Giulia is planning to sell her car, which is fueled by gasoline

Question

Giulia is planning to sell her car, which is fueled by gasoline (petrol) and averages 20 miles per gallon (mpg), and purchase a diesel-fueled car that averages 30 mpg. She estimates that her future cost per gallon of diesel fuel will be 5% higher than is her present cost per gallon of gasoline. She wishes to estimate (1) the annual cost of fuel for her new car if she maintains her present annual total miles driven and (2) the annual total miles she can drive her new car if she maintains her present annual expenditure on fuel.

Let x represent Giulia's present annual cost per gallon of gasoline in US dollars, and let y equal her present annual total of miles driven. Select for Cost an appropriate expression for Giulia's estimate of (1) above, and select for Miles an appropriate expression for her estimate of (2) above. Make only two selections, one in each column.

ID: 700227

egmat · Accepted Answer

An amazing question that shows how using a simple concept like ‘unitary method’ one can make a difficult question – by adding various different layers of complexity.
1. Layer 1 – Applying unitary method twice.
2. Layer 2 – Incorporating translation to understand the scenario on which unitary method is to be applied.
3. Layer 3 – Incorporating variables instead of numbers.

Watch the solution and tell me which layer(s) made this question difficult for you!

https://www.youtube.com/watch?v=x3G8Xg-u8wU

Happy Learning!

chetan2u · Answer

Giulia is planning to sell her car, which is fueled by gasoline (petrol) and averages 20 miles per gallon (mpg), and purchase a diesel-fueled car that averages 30 mpg. She estimates that her future cost per gallon of diesel fuel will be 5% higher than is her present cost per gallon of gasoline.

Let us write the details about gasoline car as given. 
Mileage = 20 mpg
Cost = x per gallon
Distance traveled = y miles
 What we can infer => Gallons used =   \frac{y}{20} , cost on fuel =  \frac{y}{20} * x

Let us now correlate with diesel car. 
Mileage = 30 mpg
Cost = 1.05x per gallon

Let us answer the details asked about diesel car

Additional information = > Miles driven = y miles
 What we can infer => Gallons used =   \frac{y}{30} , cost on fuel =  \frac{y}{30} * 1.05x

Additional information = > Cost on fuel =  \frac{y}{20} * x 
What we can infer => Gallons purchased =  \frac{total \ \ cost}{cost \ \ per \ \ gallon}= \frac{\frac{y}{20}*x}{1.05x}=\frac{y}{20*1.05} , 
 Total miles = Total Gallons * average =  \frac{y}{20*1.05} * 30 = \frac{y*30}{20*1.05}=\frac{3}{2}*\frac{y}{1.05}

MartyMurray · Answer

Giulia is planning to sell her car, which is fueled by gasoline (petrol) and averages 20 miles per gallon (mpg), and purchase a diesel-fueled car that averages 30 mpg. She estimates that her future cost per gallon of diesel fuel will be 5% higher than is her present cost per gallon of gasoline. She wishes to estimate (1) the annual cost of fuel for her new car if she maintains her present annual total miles driven and (2) the annual total miles she can drive her new car if she maintains her present annual expenditure on fuel.

Let x represent Giulia's present annual cost per gallon of gasoline in US dollars, and let y equal her present annual total of miles driven. Select for Cost an appropriate expression for Giulia's estimate of (1) above, and select for Miles an appropriate expression for her estimate of (2) above. Make only two selections, one in each column.

Cost

To find (1) the annual cost of fuel for her new car if she maintains her present annual total miles driven, we need to multiply cost per gallon by total gallons.

Cost per gallon for her new car = current cost * 1.05 = x * 1.05

Total gallons used by her new car annually = miles driven/miles per gallon = y/30

Annual cost of fuel for her new car = x * 1.05 * y/30 = 1.05xy/30

Miles

To find (2) the annual total miles she can drive her new car if she maintains her present annual expenditure on fuel, we can just take her current miles and adjust for her new miles per gallon and new cost per gallon.

The passage says that her current car "averages 20 miles per gallon" and that the new car "averages 30 mpg." So, for every gallon of gas, she will go 30/20 as far in the new car as she does in her current car. According, to adjust for the new miles per gallon, multiply by 30/20 = 3/2.

To adjust for the new cost per gallon, divide by 1.05 since each gallon she buys will now cost 5% more.

So the new annual total miles she can drive = 3/2 * old total annual miles/1.05 = 3/2 * y/1.05

Correct answer:  1.05xy/30, 3/2 * y/1.05

cathyjiang04 · Answer

I can't understand how to calculate new annual miles. Why is "To adjust for the new miles per gallon, multiply by 30/20 = 3/2？"

MartyMurray · Answer

Hi  cathyjiang04 .

I've added to the explanation more detail on why we multiply by 3/2. Please see whether that information answers your question, and let me know if you need further explanation.

cathyjiang04 · Answer

Actually I Can't quite understand why " multiply by 3/2. "

MartyMurray · Answer

You can think about it intuitively.

We are trying to figure out how far she can go spending the same amount of money.

One factor in that calculation is how far she can go per gallon of fuel.

In her current car, she can go 20 miles per gallon of fuel.

In her new car, she will go 30 miles per gallon of fuel.

So, the ratio of how far she can go per gallon of fuel = new car mpg/current car mpg = 30/20 = 3/2.

Thus, to adjust for how far she can go on the same amount of money, one thing we do is multiply by 3/2.

