lpetroski wrote:
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]
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) × 0.75Example 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!
Shweta Koshija
Quant Product Creator,
e-GMAT
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14 Feb 2023, 22:25
