Question 46·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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A digitization center uploaded a collection of files in two stages with no pauses. During the first stage, files were uploaded at a constant rate of $18$ megabits per second. During the second stage, files were uploaded at a constant rate of $30$ megabits per second. The average upload rate for the entire collection, calculated as total data uploaded divided by total upload time, was $25.5$ megabits per second. What fraction of the total data was uploaded during the first stage?

Your answer

A two-stage average rate is total data divided by total time, so each stage contributes its rate times its own time. If no total time is given, scale the stage times to add to $1$ , then use a Desmos regression to find the first stage’s time fraction. A fraction of the time is not necessarily a fraction of the data when the rates differ.

Hints

Hint 1
A rate in megabits per second times a time in seconds gives megabits. Scale the whole upload time to $1$ second. If the first stage takes $x$ second, how long does the second take?
Hint 2
The average rate uses all the data divided by all the time. With a total time of $1$ second, the number of megabits uploaded must equal the given average rate.
Hint 3
A fraction of the total data needs the first stage’s megabits on top and both stages’ megabits on the bottom. The fraction of time spent in the first stage isn’t yet the answer.

Step-by-step

Scale the time, then find the data fraction
Step 1
Make the total time one second
Scale the total upload time to $1$ second. Scaling both stages’ times also scales both data amounts by the same factor, so neither the average rate nor the requested fraction changes. Let $x$ be the first stage’s time in this scaled upload. With no pauses, the second stage lasts $1-x$ second.
Step 2
Find each stage’s data
Multiply each rate by its stage’s time. Megabits per second times seconds gives megabits, so the first stage uploads $18x$ megabits and the second uploads $30(1-x)$ megabits.
Step 3
Write the average-rate equation
The average rate is total data divided by total time. The scaled total time is $1$ second, so the given rate means:
$18x+30(1-x)=25.5.$
Don’t average $18$ and $30$ equally: the stages need not last equally long.
Step 4
Find the first stage’s time fraction
Type $18x_1+30(1-x_1)\sim25.5$ in Desmos. Use $x_1$ for the unknown and $\sim$ instead of $=$ to have Desmos find where the sides match. Under PARAMETERS, Desmos shows $x_1=0.375$ . That is the first stage’s fraction of time, not its fraction of data.
Step 5
Convert the time fraction to a data fraction
The first stage’s data is $18x_1$ megabits, and the total data is $25.5$ megabits on the scaled $1$ -second timeline. Type $\frac{18x_1}{25.5}$ on the next Desmos line. It shows about $0.2647058824$ ; tap the fraction button to see $\frac{9}{34}$ . So the first stage uploaded $\frac{9}{34}$ of the total data. Grid in 9/34.

Question 46·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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