Super-Hard SAT Math Question 109 | Free SAT Prep | Aniko

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Question 109·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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A taxi company charges a fare that includes a fixed pickup fee plus an amount proportional to the distance traveled. Before midnight, a 3-mile ride costs $11 and an 8-mile ride costs $21. After midnight, only the proportional (per-distance) part of the fare increases by 15%, but the pickup fee stays the same. © Аnік‌o

Which choice is the fare for a 12-kilometer ride after midnight? (1 mile $= 1.6$ kilometers.)

When a fare has a fixed part plus a per-distance part, write an equation for each given ride and solve for the fixed fee and unit rate. Convert the new distance to the same unit, then apply any percent change to the part of the fare the problem specifies (here, only the per-distance rate), and finally add the fixed fee. ЅАT рr⁠eр⁠ by Аni⁠кο⁠.‌аi

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Set up the model

Let $b$ be the pickup fee and $r$ be the before-midnight dollars-per-mile rate. Write one equation for each ride.

Use subtraction to eliminate a variable

Subtract the two equations to solve for $r$ (the pickup fee cancels). F⁠rom а‍nікo⁠.a‌і

Don’t forget the unit conversion

Convert 12 kilometers to miles using $1$ mile $=1.6$ kilometers.

Be careful about what increases by 15%

Only the per-mile part increases; the pickup fee stays the same.

Desmos Guide

Solve for the pickup fee and rate

In Desmos, enter b+3r=11 and b+8r=21, then find the intersection point to get the solution values for $(b,r)$ .

Convert 12 km to miles

Enter m=12/1.6 to get the distance in miles.

Compute the after-midnight fare

Enter F=b+(1.15*r)*m and read the value of F. Pοw⁠er‌ed by ⁠Аniko

Step-by-step Explanation

Define variables and write equations

Let $b$ be the pickup fee (in dollars) and let $r$ be the before-midnight rate (dollars per mile).

From the two rides:

b+3r=11

b+8r=21

Solve for the per-mile rate

Subtract the first equation from the second:

(b+8r)-(b+3r)=21-11

5r=10 \Rightarrow r=2

Solve for the pickup fee

Substitute $r=2$ into $b+3r=11$ :

b+3(2)=11 \Rightarrow b+6=11 \Rightarrow b=5

Convert 12 kilometers to miles

Using $1$ mile $=1.6$ kilometers,

12\text{ km}=\frac{12}{1.6}=7.5\text{ miles}

Apply the after-midnight rate change and compute the fare

After midnight, only the per-mile part increases by 15%, so the new rate is

2(1.15)=2.3

Fare after midnight:

5+2.3(7.5)=5+17.25=22.25

So the fare is $22.25. From ⁠аn‌iко.а‍i

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation