Here’s a question from the May 2023 (US) SAT, non-calculator section:

Alice took 60 minutes to complete a task on her first trial. The time it took Alice to complete the task decreased by 10% of the previous time for each additional trial. Approximately how many minutes will it take Alice to complete the task on her fifth trial?

A) 50
B) 42
C) 39
D) 35

I know we can solve this by subtracting 10% of the current value 5 times, but is there a more direct way to solve, maybe as a function?


Yes, an exponential function will work here. The Alice’s first trial is 60 mins, and then each subsequent trial gets shorter by 10%. So you could say that x is the number of trials and then say t(x)=60(1-0.1)^{x-1}=60(0.9)^{x-1}. Note that the x-1 is because you don’t want to start subtracting 10% until trial 2–you know trial 1 was 60 minutes, so you want the exponent for trial 1 to be zero.

You can now evaluate for t(5):

t(5)=60(0.9)^{5-1}\\t(5)=60(0.9)^4\\t(5)\approx 39.4

All that being said, this being a no-calculator question, most folks will probably find it easier to do the arithmetic one step at a time:

Trial 1: 60
Trial 2: 60(1-0.1)=60-6=54
Trial 3: 54(1-0.1)=54-5.4=48.6
Trial 4: 48.6(1-0.1)=48.6-4.86\approx 43
Trial 5: 43(1-0.1)=43-4.3\approx 39

See how I started approximating when the decimals started to grow? Close enough is QUITE good enough for a question like this.

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