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.

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

A fixed fee plus a charge per mile is a linear model: the fee stays constant, while the per-mile charge grows with distance. Use a Desmos regression to find both parts from the two rides. Then increase only the per-mile rate and convert the requested distance to miles. Multiplying by kilometers per mile instead of dividing reverses that conversion.

Hints

Hint 1
The pickup fee is paid once, while the per-mile charge repeats. Let $f(x)=ax+b$ be the fare for $x$ miles. Which two distance-and-fare pairs can you use to find both $a$ and $b$ ?
Hint 2
A $15\%$ increase makes the new per-mile rate $115\%$ of the old rate, or $1.15$ times as much. The pickup fee stays unchanged, even though it is part of the total fare.
Hint 3
The fare model takes its distance in miles. Since each mile contains $1.6$ kilometers, divide the trip’s kilometer distance by $1.6$ before putting it into the model.

Step-by-step

Fit the fare, then change its rate
Step 1
Turn the rides into equations
Let $f(x)$ be the before-midnight fare for $x$ miles. The rate $a$ is charged for each mile, and the pickup fee $b$ is paid once, so the two rides give $f(x)=ax+b,\qquad f(3)=11,\qquad f(8)=21$ .
Step 2
Find the original rate and pickup fee
Type $f(x)=ax+b$ , then $[f(3),f(8)]\sim[11,21]$ . The $\sim$ tells Desmos to find values that fit both rides. Under PARAMETERS, it shows $a=2$ and $b=5$ . So the original rate is $2 per mile, and the pickup fee is $5. The $11 fare isn't the pickup fee; it includes $3$ miles.
Step 3
Increase only the per-mile rate
A $15\%$ increase makes the per-mile rate $115\%$ , or $1.15$ times, its old value. Type $2(1.15)$ ; Desmos prints $2.3$ , so the new rate is $2.30 per mile. Change the rate, not the fixed fee. The pickup fee stays $5.
Step 4
Convert the trip to miles
The rate is per mile, but the trip is in kilometers. Since $1$ mile is $1.6$ kilometers, divide the $12$ kilometers by $1.6$ to get miles: $\frac{12}{1.6}=7.5$ . Multiplying instead would convert in the wrong direction.
Step 5
Build the after-midnight fare
Let $g(x)$ be the after-midnight fare for $x$ miles. Keep the $5 pickup fee and use the new rate of $2.30 per mile. Type $g(x)=5+2.3x$ in Desmos; this gives you the rule to evaluate for the trip.
Step 6
Find the fare for the trip
Type $g(7.5)$ , using the trip’s distance in miles. Desmos prints $22.25$ , which includes both the pickup fee and the distance charge. The fare for the $12$ -kilometer ride after midnight is $22.25. Choice B.

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

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