.

VeritasKarishma please help solve this via weighted average method

VeritasKarishma,
Thank you
Learning this method has been the best thing.

I agree. This method has one of the best ROI.
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Alternate Method (Also using Weighted Averages)

Method 2:

If the truck went with 55 miles/hr for the whole journey, then it would have gone 4.5 m/g *120 g/tank = 540 miles/tank -> Speed 2
If the truck went with 60 miles/hr for the whole journey, then it would have gone 3.5 m/g *120 g/tank = 420 miles/tank -> Speed 1

Using the w1/w2 formula (where average = 500 miles/tank; weight = tank)

w1/w2 = (A2 - Aavg)/(Aavg - A1) = (540 - 500)/(500 - 420) = 40/80 = 1/2

So while travelling at 55 mph, 2/3 tank * 120 g/tank = 80 g was consumed.
To use 80 gallons, he must have traveled 80*4.5 = 360 miles

Method 3: Slowest & Too many calculations

Let distance travelled by 55miles/hr speed be X
Then distance travelled by 60miles/hr speed be 500-X
500m/120g = 500m/((Xm/4.5m/g) + (500-X)m/(3.5m/g))
1/120 = 500/(2X/9 + 2(500-X)/7)
(14x + 500*18 -18X)/63 = 120
-4X = 120 * 63
4X = 500*18 - 120*63 = 4 (125*18 - 30*63)
X = 125*18 - 30*63
X = 360

We can let x = the distance traveled at 55 mph. We can create the equation:

120 = x/4.5 + (500 - x)/3.5

120 = 10x/45 + (5000 - 10x)/35

Multiplying by 315, we have:

37,800 = 70x + 9(5000 - 10x)

37,800 = 70x + 45,000 - 90x

37,800 = 45,000 - 20x

20x = 7200

x = 360

Answer: E
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Alternative Method: We can use the answer choice to determine the correct answer.

Let's pick B first:
If the truck traveled 200 miles at 55 mph, then it traveled 300 miles at 60 mph.

That means the truck used 200/4.5 gallons of gas at 55 mph, and 300/3.5 gallons of gas at 60.
When you reduce those fractions, you get 400/9 and 600/7
Adding those together, you get approximately 44+86 = 130 total gallons, which is too much.
This means that the truck drove MORE than 200 miles at 55 mph, so A and B are out

Let's pick D next:
If the truck traveled 300 miles at 55 mph, then it traveled 200 miles at 60 mph.
That means the truck used 300/4.5 gallons of gas at 55 mph, and 200/3.5 gallons of gas at 60.
When you reduce those fractions, you get 600/9 and 400/7
Adding those together, you get approximately 67+57 = 124 total gallons, which is STILL too much.

So E has to be the correct answer!

This question can be solved in 15 seconds.

We can simply use the answers.

Since 120 is an integer, the distances traveled 55mph and 60mph shall be divided by 4.5 and by 3.5, accordingly.

The only answers which may fit are A and E.

Now, we look for the distance traveled 55mph, and E is the only answer which is divided by 9 (sum of digit is divisible by 9 - 4.5*2).

Total G = 120 g
Total Dist = 500 m
S1 = 55 mph; C1 = 4.5 mpg, let dist covered = X miles .................................. 1
S2 = 60 mph; C2 = 3.5 mpg, dist covered be = 500- X miles........................... 2

SO, FUEL USED FOR Phase 1 = X/ 4.5
&, FUEL USED FOR Phase 2 = (500 -X)/ 3.5

Now, X/ 4.5 + (500 -X)/ 3.5 =120

solving for X gives, X = 360 miles

So E has to be the correct answer!

Hi!

I really don't know why I am getting 1/3 instead that your result 1/6 in the highlighted portion

What am I missing?

Sharing the method to solve backwards from answer options

1. If we divide 500 by 120 then we get 4.16 which is closer to 4.5 mpg than 3.5 mpg. This means that the truck traveled more than 50% of the distance at 55 mph than 60 mph
2. As we are asked to find the distance traveled at 55 mph, we now know that it will be > 250, hence options A, B and C can be eliminated straight away
3. Now, lets check D and E

Lets start with option E
1. If 360 miles was traveled at 55 mph, then 360/4.5 = 80 gallons of fuel was utilized when traveling at 55 mph
2. If the remaining 140 miles were traveled at 60 mph, then 140/3.5 = 40 gallons of fuel was utilized when traveling at 60 mph

Summing up the above values gives us 80 + 40 = 120 gallons, which matches the information provided in the question. As we have arrived at the correct answer, we do not need to check option D

Ans. E
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500 miles / 120 gallons of diesel fuel = 4.16

4.16 - 3.5 = 0.66
4.5 - 4.16 = 0.34

2/3 were used at 55 mph and 1/3 were used at 60 mph:

80 gallons * 4.5 = 360 miles
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This is a MIXTURE PROBLEM.
4.5 miles per gallon at a lower speed is combined with 3.5 miles per gallon at a higher speed to yield a MIXTURE with 500 miles per 120 gallons.

The following approach is called ALLIGATION -- a great way to handle mixture problems.
Let L = the lower speed and H = the higher speed.

Step 1: Convert to blue values to fractions over a common denominator
L = 4.5 miles per gallon \(= \frac{9}{2} = \frac{27}{6}\)
H = 3.5 miles per gallon \(= \frac{7}{2} = \frac{21}{6}\)
Whole trip = 500 miles per 120 gallons \(= \frac{500}{120} = \frac{25}{6}\)

Step 2: Plot the 3 numerators on a number line, with the numerators for L and H on the ends and the numerator for the whole trip in the middle
L 27---------25---------21 H

Step 3: Calculate the distances between the numerators.
L 27----2----25----4----21 H

Step 4: Determine the ratio in the mixture.
The ratio of L to H is equal to the RECIPROCAL of the distances in red.
L:H = 4:2 = 2:1

Implication of the resulting ratio:
Of every 3 gallons, 2 gallons are used at the lower speed, while 1 gallon is used at the higher speed.
Thus, 2/3 of the 120 gallons are used at the lower speed:
\(\frac{2}{3}*120 = 80\) gallons
Distance traveled at the lower speed = (4.5 miles per gallon)(80 gallons) = 360 miles


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