For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75

Lets take a value, say r= 3.25
The company charges 2 + 3/(1/2) *0.75 + 0.75= 2+ 0.75*(7)

The answer should be 2 + 0.75*(7)

We can easily see that Choice E is exactly that since 2+ 0.75[2r]= 2+0.75[6.5] = 2+0.75*7

Can you please tell me if one can solve this without substituting values?

How do we obtain the '2r'?

Hi rajsekhark,

There is a way to logically get to the correct answer without TESTing VALUES, but you have to note that R COULD be a NON-INTEGER (and that distinction is necessary to select the correct answer). You also have to acknowledge that this is a 'Symbolism' question - so you have to account for how the Symbol 'works' in the context of the correct answer.

From the beginning of the prompt, we know that the total cost of a taxi ride will be $2.00 + some additional charge. That extra charge is $0.75 per half-mile OR fraction of a half-mile. This means that any distance from 0 miles to 1/2 mile will cost the SAME. Any distance from approximately .5000001 miles to 1 mile will cost the SAME, etc. Thus, if you travel even a little more than a 1/2 mile increment, then there will be an additional $0.75 added to the charge.

Many GMATers would only be thinking about integer values for R - and since every mile is made up of 2 "1/2 mile increments", it would be logical to just refer to the additional charge as ($0.75)(2R).

However, take a good look at Answers D and E. If R is an integer, then those two answers will lead to identical results.... but there can't be 2 correct answers.... so what is the actual difference between Answer D and Answer E?

It's the fact that R isn't necessarily an integer (R could be a non-integer) and we have not accounted for the effect that the Symbol has on the calculation. When using the Symbol as described in the prompt, [1.5R] doesn't correctly account for the possibility of an extra increment of distance that is less than 1/2 a mile, but [2R] DOES.

GMAT assassins aren't born, they're made,
Rich

Thank you for your help!

My doubt was regarding the non-integer value of R, which would be an addition to R; I do not understand why we are rounding it to 2R, when "If, for every number x, [ x ] is defined to be the least integer greater than or equal to x" is already elevating the additional miles travelled to an integer status.

What am I overlooking/assuming wrongly here?

Fixed fee: $2.00

Variable fee: $0.75 for 0.5 miles or fraction thereof

No of miles: r

Now 0.5 miles = r then 1 full mile = 2r

We are dealing with least integer greater than or equal to [2r]

Total cost: 2.00 + 0.75 [2r]

Answer E

We need to determine the number of 1/2 miles in r. The only way of "obtaining" [2r] I can see is by observation. We will consider the cases when r is an integer, when r is a decimal where the decimal part is between 0 and 1/2, and when r is a decimal where the decimal part is between 1/2 and 1.

If r is an integer, there are exactly 2r half miles in r. The charge would be 2r(0.75). In this case, [2r] = 2r.

If r = k + x where k is an integer and 0 < x < 1/2, then the charge would be (2k + 1)(0.75) (since there are 2k half miles and one fraction of 1/2 mile in k + x). Write [2r] = [2k + 2x]. Since x is between 0 and 1/2, 2x is less than 1. Thus, [2k + 2x] = 2k + 1. In this case, [2r] = 2k + 1.

Finally, if r = k + x where k is an integer and 1/2 ≤ x < 1, then the charge would be (2k + 2)(0.75) (since there are 2k + 1 half miles and one fraction of 1/2 mile in k + x). Again writing [2r] = [2k + 2x], we see that [2r] = 2k + 2 since 2x is greater than or equal to 1.

As we can see, the charge is [2r]*(0.75) dollars in each case.

