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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 06 Jul 2020, 17:43
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Hi Nityanshu1990,

Yes - you are correct. The original post that you're referring to is several years old - and a number of posters (including yourself) have caught the error in those calculations.

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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 06 Jul 2020, 18:23
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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]
Show more
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
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 09 Oct 2020, 06:45
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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?
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Can you please tell me if one can solve this without substituting values?

How do we obtain the '2r'?
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 09 Oct 2020, 14:00
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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?


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

How do we obtain the '2r'?
Show more

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.

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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 09 Oct 2020, 21:14
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EMPOWERgmatRichC


Quote:

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.

Show more

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?
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 10 Oct 2020, 10:34
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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
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 15 Oct 2020, 12:35
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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?


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

How do we obtain the '2r'?
Show more

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.
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 19 Oct 2020, 22:45
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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.
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 27 Dec 2020, 10:45
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[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
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 06 Nov 2021, 10:57
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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]

Hi,
Fixed fee= 2.00
fee dependent on miles = 0.75 per 1/2 mile or fraction there off
no of miles = r
no of 1/2 miles= 2r..
and to convert 2r into integer mile [2r]..
total cost on miles= 0.75[2r]..
total = 2.00 + 0.75 [ 2r]
E
Show more

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?

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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 06 Nov 2021, 12:44
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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.

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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 06 Nov 2021, 20:39
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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]

Hi,
Fixed fee= 2.00
fee dependent on miles = 0.75 per 1/2 mile or fraction there off
no of miles = r
no of 1/2 miles= 2r..
and to convert 2r into integer mile [2r]..
total cost on miles= 0.75[2r]..
total = 2.00 + 0.75 [ 2r]
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
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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
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 06 Nov 2021, 23:07
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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.

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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?
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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?
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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.
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 07 Nov 2021, 12:31
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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.

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Rich

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?
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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)

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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 07 Nov 2021, 12:38
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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

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?

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
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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
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 11 Nov 2021, 23:24
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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]
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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.
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 14 Dec 2021, 08:59
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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
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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 14 Feb 2023, 22:25
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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]
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GIVEN – Understand the given information - TRANSLATE:
  • For each trip, a taxi company charges:
    • a fixed fee of $2.00 plus ---- (1)
    • $0.75 for each 1/2 mile or fraction of 1/2 miles ----- (2)
  • For every number x, [x] is defined to be the least integer greater than or equal to x.


TO FIND – Understand the question - TRANSLATE:
  • Company's charge, in dollars, for a trip that is r miles long.


SOLUTION:
From the given information, (1) and (2) above, we know that the charges of the taxi company have two components – a fixed component that is irrespective of the length of a trip (miles travelled) and a variable component that completely depends on the length of the trip.

We will try to find both these components for the ‘r’ mile trip in question. Let’s call the components FC (Fixed charge) and VC (Variable charge).


FIXED CHARGE:
Since the fixed component is always “fixed” as the name suggests, we already know that FC = $2. (From (1) in the Given section) --------- (3)


UNDERSTAND VARIABLE CHARGE - TRANSLATION:
We need to be a little more careful here. Let’s bring back our given information (Point (2)).
  • For each trip, a taxi company charges:
    • a fixed fee of $2.00 plus ---- (1)
    • $0.75 for each 1/2 mile or fraction of 1/2 miles ----- (2)

This “or fraction of ½ mile” makes this question way more interesting. Let me explain:
“$0.75 for each 1/2 mile or fraction of 1/2 miles” can really be seen as composed of two parts
  • $0.75 for each 1/2 mile, and
  • $0.75 for each fraction of 1/2 mile

The first part is straightforward. It just means that we find the number of complete ½-miles and the charge is $0.75 for each such ½-mile. Let’s just move to the more interesting part - “$0.75 for each fraction of 1/2 miles.”

What does this mean? Careful translation reveals the meaning as - Even if you do NOT cover a complete ½-mile, you will still have to pay $0.75 which is for a complete ½-mile!

So, whether it is 0.1 mile or 0.25 mile or anything smaller than 0.5 mile, you will pay for a full ½-mile: $0.75. Let’s take two examples and see how variable charge will be calculated:


Example 1: Consider a 5-mile journey. Then, there still are (5 × 2) = 10 half-miles. This example has NO fractional half-miles, so it’s just straightforward: VC = (10) × 0.75
Example 2: Consider a 5.2-mile journey. Then, there are (5.2 × 2) = 10.4 half-miles. This means that there are 10 complete ½-miles and 0.4 fractional half-mile. Now, even though 0.4 is not a complete ½-mile, it will be charged like one, that is, it will be charged equal to 1 complete ½-mile. In total, the charge will be for (10 + 1) ½-miles.
So, VC = (10 + 1) × 0.75 = [10.4] × 0.75 (Per definition of [x] given in the question.)


Note: In example 1 as well, we can write VC = [10] × 0.75 (since 10 = [10]) ---- To be used in the final solution!


FIND TOTAL CHARGE:
  • From both examples seen above, we can conclude the VC = [number of miles × 2] × 0.75.
    • And hence, for an r-mile trip, VC = [2r] × 0.75.
  • Also, from (3), we already have FC = $2.

Combining these, we get Total charge = 2 + [2r] × 0.75.


Correct Answer: Choice E

Hope this helps!


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For each trip, a taxi company charges a fixed fee of $2.00 plus $0.75 08 Jul 2023, 04:54
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1/2 mile = 0.75
1 mile = 1.5 or 0.75 * 2
for r mile = r * 0.75 * 2

so answer ==> 2.00 + 0.75 [2r] E
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