At the same time, she won't be able to buy as many gallons because the cost will be higher. So, she won't simply go 3/2 as far in the new car. Rather, since the cost of diesel fuel is 5% higher, she amount of fuel she'll be able to purchase = current amount/1.05.

So, the distance she'll be able to go = 3/2 * current distance/1.05 = 3/2 * y/1.05.

KarishmaB · Answer

(1) the annual cost of fuel for her new car if she maintains her present annual total miles driven

We know that Total Cost = Total Quantity Used * Price per unit of quantity
This is all that is used to solve the entire question.

Diesel: 
Price per unit of quantity = 1.05x per gallon (5% more than x)

Total Quantity of Fuel Used  = \frac{1}{30} \frac{gallons}{mile} * y (miles) = \frac{y}{30} gallons

Total Cost = \frac{y}{30} * 1.05x = \frac{1.05xy}{30} 
 ANSWER

(2) the annual total miles she can drive her new car if she maintains her present annual expenditure on fuel.

Using the same concept as above for petrol, 
Current Total Cost  = x * (\frac{1}{20})*y

If this total cost has to be maintained,  \frac{xy}{20} = 1.05x*(\frac{1}{30})*N  where N is the New number of miles that she can drive
 N = \frac{3}{2}*\frac{y}{1.05} 
 ANSWER

sayan640 · Answer

Form a table...

miles per gallon   |     cost per gallon    |        total miles driven   |       gallons needed for total travel           | total cost
20 mpg               |         x$                 |               y                   |                    y/20                                  |       xy / 20

30 mpg               |      1.05x $             |         y ( maintains same)|                  y/30                                   |    1.05 xy/30

For the first blank the answer is :- 1.05 xy / 30

miles per gallon |   cost per gallon          |      total miles driven   |             gallons needed for total travel       |total cost
20 mpg             |       x$                       |         y                       |                  y/20                                      |  xy / 20

30 mpg             |    1.05x $                   |        p                        |                  p/30                                      | 1.05xp/30

Hence , xy/20 = 1.05 xp / 30 (As the annual cost needs to be kept same)
p = 3/2 * y/1.05 ( Answer)

Gemmie · Answer

...................Now.............................Planned

MPG............. \frac{20 miles}{1 gallon} .................... \frac{30 miles}{1 gallon}

Cost..............$ x..............................$ 1.05 x

(1) the annual cost of fuel for her new car if she maintains her present annual total miles driven

Planned Gallons used =  y \div \frac{30 miles}{1 gallon} = \frac{y}{30}  (gallons)

Planned total expenditure =  \frac{1.05x * y}{30}  ($)

(2) the annual total miles she can drive her new car if she maintains her present annual expenditure on fuel.

Current gallons used =  y \div \frac{20 miles}{1 gallon} = \frac{y}{20}  (gallons)

Current/Planned total expenditure =  \frac{xy}{20}  ($)

Planned gallons used =  \frac{xy}{20} \div 1.05x = \frac{y}{20 * 1.05}  (gallon)

Planned miles driven:  \frac{y}{20 * 1.05} * \frac{30 miles}{1 gallon} = \frac{3y}{2 * 1.05}

nisen20 · Answer

https://gmatclub.com/forum/download/file.php?mode=view&id=78200

tabishjaved · Answer

Solver mindset will take 4/5 minutes.

vs
Attack mode via just analyzing the options:
Cost -> xy variable needed along with avg values -> option DEF
Miles -> ONLY y variable needed -> option ABC

For Cost: 
5% factor is increasing the cost -> it should be in the Numerator
20,30 factor is for avg. greater avg should lower the cost -> 30(the avg for Diesel) should be in Denominator
=> Option D: 1.05xy/30

For Miles: 
5% factor is increasing the cost and decreasing the distance -> it should be in the Denominator
2/3 or 3/2 factor is for avg., which should increase the distance -> Use 3/2 and not 2/3
=> Option B: 3/2 * y/1.05

Julioo · Answer

The problem says "Let x represent Giulia's present annual cost per gallon of gasoline in US dollars". But the word "ANNUAL" really confused me and I could not solve the problem. After reviewing the problem, my understanding is that x represent Giulia's present cost per gallon, that is, without the word "ANNUAL". Cost per gallon already means a unit price. That price does not depend on time, it applies whenever we buy a gallon. So, adding “ANNUAL” suggests time-based aggregation, which confused me. So, given that per-gallon prices don’t need a time qualifier, to me, as written, it is unclear. What do you think?

miag · Answer

Hi, You’re right that cost per gallon doesn’t need a time qualifier. In this question, annual is not modifying the unit; it’s indicating that this per-gallon price is assumed to apply over the same one-year period as the miles driven. The unit of x is still dollars per gallon. x is the per-gallon gasoline price that applies during the year under consideration y is the total miles driven during that same year So annual is a contextual qualifier, not a mathematical one. Hope this helps!

Cost	Miles
		\(\frac{2}{3} * 1.05x\)
		\(\frac{3}{2} * \frac{y}{1.05}\)
		\(\frac{3}{2} * 1.05y\)
		\(\frac{1.05xy}{30} \)
		\(\frac{1.05xy}{20} \)
		\(\frac{20xy}{1.05} \)

Giulia is planning to sell her car, which is fueled by gasoline

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Giulia is planning to sell her car, which is fueled by gasoline

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