Noticing the Answer Choices, the Fixed Fee of $2.00 is the Same for every A.C. ----- so focus just on the .75 cents per 1/2 mile or every fraction of 1/2 mile


Test a few Sample r-miles to Understand the Question First:

Case 1:
if r = 1.1 miles -----> How many 1/2 miles OR fraction of 1/2 miles would be in 1.1?

we would have 2 Full HALF miles + 1 portion of a HALF mile = 3 Charges of $.75****




we are also given that [r] ------> Round UP to the Closest Integer regardless of the Decimal


In order to find every 1/2 mile that is in 1.1, what if we divided ------> (1.1) / (1/2 mile)

(1.1)
______
1
_
2

= 1.1 * (2) = 2.2

[2.2] = 3 -----> which is the amount of $.75 charges we found above by breaking 1.1 apart




Case 2:
Try r = 2.6 -----> How many 1/2 miles OR Fraction of 1/2 miles would be in 2.6?

we would have 5 Full HALF Miles + 1 Fraction of a HALF Mile = 6 Charges of $.75 cents***


again, lets try finding how many 1/2 miles are in r = 2.6 by DIVIDING ------> (2.6) / (1/2 miles)

(2.6 miles)
__________
1
_
2
=
(2.6)(2) = 5.2


Using the Given Function Symbol [X] ------> [5.2] = 6 = which means 6 Charges of $.75 cents, which what we found to be the Truth above



Therefore, no matter what Decimal or Integer r-miles will be, if we Divide (r) by 1/2 miles ---- which is the Equivalent of DOUBLING (r)----- and then put that Result in the Given Symbol Function [X], we will always get the Correct Result of the Amount of $.75 cent charges that should be made


Total Charge = $2.00 Fixed Fee + [ (r) / (1/2)] =

$2.00 + [2r]

-E-


Note: you can test an Integer Value for (r) miles just to make sure as well. The answer will remain the same.

[quote="lpetroski"]For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 for each 1/2 mile or fraction of 1/2 miles. If, for every number x, [ x ] is defined to be the least integer greater than or equal to x, then which of the following represents the company's charge, in dollars, for a trip that is r miles long?

A. 2.00 + [ \(\frac{0.75r}{2}\) ]
B. 2.00 + 0.75[ \(\frac{r}{2}\) ]
C. 2.00 + 0.75[ r ]
D. 2.00 + [1.5r ]
E. 2.00 + 0.75 [ 2r]


MY SOLUTION

Let's assume that he makes a 1mile trip. Therefore r = 1
If r =1, then it means that there are TWO half-miles in the 1mile trip for which he earns $0.75 on each.
Keep in mind that the $0.75 is NOT for additional half miles travelled. But rather each 1/2 mile travelled

then it means his total income will be the fixed $2 plus $0.75 for each 1/2 mile
$2.00 + $0.75 + $0.75 = $3.5


Let's note the following:
1. The first part of the solution is $2 so basically we want the 2nd part of the solution to be $1.5
2. [X] simply means the next whole number of a decimal . Eg. [1.1] = 2 , [1] = 1 , [1.9] = 2

Now let's go to the answer choices and see which gives us $3.5


A. 2.00 + [ \(\frac{0.75r}{2}\) ]
Note that [ 0.75*1 /2 ] = [ 0.375] = 1
Total amount = 2+1 = 3.
Not 3.5

B. 2.00 + 0.75[ \(\frac{r}{2}\) ]
Note that 0.75 * [1/2]
= 0.75 * 1 = 0.75
Total value = 2 + 0.75 = 2.75
Not 3.5

C. 2.00 + 0.75[ r ]
Note that 0.75 * [1] = 0.75 * 1 = 0.75
Total value = 2 + 0.75 = 2.75
Not 3.5

D. 2.00 + [1.5r ]
Note that [1.5 * 1] = [1.5] = 2
Total value = 2+2 = 4
Not 3.5

E. 2.00 + 0.75 [ 2r]
Note that 0.75 * [2*1]
=0.75 * [2]
= 0.75 * 2 = 1.5
Total value = 2 + 1.5 = 3.5
Bingo!

Answer: E

the dependent fee is 0.75 per half mile
So, 1/2 mile 0.75
r mile ?

Cross multiplying we get 1.5r
So, the equation becomes 2+ 1.5r
Hereafter, how to decide between D and E?

chetan2u Bunuel VeritasKarishma

Hi Chitra657,

There is a way to logically get to the correct answer to this question without TESTing VALUES, but you have to note that R COULD be a NON-INTEGER (and that distinction is necessary to select the correct answer). You also have to acknowledge that this is a 'Symbolism' question - so you have to account for how the Symbol 'works' in the context of the correct answer.

From the beginning of the prompt, we know that the total cost of a taxi ride will be $2.00 + some additional charge. That extra charge is $0.75 per half-mile OR fraction of a half-mile. This means that any distance from 0 miles to 1/2 mile will cost the SAME. Any distance from approximately .5000001 miles to 1 mile will cost the SAME, etc. Thus, if you travel even a little more than a 1/2 mile increment, then there will be an additional $0.75 added to the charge.

Many GMATers would only be thinking about integer values for R - and since every mile is made up of 2 "1/2 mile increments", it would be logical to just refer to the additional charge as ($0.75)(2R).

However, take a good look at Answers D and E. If R is an integer, then those two answers will lead to identical results.... but there can't be 2 correct answers.... so what is the actual difference between Answer D and Answer E?

It's the fact that R isn't necessarily an integer (R could be a non-integer) and we have not accounted for the effect that the Symbol has on the calculation. When using the Symbol as described in the prompt, [1.5R] doesn't correctly account for the possibility of an extra increment of distance that is less than 1/2 a mile, but [2R] DOES.

GMAT assassins aren't born, they're made,
Rich

We have to convert miles to next greater integer, so 2r has to be taken to next integer wherever required.

However, in D, we are taking the variable amount to next integer. That is not correct.
Say, r=1, so charges for 2r miles is 2*0.75=1.5
Total 2+1.5=3.5
But 2+[1.5r] will give 2+[1.5*1]=2+2=4

In the portion that I've highlighted above, you say that if we take r to be an integer then D and E will give same results.
So, lets say r=1

In d) 2+ [1.5(1)] = 2+2 = 4
In E) 2+ 0.75[2(1)] = 2+1.5 = 3.5

So these are not identical answers. Where did I go wrong?

When it is odd integer, D and E will always give different options.
It is only when r is even that you will get same answer. REASON: you will get integer for both [1.5r] and 0.75[2r], and you are not increasing any value.

Hi Chitra657,

You made a small math mistake in your work on Answer D.

(1.5)(1) = 1.5.... so 2 + 1.5 = 3.5 (not 4)

GMAT assassins aren't born, they're made,
Rich

But in D, the 1.5r is in square brackets. [] this is essentially rounding up a number right? So, 1.5(1) = 1.5 [] bracket will round it up to 2. So then it wont be identical. Could you explain a bit more

Total Charge = Fixed Charge + Variable Charge * Number of half miles

Case I:
So if total miles travelled is say 2,
Total Charge = 2 + 0.75 * 4

Case II:
What if total miles travelled is say 2.3?
Then Total Charge = 2 + 0.75 * 5 (because we have 4 half miles to get 2 miles and the same 0.75 is charged for the last 0.3 miles too)

Case III:
What if total miles travelled is say 2.6?
Then Total Charge = 2 + 0.75 * 6 (because there are 5 half miles to make 2.5 miles and the last 0.1 leftover for which we charge another $0.75)

So number of half miles (including the last fraction of a half mile) changes the total.
When r is an integer, number of half miles is simply 2r. (Case I)
When r is a decimal such as 2.3 (less than or equal to 2.5), 2r gives us 4.6 which needs to be rounded up to 5. (Case II)
When r is a decimal such as 2.6 (greater than 2.5), 2r gives us 5.2 which again needs to be rounded up to 6. (Case III)

Hence the expression becomes 2 + 0.75*[2r], for all values of r.

Answer (E)

Chitra657 - Try putting these values in (D) to see why they do not work.

The taxi company always rounds up the last 1/2 a mile. So if the total distance was 4.7 miles, you would be charged for 5 miles (or 10 half miles)

This is exactly what the [x] function is doing: rounding to the least integer greater than or equal to x.

D is a trap because we want to round the last 1/2 mile -- not the money charged.

E is the answer.

Using 4.7 miles to check:

10 * 0.75 + 2 = 9.50

2.00 + 0.75 [2(4.7]
2.00 + 0.75 * 10 = 9.50